Those Fascinating Numbers [New ed.] 0821848070, 9780821848074

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Those Fascinating Numbers Jean-Marie De Koninck

Lee NDE

ee

2

Those Fascinating Numbers

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Those Fascinating Numbers Jean-Marie De Koninck

Translated by Jean-Marie De Koninck

ess DONDE

AMERICAN MATHEMATICAL SOCIETY

Providence, Rhode Island

LIBRARY UNIVERSITY FRANKLIN PIERCE 461 RINDNGE, NH 03

This work was originally published in French by ELLIPSES under the title: Ces nombres qui nous fascinent, ©2008 Edition Marketing S.A. The present translation was created under license for the American Mathematical Society and is published by permission. Translated by Jean-Marie De Koninck. Cover image by Jean-Sébastien Bérubé.

2000 Mathematics Subject Classification. Primary 11-00, 11A05, 11A25, 11A41, 11A51, 11K65, 11N05, 11N25, 11N37, 11N56.

For additional information and updates on this book, visit

www.ams.org/bookpages/mbk-64

Library of Congress

Cataloging-in-Publication

Data

Koninck, J.-M. de, 1948-

[Ces nombres qui nous fascinent. English] Those fascinating numbers / Jean-Marie De Koninck ; translated by Jean-Marie De Koninck.

p. cm. Includes bibliographical references and index. ISBN 978-0-8218-4807-4 (alk. paper) 1. Number theory. I. Title.

.-QA241.K686 512.7-de22

2009 2009012806

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is permitted only under license from the American

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© 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www. ams. org/ NONE) 1 @ yah 3} 2 al

14 13 12 11 10 09

Contents

Intel aCenay amen ee mere INO UG LON) Spe gente Meme

ft 22

ate

tieheeter

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A Be ne

LEAT Als UINCIONS comment cs aati arertnne Ree Mee

Frequently ised stheorems and conjectures

Pip CseataSc Ita LINCan iN DCrs...c 7am

AEPENS. . Gos@ asesleet a ences elegy eho

al

Bn iG ee cao.

mine ea xiii

wess conc e ees occe neces eseens Xvil

Cen

nak

cael. gS Ae

ae ae 1

gata. nurs ene tarts ata eencs dine uate ne aes 409

ete peer ee

elegance

cee cig tab anya

eree Ae LG cree eae: XV

.iix scence

PADDENCIx mune phinecenitiMn Persson OQUMAr

BIDICRTA DY eee

oes waereaiuetw okie backs @lwe ix

er

pe

vil

er ee ts es Teena

ere ee eg

area ee

rt

eke

413

ee see 425

ae

wo a

o

a o7) weal wiprming "|

ul

saehecadt

= 15

a

Preface

One day, in 1918, G.H. Hardy, the great English mathematician, took what he thought was an ordinary cab ride to go visit his young protégé at the hospital, the Indian mathematician S. Ramanujan. To break the ice, Hardy mentioned that the number 1729 on his taxicab was a rather dull number. Ramanujan immediately replied that, on the contrary, it was a very fascinating number since it was the smallest positive integer which could be written as the sum of two cubes in two distinct ways: 1729 = 12° + 13 = 10° 4 9%. This anecdote certainly shows the genius of Ramanujan, but it also stirs our imagination. In some sense, it challenges us to find the remarkable characteristics of other numbers. This is precisely the task we undertake in this project. The reader will find here “famous” numbers such as 1729, Mersenne prime numbers (those prime numbers of the form 2? — 1, where p is itself a prime number) and perfect numbers (those numbers equal to the sum of their proper divisors); also “less famous” numbers, but

no less fascinating, such as the following ones: e 37, the median value of the second prime factor of an integer; thus, the probability that the second prime factor of an integer chosen at random is smaller than 37 is approximately $3 e 277, the smallest prime number p which allows the sum

a Deus

ct ala eae cle lde at veer cefp

(where the sum is running over all the prime numbers < p) to exceed 2; ’

e 378, the smallest prime number which is not a cube, but which can be written

as the sum of the cubes of its prime factors: indeed, 378 = 2:3°-7 = 22+33+73; e 480, possibly the largest number n such that n(n + 1)...(n+ 5) has exactly the same distinct prime factors as (n + 1)(n + 2)...(n +6); indeed,

ARMAS tan AS8 2825 117 13.23.4637 072 941, ASI ASD wane 4860 =. 2-930.507- 117-1323. 37-97. 241:

Xx

PREFACE

e 736, the only three digit number abc such that abc = a+b°; indeed, 736 = 74+3°;

e 1782, possibly the only integer n > 1 for which |» Se p|n

e 548 834, the only number

Se d; d|n

> 1 which can be written as the sum of the sixth

powers of its digits: indeed, 548 834 = 5° + 4° + 8° + 8° + 3° + 4°; e 11859210, the smallest number n for which P(n)4|n and P(n + 1)4|(n + 1), where P(n) stands for the largest prime factor of n (here P(n) = 11 and P(n +1) = 19); the second smallest known number n satisfying this property is n = 632 127 050 601 113 666 430 (here P(n) = 2131 and P(n + 1) = 3691);

e 89460 294, the smallest number n (and the only one known) for which 6(n) = GB(n+1) = B(n+2), where 3(n) stands for the sum of the distinct prime factors of 7;

e 305 635 357, the smallest composite number n for which o(n + 4) = a(n) + 4, where o(n) stands for the sum of the divisors of n; e 612220032, the smallest number n > 1 whose sum of digits is equal to ¥/n;

e 3262811042, possibly the only number which can be written as the sum of the fourth powers of two prime numbers in two distinct ways: 3262811042 =

74 + 2394 = 1574 + 227%; = ine? e 3569 485 920, the number n at which the expression

reaches its max-

n imum value, namely 2.97088..., where w(n) stands for the number of distinct prime factors of n and Q(n) stands for the number of prime factors of n counting their multiplicity.

Various numbers also raise interesting issues. For instance, does there exist a number which is not the square of a prime number but which can be written as the sum of the squares of its prime factors? Given an arbitrary integer k > 2, does

there exist a number n such that P(n)*|n and P(n+1)*|(n +1)? For each integer k > 2 which is not a multiple of 3, can one always find a prime number whose sum of digits is equal to k? These are some of the numerous open problems stated in this book, each of them standing for an enigma that will certainly feed the curiosity of the reader. Actually my hope for this book is to encourage many to explore more thoroughly some of the questions raised all along this book.

There are currently several books whose main purpose is to exhibit interesting properties of numbers. This book is along the lines of these works but offers more features. For instance, one will find — mainly in the footnotes — short proofs of key results as well as statements of many new open problems.

Finally, I would like to acknowledge all those who contributed to this manuscript. With their precious input, suggestions and ideas, this project was expansive

but enjoyable. Thanks to Jean-Lou De Carufel, Charles Cassidy, Zita De Koninck, Eric Doddridge, Nicolas Doyon, Eric Drolet, David Grégoire, Bernard Hodgson,

PREFACE

xi

Imre Katai, Patrick Letendre, Claude Levesque, Florian Luca, Michael Murphy, Erik Pronovost and Jér6me Soucy.

This edition is a translation of my French book Ces nombres qui nous fascinent published by ELLIPSES in 2008. Anyone enjoying this book is welcome to send me suggestions and ideas which could improve and enlighten this project.

Jean-Marie De Koninck Département de mathématiques et de statistique Université Laval Québec G1V O0A6 CANADA

[email protected]

a

i ase

Notations

In this book, unless indicated otherwise, integer”.

by “number”

we mean

a “positive

The sequence pj, p2,p3,... stands for the sequence of prime numbers

2, 3, 5,

... Thus px stands for the k‘” prime number. Unless indicated otherwise, the letters p and q stand for prime numbers.

By a|b, we mean that a divides b. By a Jb, we mean that a does not divide b. Given a positive integer k, by p”||n, we mean that p*|n but that p*t! Jn. When we write S f(p), we mean the infinite sum f(2) + f(3) + f(5) + f(7) + P ...+f(p) +.... Similarly we write Se f(p) to indicate that the summation pe

runs over all primes p < a.

The expressions II f(p) and II f(p) are analogue to the ones mentioned just P

Px

above, except that this time they stand for products and not summations.

By Se f(d), we mean

that the summation

runs on all divisors d of n; by

d|n

ye f(p), we mean that the summation runs over all prime factors p of n. We p|n

use the corresponding notations for the products, that is II f(d) and II f(p). d|n

pin

We denote by y the Euler constant, which is defined by N

y= lim (>.*— log x)= 0.5772156649.... N-oo

(ail

Given an integer b > 2 and a number n whose digits in base b are dj, d2,...,d;, we sometimes use the notation n = [d;,d2,...,d;|,. If the base is not mentioned, it should be understood that we are working in base 10.

xiii

XIV

NOTATIONS

e The factorization of a number usually appears in the form a Ti tt Goa2 eae

ar

ee

where gi < q2 < ... < gy are the r distinct prime numbers dividing n and where Q1,@2,...,@, are positive integers. For instance, we write

11560 = 2°- 5-177. It may happen that the factorization of a large number takes on a particular form, such as

(ee

57 Pisa:

in this case, the expression P,3g stands for a (known) 136 digit prime number which there is no need to write at length, since it can be obtained explicitly by simply dividing 1 by 257. Another possible situation could occur, as for instance:

;

127 + 1 = 8253953 - 295278642689 - Cos8; in this case, the expression Co5g stands for a composite 258 digit number for which no non trivial factorization is known.

e In order to compare the size of certain expressions in the neighborhood of infinity, we use various notations, some of which have been introduced by Landau,

namely O(...) and o(...). Hence, given two functions f and g defined on [a, oo) (where a > 0), we write:

(i) f(z) = O(g(x)) if there exist two constants M > 0 and zo for which \(z) |)=< Mig(a) tortall a) > 263) in particulary f(z) =O) air fi@)ias a bounded function; moreover, instead of writing f(x) = O(g(zx)), we sometimes write f(x) < g(x); thus we have

G= O(a");

sine = O(1)"

logx = O(c"),

eee

(ii) f(x) = o(g(z)) if, for each e > 0, there exists a constant xo = ro(€) such that |f(x)| < eg(x) for all x > xo; thus we have I

2 7 Ot);

sina, = 0(2),

loga =o0(z),

«* =ole).

(iii) f(v) = Q(g(a)) if there exist two constants M > 0 and zo such that |f(x)| > M|g(x)| for all x > xo; instead of writing f(r) = Q(g(zx)), we sometimes write f(x) >> g(x); thus we have

P= 06/2),

V& = O(log x),

e* = (a7),

eo Ses

(iv) f(z) ~ g(x) to mean that Jim a = 1; thus, as 2 — oo, we have

Sori z

Be vert Wieg

Pe

eres

(v) f(x) © g(x) to mean that we have both f(x) < g(x) and g(x) « f(z).

The Main

Functions

[x], the largest integer < x B=

Dy ap, the sum of the prime factors of n with multiplicity p*||n

By(n) = ~ p”, the sum of the largest prime powers dividing n p@||n

P(n) = max{p : p|n}, the largest prime factor of the number n > 2

p(n) = min{p: p|n}, the smallest prime factor of the number n > 2 Bn) = S p, the sum of the distinct prime factors of n pin

i, (i) —

YS p = B(n) — P(n), the sum of the prime factors of n except for the p|n p 2, that is the dis-

log y(n) nin

P(m) log? x ( ipa x

xL (|

where r is any given fixed integer.

The

Chinese

Remainder

Theorem

Let m1,™m2,...,m, be co-prime integers, and let aj, a2,...,a, trary integers. Then the system of congruences

n

=

a,

(mod mz)

n

=

a2

(mod mz)

n

=

a,

(mod ™m,)

be arbi-

has a solution given by

where

m =m

ence (m/mj;)b;

mM2...m,

= 1

and where each b; is the solution of the congru-

(mod m).

xvil

:

XVill

FREQUENTLY

USED

Hypothesis

H (or Schinzel

THEOREMS

AND

CONJECTURES

Hypothesis)

Let € > 1 and fi(x),.... fe(x) be irreducible polynomials with integer coefficients and positive leading coefficient. Assume that there are no

integers > 1 dividing the product f1(n)...fe(n) for all positive integers mn.

Then there exist infinitely many positive integers m such that all

numbers f;(m),..., fe(m) are primes. This conjecture was first stated in 1958 by A. Schinzel and W. Sierpinski [181]. The abc Conjecture

Let ¢ > 0. There exists a positive constant M = M(e) such that, given any co-prime integers a,b,c verifying the conditions 0 < a < b < c and a+b=c, we have 1+e

c 0. There exists a positive constant M = M(e) such that, given any co-prime integers a,b,c satisfying a + b = c, we have l+te

max{|a|,|d|,\el}< M-|{

|] p plabc

The abe Conjecture was first stated in 1985 by D.W. Masser and J. Oesterlé.

Those Fascinating Numbers

the only number which divides all the others.

the only even prime number.

the prime number which appears the most often as the second prime factor of an integer, and actually with a frequency of z (see the number 199 for the list of those prime numbers which appear the most often as the k*” prime factor of an integer, for any fixed k > 1).

the smallest Mersenne prime (3 = 2?—1): a prime number is called a Mersenne prime if it is of the form 2? — 1, where p is prime (see the number 131071 for the list of all Mersenne primes known as of May 2009); the prime number which appears the most often as the second largest prime factor of an integer, that is approximately (1 + log2 + 3 log 3)xz/logx times

amongst the positive integers n < x (see J.M. De Koninck [44]); the smallest triangular number > 1: a number n is said to be triangular if there exists a number & such that t

n=1+2+3+...4k=

k(k as 1)

2

e

e

e

e

e

e

e

JEAN-MARIE

DE KONINCK

e the smallest number r which has the property that each number can be written

in the form 2? + 73 +...+ 27, where the z;’s are non negative integers; the problem consisting in determining if, for a given integer k > 2, there exists a number r (depending only on k) such that equation

(*)

Me

es

ee

has solutions for each number n, is due to the English mathematician E. Waring who, in 1770, stated without proof that “each number is the sum of 4 squares, of 9 cubes, of 19 fourth powers, and so on”; if we denote by g(k) the smallest number r such that equation (*) has solutions for each number n, Lagrange proved in 1770 that g(2) = 4, Wieferich and Kempner proved around 1910

that g(3) = 9, while R. proved in 1986 that g(4) (where [x] stands for the Dickson [65])'; hence by for

Balasubramanian, J.M.Deshouillers & F. Dress [12] = 19; it is conjectured that g(k) = 2* + [(3/2)*] — 2 largest integer < x) for each integer k > 2; see L.E. using this formula, we find that the values of g(k),

k= 1,2,..., 20, are respectively 1, 4,°9>19,.37,,

73, 143, 279, 548, 1079;

2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899 (see the book of Eric Weisstein [201], p. 1917).

the smallest Wilson prime:

a prime number p is called a Wilson prime if it

satisfies the congruence (p — 1)! = —1 (mod p*): the only known Wilson primes are 5, 13 and 563; K.Dilcher & C.Pomerance [68] have shown that there are no other Wilson primes up to 5 - 108.

the smallest perfect number: a number the sum of its proper divisors, that is numbers starts as follows: 6, 28, 496, said to be k-perfect if a(n) = kn: if we number, then ng = 6, ng = 120, nq = 154 345 556 085 770 649 600;

n is said to be perfect if it is equal to if a(n) = 2n; the sequence of perfect 8128, 33550336, ...; a number n is let ng, stand for the smallest k-perfect 30240, ns = 14182439040 and ng =

the smallest unitary perfect number: a number n is said to be a unitary perfect

number if

ye

d = 2n, where (d, n/d) stands for the greatest common divisor

d|n

(d,n/d)=1

of d and n/d; only five unitary perfect numbers are known, namely 6, 60, 90,

87 360 and 146 361 946 186 458 562 560000 = 218 -3.54.7-11-13-19-37-79-

109 - 157 - 313: this last number was discovered by C.R. Wall [198] (see also R.K. Guy [101], B3); "In 1936, S. Pillai [161] proved that if one writes 3* = q2* +r with 0 < r < 2*, then Gls) =

2* + [(3/2)*] — 2 provided r+ q < 2°.

THOSE

e the only triangular number

FASCINATING

>

1 whose

NUMBERS

square

3

is also a triangular number

(W. Ljunggren, 1946): here 67 = 36 =1+243+...+8.

e one of the two prime numbers (the other one is 5) which appears most often as

the third prime factor of an integer (1 time in 30); e the second Mersenne prime: 7 = 2? — 1.

e the third number n such that t(n) = ¢(n): the only numbers satisfying this equation are 1, 3, 8, 10, 18, 24 and 30;

e the number of twin prime pairs < 100 (see the number 1 224).

e the only square which follows? a power of 2: 2? + 1 = 3?; e the only perfect square which cannot be written as the sum of four squares

(Sierpinski [185], p. 405); e the smallest number r which has the property that each number can be written

as z?+23+...+23, where the z;’s are non negative integers (see the number 4).

e one of the five numbers (the others are 1, 120, 1540 and 7140) which are both triangular and tetrahedral (see E.T. Avanesov [8]): a number n is said to be tetrahedral if it can be written as n = ¢m(m + 1)(m + 2) for some number m: it corresponds to the number of spheres with same radius which can be piled up in a tetrahedron;

e the fourth number n such that t(n) = ¢(n) (see the number 8).

2Much more is known. Indeed, according to the Catalan Conjecture (first stated Catalan [31] in 1844), the only consecutive numbers in the sequence of powers 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125,... are 8 and 9; this conjecture was recently proved

by

by Preda Mihailescu [135].

4

JEAN-MARIE

DE KONINCK

e the smallest prime number p such that 3?-! = 1

(mod p*): the only other

prime number p < 2°? satisfying this congruence is p = 1006003 (see Riben-

boim [169], p. 347)3;

1 ) 7 to exceed 3 (see the number

e the smallest number n which allows the sum

1 1 such that o(7(n)) = n; e the smallest sublime number: we say that a number n is sublime if T(n) and a(n) are both perfect numbers: here 7(12) = 6 and o(12) = 28; this concept was introduced by Kevin Ford; the only other known sublime number is 2!76(2°! —

1)(23+ — 1)(21® — 1)(2” — 1)(2° — 1)(2° — 1).

e the second Wilson prime (see the number 5); e the prime number which appears the most often as the fourth prime factor of

an integer, namely 31 times in 5005 (see the number 199); e the smallest prime number p such that 23?-! =1

(mod p?): the only prime

numbers p < 23” satisfying this congruence are 13, 2481757 and 13 703077 (see Ribenboim [169], p. 347); e the third horse number: we say that n is a horse number if it represents the number of possible results accounting for ties, in a race in which k horses

participate; thus, if Hj is the k*” horse number, one can prove‘ that

n-£e (E39) j=0

J

the first 20 terms of the sequence (Hy),>1 are 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632573, 28091567595, 526858348381, 10641342970443, 230283190977853, 5315654681981355, 130370767029135901, 3385534663256845323, 92801587319328411133 and 2677687796244384203115. 3 As is the case for the Wieferich primes (see the number 1093), it is not known if this sequence of numbers is infinite. 4A formula established by Charles Cassidy (Université Laval).

THOSE

FASCINATING

NUMBERS

5

e the smallest solution® of a(n) = a(n + 1); the sequence of numbers satisfying this equation begins as follows: 14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, ...; e the eure Catalan number: n

Catalan numbers®

are the numbers

of the form

n+1\n

e the third smallest solution of ¢(n) = ¢(n + 1); the sequence of numbers satisfying this equation begins as follows: 1, 3, 15, 104, 164, 194, 255, 495, 584, 975,

2204, 2625, 2834, 3255, 3705, 5186, 5187, 10604, 11715, 13365, 18315, 22935, 25945, 32864, 38804, 39524, 46215, 48704, 49215, 49335, 56864, 57584, 57645, 64004, 65535, 73124, ...: R. Baillie [10] found 391 solutions n < 2-108 ;’ e one of the three numbers n such that the polynomial 7° —x2+n can be factored: the other two are n = 22440 and n = 2759640: here we have x° —x+15 =

(x7 +2 + 3)(x° = x? — 22 +5); see the number 22 440; e the value of the sum of the elements of a diagonal, a row or a column of a 3 x 3 magic square: for a k x k magic square with k > 3, the common value is k(k* + 1)/2, which gives place to the sequence whose first terms are 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, ... (see Sierpinski [185],

p. 434).

e the only number n for which there exist two distinct integers a and b such that =a= 6") herea = 2, b=A: e the smallest perfect square for which there exists another perfect square with

the same sum of divisors: o(16) = o(25) = 31.

5 This sequence of numbers is probably infinite, but no one has yet proved it. 6Catalan numbers appear when one wants to find in how many ways it is possible to partition a convex polygon in triangles by drawing some of its diagonals. TP. Erdés, C. Pomerance & A. Sark6zy [79] provide a heuristic argument which suggests that, for

each fixed € > 0, equation ¢(n) = ¢(n + 1) has at least x1~* [180] believes that it may be possible that equation ¢(n) = solutions, but he conjectures that for each even integer k > 2, many solutions. Let us add that equation ¢(n) = ¢(n +k) and divisible by 3; thus by letting E, be the set of solutions

solutions n < x. However, A. Schinzel ¢(n + 1) has only a finite number of equation ¢(n) = ¢(n+k) has infinitely has very few solutions when k is odd n < 108 of d(n) = ¢(n +k), we have

Es = {3,5}, Eo = {9,15}, Fis = {13, 15, 17,21}, Bo: = {21,35} and E27 = {27,45,55}, while the

cardinality of each of the other sets Ex, 1 o is defined implicitly by the expansion

(see E.W. Weisstein [201], p.111).

the third Mersenne prime: 31 = 2° — 1 (Euler, 1750); the sixth prime number px such that p;p2...pzx +1 is prime (see the number 379);

1 the smallest number n which allows the sum DD 7 to exceed 4 (see the number w 2, one can find infinitely many numbers n such that f(n) = f(n+1) =... = f(n+k—1), where f(n) stands for the product of the exponents in the factorization of n, which is clearly a function very similar to the 7(n) function.

THOSE

FASCINATING

NUMBERS

3

is located in such a manner that the sum of the numbers of the houses to its left is equal to the sum of the numbers to its right: the number on the maire’s house is therefore the solution of equation 1+2+...+(w—1) =

(w+1)+(w+2)+...+(w+s):

the smallest solution is given by w = 6

and w+ s = 8, while the next six are w = 35 and w+ s = 49, w = 204 and w+s = 288, w = 1189 and w+s = 1681, w = 6930 and w+ s = 9800, w = 40391 and w+ s = 57121, w = 235416 and w + s = 332928; it is possible to prove that the aligned houses problem has infinitely many solutions; on the other hand, it is interesting to observe that there exists a connection between

this problem and the numbers n for which n and n+ 1 are powerful!?.

[36| 8(8+1) _ 2. e the smallest triangular number > 1 which is also a perfect square: Movi» =1(5 3 the sequence of numbers satisfying this property begins as follows: 1, 36, 1225, 41616, 1413721, 48024900, 55420693056, ...; there exist infinitely many such numbers!4;

e the largest solution n of equation >»?T(d) =n: the only solutions of this equad\n

tion are 1, 3, 18 and 36.

e the median value of the second prime factor of a number: indeed, one can show that the probability that the second prime factor of a number is < 37 is equal to 0.500248... + 4; the median value of the third prime factor of a number is 42719, while that!> of the fourth one is 5737 850 066 077;

e the smallest irregular prime number (see the number 59); e the smallest number r which has the property that each number can be written

as 2? +23 +...+ 22, where the z,’s are non negative integers (Chen, 1964): see the number 4; 13Indeed, since

1+2+...+(w—-1)=(wt+1)+(w+2)+...+(w+s), it is easy to see that for any other solution (w,s) of the aligned houses problem, we have

(x)

(w+ s)(w+s+1) = 2w?.

This is why, since (w+s,w+s+1)

=1, it follows from (*) that if w is odd, the numbers w +s and

w-+s-+1

This explains why we find the numbers 288, 9800 and 332928

must both be powerful.

amongst the numbers n for which n and n + 1 are powerful (see the number 288). 14Indeed, this follows from the fact that the diophantine equation n(n + 1) = 2m? has a solution m if and only if m appears in the sequence (a@n)n>o defined by ap = 1, a1 = 6 and, for n > 2, by Qn = 6an—1

— Gn—-2.-

15In 2002, J.M. De Koninck & G. Tenenbaum

[63] proved that if pj stands for the median value

of the k*” prime factor of an integer, then loglogp,

=

{log (be) — i}

= k — 6+ O(1/Vk), where b =

3 +y-

0.59483 (here y © 0.557215 stands for Euler’s constant); on the other

hand, they proved that if the Riemann Hypothesis is true, then pz ~ 7.887- 107e:

14

JEAN-MARIE

DE KONINCK

e the smallest number n > 1 such that n?+3 is a powerful number: here 377+3 = 2? . 73: the sequence of numbers satisfying this property begins as follows: 37, 79 196, 177833, ...; Florian Luca proved that this sequence is infinite!®, and in fact his argument reveals in particular two additional solutions of n?74+3=m, where m is powerful, namely

96679390107 +3 = eT 2524807950507510523° +3 = —

93469044701079780103 GloeM007 28511 6374655186945935706615682630955733532 27.77.19? .732152605848617;

Patrick Letendre observes that more generally, one can prove that if there exists a powerful number of the form n? +k which is not a perfect square, then there are infinitely many of them"; e the largest solution y of the diophantine equation x? + 28 = y® (see the number

225)5 e the smallest number

>

1 which is equal to the sum

of the squares of the

factorials of its digits (in base 12): here 73 = [3,3, 112 = 3!? + 3!? + 1!? (see the numbers 145 and 40465); the set of numbers satisfying this property is of course finite and its four smallest elements are 1, 37, 613 and 519018;

e the largest number n the form n convenient

known prime number which is also a convenient number; an odd > 1 is said to be convenient if it has only one representation of

= 2? + my?, with x,y positive and (z,my) = 1; the only known prime numbers are 2, 3, 5, 7, 13 and 37 (see also the number 1 848).

e the largest even number which cannot be written as the sum of two odd com-

posite numbers!®. 16Here is Luca’s argument. First consider the Fermat-Pell equation x? — 7y? = 1. It has infinitely many solutions (tn, Yn), nm = 1,2,..., the smallest of which is (x1, y1) = (8,3). For each of these solutions (%n, Yn), consider the numbers £ = 372%n + 98yn and y = 14%7y + 37yn. One easily verifies that x? +3 = Ty. To complete the proof, one needs to have that y is a multiple of 7. But y = 2yn (mod 7), so that if yn is a multiple of 7, the same will be true for y. Since it is the case for n= 7,14, 21,..., the result follows. 17Let d be a positive integer which is not a perfect square and consider the quantities n = x2 — dy?

and m = a? — db?. One easily verifies that (x) m+n = (ax — dby)? — d(ay + bx)”. This is why, knowing numbers x9 and yo such that a4 = dyé = 1 (it is known that there are infinitely many of these), and starting with the known solution (t,z) of equation t? + k = z, with z powerful (4 perfect square), then let u? be the largest square divisor of z satisfying z = u? - v, where (wv) =1 and v > 1. This brings us to equation t? — vu? = —k, of which we already know by hypothesis a

solution (to, uo), which in turn allows us to generate infinitely many solutions using (*). 18The proof of this result is very simple. Indeed, let n = 2m be an even number. This number m is necessarily of one of the three following forms: m = 3k, m = 3k+1 or m = 3k+2. In the first case, n=2m=9+3- 2™=3 the sum of two odd composite numbers. In the second case, we can write

n = 2m = 2(3k+1) = 35+3(2k—11). In the third case, we have n = 2m = 2(3k+2) = 25+3(2k—7). We have thus settled the case of all even numbers which are multiples of 3 and > 9+ 6 = 15, that of each number

2m with m = 3k +1

and 2m > 35+ 3-3 = 44 and that of each number

2m with

m = 3k +2 and 2m > 25+3-3 = 34. Therefore, since 40 = 15 + 25 and since we can easily verify that 38 is not the sum of two odd composite numbers, the result follows.

THOSE

FASCINATING

NUMBERS

15

e the smallest number n such that 2” — 7 is prime (a question raised by Erdés in 1956): using a computer, one obtains that the other numbers n < 20000

satisfying this property are 715, 1983, 2319, 2499, 3775 and 12819.9

a(n)

9

e the smallest solution of equation —— = z} the sequence of numbers satisfying n this equation begins as follows: 40, 224, 174592, 492101632, ...

e the largest odd number which is not the sum of four non zero squares (Sierpinski [185], p. 404);

e the largest number n such that the polynomial x? + «+n is prime for each of the numbers z = 0,1,2,...,n—2; the other numbers n satisfying this property are n = 1, 2, 3, 5, 11 and 17 (see D. Fendel & R.A. Mollin [80]); e the integer part of the number yo = 41.677647, that is the conjectured value

of limsup,, ..6 Felt) where 0..(n) stands for the smallest number k such that

f*(n) =1, where iOS

1

ih Osean le

eye

if n is even,

3n+1

if n is odd,

f(r) = f(r), f?(n) = f(F(m)), fP(m) = f(f?(m)) and so on; according to the Syracuse conjecture (also called the 3x +1 problem), this sequence inevitably

reaches the number 1 (see J.C. Lagarias & A. Weiss [120}); e the smallest prime number of the form (x*+y*)/2: here 41 = (34+1+*)/2: there

exist only eight prime numbers < 10000 satisfying? this property, namely 41, 313, 353, 1201, 3593, 4481, 7321 and 8521.

e the smallest number n > 1 such that o2(n) is a perfect square: the sequence of numbers satisfying this property begins as follows: 42, 246, 287, 728, 1434, 1673, 1880, 4264, 6237, 9799, 9855,...;

e the fifth Catalan number (see the number 14); 19While performing this search, one can ignore all even numbers n, all numbers

all numbers

n = 7

(mod 10) as well as all numbers

n = 11

n =1

(mod 4),

(mod 12), since in these four cases,

we obtain respectively that 3, 5, 11 and 13 divide 2” — 7. 20Tt is interesting to mention that there exist other forms which generate infinitely many prime numbers: it is the case, for instance, for the form 2? + y* as it was proved by J. Friedlander &

H. Iwaniec [85], as well as for the form x? + 2y? as was established by D.R. Heath-Brown [110] (see also for that matter the number 3391).

16

JEAN-MARIE

DE KONINCK

the smallest solution of a(n) = a(n + 20); the sequence of numbers satisfying this equation begins as follows: 42, 51, 123, 141, 204, 371, 497, 708, 923, 992,

1034, 1343, 1391, 1484, 1595, 1691, 1826, 3266, 3317, 5015, 5152, 7367, 8003, 9132, 9287, 9494,...

the fourth prime number p such that 19?~!

= 1

(mod p’): the only prime

numbers p < 2° satisfying this congruence are 3, 7, 13, 43, 137 and 63 061 489 (see Ribenboim [169], p. 347).

the second number n such that ¢(x) = n has exactly three solutions:

the

sequence of numbers”! satisfying this property begins as follows: 2, 44, 56, 92, 10451165 140 2os the smallest number n which allows the sum

it

De m 2, if we let ng be the smallest integer n which allows the above sum to exceed k, then the sequence (n;),>1 begins as follows: 44, 236, 1853, 24692, 627 2083375 1.004,...--.

the smallest number n such that r(n) > T(n + 1) > T(n + 2): here 6 > 4 > 2; if we denote by nz the smallest number n such that tT(n) > T(n +1) >... > tT(n+k), then n; = 4, no = 45, nz = 80, ng = ns = 28974, ng = 8489103 and n7 = 80314575; the smallest number n having at least two distinct prime factors and which is such that pln = > p+ 3|n + 3: the sequence of numbers which satisfy this property begins as follows: 45, 147, 165, 357, 405, 567, 621, 637, 845, ... (see

the number 399).

21See the footnote tied to the number 24. 22One can estimate the size of the number nx with respect to k by using estimate 72 (3)

x: w(n) = 2} ~ a(loglogz)/(logz), which follows essentially from the Prime Number Theorem. Indeed, by first writing the sum as a Stieltjes integral and then integrating by parts, one obtains

Th es faa

y += [ pS nox w(n)=2

_ m2(t)

ae

ne

a

x +f silt ree

Fae

x

5

loglogt | i. (log log x) 2

tlogt

2

(log logTeac nx)? ad k as k — oo, which Jak This is why the number nz, satisfies sca means that nz © e®

THOSE

FASCINATING

NUMBERS

ily

e the rank of the prime number which appears the most often as the eighth prime

factor of an integer: pag = 199 (see the number 199).

e the prime number which appears the most often as the sixth prime factor of an

integer (see the number 199); e the fifth Hamilton number (see the number 923).

e the smallest number which is divisible by a square > 1 and is followed by two other numbers with the same property: here 48 = 2+-3, 49 = 7? and 50 = 2-5?: the sequence of numbers satisfying this property begins as follows: 48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846,...;

e the smallest number for which the product of its proper divisors is a fourth power, that is such that Il d =a‘: here d\n, d 1 and such that y(n+1)—y(n) = 1; the sequence of numbers satisfying this property begins as follows: 8, 48, 224,

960, 65024, 261120, 1046528, 4190208, ...?3 (see the number 98).

e the smallest number n divisible by a square > 1 and such that 6(n+1)—4d(n) = 1, where 6(n) = Mis p; the sequence of numbers satisfying this property begins as follows:

S13030

49, 1681, 18490, 23762, 39325, 57121, 182182, 453962, 656914,

ee

e the second solution w + s of the aligned houses problem (see the number 35).

231t is easy to see that each number n = 27+!(27—!—1), where 2”—1 and 27—! —1 are square-free, is a solution of y(n + 1) — y(n) = 1. The numbers r < 200 such that 2” — 1 and 2"~1 — 1 are both square-free are the numbers 2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 23, 26, 27, 28, 29, 32, 33, 34, 35,

SOOO NAAN 87, 88, 89, 92, IZOMISIen IGA 173, 176, 177, that equation

AGU (On ol 2.nO8 Os, 09 62, 655168, 69705) 7L74.975, 16) «7, 82, 83, 86, 93, 94, 95, 98, 99, 104, 107, 112, 113, 116, 117, 118, 119, 122, 123, 124, 125, 128, 129, Som 42 las AG OM b 2a 53 lod. LoS. 159164) 165.166. 167, 170) Lily 72) 178, 179, 182, 183, 184, 185, 188, 191, 194, 195, 196 and 197. Therefore it follows y(n + 1) — y(n) = 1 has infinitely many solutions n (with u(n) = 0). But it is not

known if there exist infinitely numbers of the form 2” — 1 which are square-free. In fact, Andrew Granville believes (private communication) that it is unlikely that one could easily prove that there exist infinitely many square-free numbers of the form 2" — 1, since if that was the case, it would follow that there exist infinitely many prime numbers which are not Wieferich primes, a result that is certainly true, but that we are so far unable to prove without assuming the abc Conjecture (see

Silverman [186]). 24There exist infinitely many numbers n divisible by a square and satisfying 6(n + 1) — 6(n) = 1. This follows from the fact that the Fermat-Pell equation 2x? — y? = 1 has infinitely many solutions.

JEAN-MARIE

DE KONINCK

the smallest number which can be written as the sum of two squares in two

distinct ways: 50 = 17+77 = 57+5?; if mn; stands for the smallest number which can be written as the sum of two squares in k distinct ways, then n2 = 50, ng

=

320,

rg

=

1105,

ms

=

n6

=

DoPAD,

n7

=

nrg

=

27 625,

(Me)

=

al 825,

N19 = 138125 and n11 = N12 = 160 225 (see also the number 1 29-3;

the smallest number n having at least two distinct prime factors and which is such that p|n => p+ 10|n + 10; the sequence of numbers satisfying this property begins as follows: 50, 242, 245, 935, 1250, 8405, ...

the smallest solution of a(n) = a(n + 4); the sequence of numbers satisfying this equation begins as follows:

51, 66, 115, 220, 319, 1003, 2585, 4024, 4183,

4195, 5720, 5826, 5959, 8004, 8374, ...; the fourth number n such that $(n)a(n) is a perfect square: here ¢(51)0(51) = 487: the sequence of numbers satisfying this property begins as follows:

1, 14,

30, 51, 105, 170, 194, 248, 264, 364, 405, 418, 477, 595, 679, ...; Niegel Boston proved that this sequence is infinite (see R.K. Guy & R.J. Nowakowski [103]);

the second number n such that o(n) = o(n + 20) (see the number 42).

the fifth Bell number, namely Bs: Bell numbers B, are defined implicitly by

et

e7—1

_

=

x”

=) Ba n=0

theirst, Bell numbers are. Bo: = 1) Bi =15,.8o = 2, B3= 5, Ba

Be = 203,87

S801, Bs —4

1b Bs = 52.

14085 — 2144 Big = 115 975;

the smallest number k > 5 such that equation o(n) — n = k has no solution?®; the largest known number?’ n such that n! ++

1 is prime; the others are 2,

4, 6 are 10.

the smallest prime number equally distant, by a distance of 6, from the preceding and following prime numbers : pis = 47, pig = 53 and p17 = 59;

the smallest number

n such that the decimal expansion of 2” contains two

consecutive zeros (see the sequence M4710 in N.J.A. Sloane & S. Plouffe [188]); 25Tt is natural to raise the following question: Does there exist a number ko such that np < ng41 for allk > ko?

26Interestingly, Erdés pointed out the existence of a large family of integers k for which equation

a(n) —n =k

has no solution.

27 Obviously, such a number n must be such that n+ 1 is prime.

THOSE

the number ORD):

FASCINATING

NUMBERS

19

of digits in the decimal expansion of the tenth perfect number

the only solution n < 10° of ¢(n)a(n) = ¢(n + 1)a(n + 1) = o(n + 2)a(n + 2) (see R.K. Guy [102]; see also the number 136); the largest Fibonacci number with only one distinct digit, a result due to F. Luca; the sequence of Fibonacci numbers (Fy)n>1 is defined as follows: ie

ol

ahi

tor each 4c:

the largest Fibonacci number which is the concatenation of two other Fibonacci

numbers (5 and 5), a result due to F. Luca & W.D. Banks [16]; the fourth and largest Fibonacci number which is triangular: the others are 1,

3 and 21 (see L. Ming [137]).

the sixth tetrahedral number (see the number 10): the sequence of tetrahedral numbers begins as follows: 1, 4, 10, 20, 35, 56, 84, 120, ...; the largest number n for which there exists

a number k < n/2 such that if we

write the binomial coefficient (7°) as a product wv, where u=

II Dp

et

=

II D;

pl(R)

Pl(z)

p v: the only other pairs (n, k) satisfying this property are (8,3), (9,4), (LO, Sd 2e5 eal2b )os(2leS)oul 30, 7), (33,13),033414),:

(36,13), (36,17) and

(56, 13) (E.F. Ecklund, R.B. Eggleton, P. Erdés & J.L. Selfridge [74]).

?

the smallest solution of a(n) = a(n + 22); the sequence of numbers satisfying this equation begins as follows: 57, 85, 213, 224, 354, 476, 568, 594, 812, 1218, 1235, 1316, 1484, 2103, 2470, 2492, 2643, 2840, 2996, 3836, 3978, 4026, 4544, 4810, 4844, 5012, 6125, 6356, 6524, 7364, 7532, 7648, 8876, 9272, 9328, ...;

the smallest number n such that >°,, n; the sequence of numbers satisfying this property begins as follows: 60, 120, 210, 420, 840, 1260, 1680, 2310, 2730, 3360, 4620, 5460, 6930, 7140, 9240, ... (see the number 3569 485 920 for w(n)

more on the behavior of the quotient cE

a

e the exponent of the ninth Mersenne prime 2° — 1 (Pervouchine, 1883; and Seelhoff, 1886); e the rank of the prime number which appears the most often as the ninth prime

factor of an integer : pg, = 283 (see the number 199); e the smallest number n such that n >

=, log Pn —

(see the number 492);?8

281¢ is interesting to mention that Ramanujan often used the approximation (x) +

=

logx —1

THOSE

FASCINATING

NUMBERS

Zall

the smallest prime number p such that Q(p+ 1) = 2 and Q(p + 2) = 3; if for each positive integer k, we let q, be the smallest prime number

q such that

GES F250 (G a2) 3 Og Bik)eek ed), then, g 13, g)= 61, qg3 = 193, gg = 15121, gs = 838561 and gg = 807 905 281; the smallest number n such that t(n) < t(n +1) < T(n + 2); it is also the smallest number n such that t(n) < r(n +1) < T(n + 2) < T(n +83): here 2 3000 not containing a given digit @ is of the order of 10—1%°. Indeed, it is clear that the probability that a number chosen at random amongst all those numbers with r digits 8

r—1

does not contain a given digit £ € [0,9] is equal to 9 x (=)

2 with r digits is approximately

with r digits,

log

10

—

. Now, the number of powers of

This is why the probability that there exists a power of 2

0g

r > 3000, not containing the digit @, for a certain ¢ € [0,9], is approximately

SUnieee acaeTy kamameanal r—1

log 2

r>3000

9

10

24

JEAN-MARIE

DE KONINCK

72 the smallest number m such that equation o(x) = m has exactly five solutions, namely 30, 46, 51, 55 and 71;

the only solution n < 10!” of o(n) = 2n +51.

the smallest number 7 which has the property that each number can be written

in the form 2§ +2$+...+2°, where the x;’s are non negative integers (see the number 4); the smallest number n such that a(n) > i7A ~ ee og” n

namely the first two

terms of the asymptotic expansion of Li(n): here we have 1(73) = 21 while Ee ae mad logn log? n

aes

73

~~ 20.9802; if we let nz be the smallest integer n such that

then Wi = nie np2— i/o, ne =

162%

na

230838

news

12 S00

ahing —

3 736 935 913 and n7 = 330645 100273; on the other hand, by examining a table of prime numbers, one can verify that 1018 < ng < 1029: the smallest number > 1 which is equal to the sum of the squares of the factorials of its digits in base 7: here 73 = [1,3,3]7 = 1!? + 3!? + 3!?; the only numbers satisfying this property are 1, 73 and 1051783 (see also the number 582).

the smallest solution of o(n + 7) = a(n) +7; the only other solution n < 10° is 531 434.

the fourth horse number (see the number 13).

the largest known number n such that the two corresponding numbers (n!)? + n! +1 and (n!)? +1 are prime: these two prime numbers both have 223 digits; the other known numbers n satisfying this property are 1, 2, 3 and 4;

the largest two digit number n which is automorphic, that is whose square ends

with n: here 76% = 5776; the automorphic numbers smaller than 107 are 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625 and 7109376.

THOSE

FASCINATING

NUMBERS

28

77] e the largest number which cannot be written as the sum of positive integers whose sum of reciprocals is equal to 1; thus 78 =2+6+8+10+4+12+40

and

tory 5G

tes oa

x

ae

= 1

(R.L. Graham [95]); the smallest champion number whose smallest prime factor is 7: we say that n

is a champion number if D(n) > D(m) for each positive integer m < n, where D(n) = r(n!) — r((n — 1)!) (see A. Ivié & C. Pomerance [112}).

the smallest solution of o2(n) = o2(n + 13); the number of digits in the decimal expansion of the Fermat number 2

il

the number of pseudoprime numbers in base 2 smaller than 10’; we say that an odd composite number n is a pseudoprime number in base 2 if 2”~! = 1

(mod n); see the number 245; the largest known number k such that the decimal expansion of 2° does not contain the digit 8; indeed, using a computer, one can verify that

28 — 302231454903657293676544 does not contain the digit 8, while each number

2", for k = 79,80,...,3000,

contains it (see the number 71);

the smallest number which cannot be written as the sum of less than 19 fourth

powers: here 79 = 15-1*+4.-

24;

the second prime number p such that 317~!' = 1

(mod p?): the only prime

numbers p < 2°” satisfying this congruence are 7, 79, 6451 and 2806 861 (see Ribenboim [169], p. 347). }

the smallest number n such that r(n) > 7(n +1) > T(n +2) > T(n +3): here 10 > 5 > 4 > 2 (see the number 45); the second number

for which the product of its proper divisors is a fourth

power, that is such that

II

d =a": here

d\n, d 1 whose sum of digits is equal to ./n; the smallest number which can be written as the sum of three cubes and as the

sum of four cubes: 81 = 32+ 3° + 3° = 1° + 22 4 23 4+ 4°.

the smallest number which can be written as the sum of four cubes as well as

the sum of five cubes: here 82 = 13 + 33+ 3° +3? = 19+ 194 23 + 234 4°.

1

the smallest number n which allows the sum es 7 to exceed 5: John V. Baxley 1 1 which divides og(n); the sequence of numbers satisfying this property begins as follows: 84, 156, 204, 364, 476, 514, 1092, 1428, 2316, 2652, 2892, 6069, 6188, 6748,... 3?

e the smallest Smith number n (see the number 22) such that n — 1 is also a

Smith number.

e the largest known number k such that the decimal expansion of 2° does not contain 0 as a digit; indeed, using a computer, one can verify that

286 — 77371252455336267181195264 does not contain the digit 0, while each number

2", for k = 87,88,...,3000

contains it (see the number 71).

e the smallest solution n > 1 of o(¢(n)) = a(n); the sequence** of numbers satisfying this equation begins as follows: 21982, 22436, 25978, ...

1, 87, 362, 1257, 1798, 5002, 9374,

e the total number of narcissistic numbers: an r digit number is said to be narcissistic if it is the sum of the r*” powers of its digits: hence if d,,d2,...,d>

are the r digits of such a number n, then n = dj +d, +...+d"; here is the

list?+ of all narcissistic numbers: 321 is worth mentioning that it is known since Erdés that for each integer k > 2, there exist

infinitely many numbers (2006), 372-373).

n such that n|o,(n) (see Problem

11090, Amer.

Math.

Monthly 113

33]t would be interesting if one could prove that this equation does indeed have infinitely many

solutions.

Observe on the other hand that the corresponding equation ¢(a(n)) = ¢(n) could be

proved to have infinitely many solutions if one could prove that there exist infinitely many prime ce p = 2q—1, where q is prime: indeed, if n is of the form n = 2p with p prime and if

= 2q — 1 with q prime, q > 3, then ¢(o(n))= ¢(38(p + 1))= $(8-2-q)= 2(q — 1) while Hea ¢(p) = p — 1 = 2(q—1). De Koninck and Luca (see [57]) have shown that the number of integers n < x such that o(¢(n))= o(n) is < z/(log? z). 34Tn order to find the r digit narcissistic numbers (given that such numbers exist!) for a given positive integer r < 39 (and thus construct the table below), we proceeded as follows. Let n be the quantity a,1"+a22"+...+a99",

where the a;’s are non negative integers such a

aj+a2+...tag =

r. If this number n has exactly r digits and if it is equal to the sum of the r*” powers of its digits, then it qualifies as a narcissistic number.

JEAN-MARIE

DE KONINCK

number of | the narcissistic numbers

digits

(Pas Ueonaen J 153, 370, 371, 407 1634, 8208, 9474 54748, 92727, 93084 548834 1741725, 4210818, 9800817, 9926315 a 24678050, 24678051, 88593477 146511208, 472335975, 534494836, 912985153 10 11

14 16 17 19 20 21 23

™ 24

25

4679307774 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914 28116440335967 4338281769391370, 4338281769391371 21897142587612075, 35641594208964132, 35875699062250035 1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826 63105425988599693916 128468643043731391252, 449177399146038697307 21887696841122916288858,

27879694893054074471405, 27907865009977052567814, 28361281321319229463398, 35452590104031691935943 174088005938065293023722, 188451485447897896036875, 239313664430041569350093 1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380, 3706907995955475988644381, 4422095118095899619457938 121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787, 174650464499531377631639254, 177265453171792792366489765 14607640612971980372614873089, 19008174136254279995012734740, 19008174136254279995012734741, 23866716435523975980390369295 1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915

i

THOSE

FASCINATING

NUMBERS

29

number of | the narcissistic numbers digits:) p)i 0

32 33

17333509997782249308725103962772 186709961001538790100634132976990, | (|: 186709961001538790100634132976991 eee 1122763285329372541592822900204593 12639369517103790328947807201478392, 12679937780272278566303885594196922 37 1219167219625434121569735803609966019

38 39

L

| 12815792078366059955099770545296129367 | |115132219018763992565095597973971522400, 115132219018763992565095597973971522401

e the smallest solution of o(n) = a(n + 30): the sequence of numbers satisfying this equation begins as follows: 88, 161, 164, 209, 221, 275, 279, 376, 497, 581,

107, 869, 910; 913...

e the exponent of the tenth Mersenne prime 2°? — 1 (Powers, 1911); e the smallest prime number amongst those which appear more often as the third prime factor of an integer than as the second prime factor (in this case, when 89 appears in the factorization of a number, it is the third prime factor of that number in 31.6% of the cases, while it is the second one in 27.9% of the cases,

the fourth in 17% of the cases, the fifth one in 6% of the cases and the sixth one in only 1% of the cases): this result can be obtained from those published in a paper by J.M. De Koninck & G. Tenenbaum [63]; e the only solution F;, (where F,, stands for the n*” Fibonacci number) of*°

see B.B.M. de Weger [200];

e the smallest number n > 9 satisfying n = d; + d2?+d4+...+ 2”’, where d,,d2,...,d, stand for the digits of n; the only other eons inibes satisfying this aoe ey is 6603; e the fifth prime Fibonacci number: the sequence of such numbers begins as follows 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433 494437, 2971215073, 99 194 853 094 755 497, 1 066 340 417 491 710 595 814572 169, 35This result can be compared with that of Cohn (see the number 144) according to which 144 is the only Fibonacci number which is a perfect square. Indeed, it is easy to establish that philcm =

py mk+1

:

— m2—m-1

for each m > 2.

al

Thus,

4

the

propert

nes

=

Fy

= a all

k=0

the relation F, =m?

10k+1

k=0

—m-—

1, whose only solution is m = 10, Fn = 89.

is equivalent to

30

JEAN-MARIE

DE KONINCK

19 134 702 400 093 278 081 449 423917 ....; if we let F, stand for the aie bonacci number, and if F; is prime, then one can show that k is either 4 or a prime number: this helps in establishing that the set of numbers k < 100000 for which F;, is prime is {3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839}; no one has yet proved that there are infinitely many prime Fibonacci numbers;

the fourth prime number with at least two digits and whose digits are consec-

utive integers (see the number 67).

the only number < 10° which is not perfect but which is equal to the sum of its proper deficient divisors: 90 = 1+2+3+45+9+410+ 15+ 45; a number n

is said to be deficient if o(n) < 2n; the largest solution of ¢(x) = 24, the others being 35, 39, 45, 52, 56, 70, 72, 78 and 84;

the third unitary perfect number (see the number 6); the only number which is equal to the sum of its digits added to the sum of the squares of its digits.

the smallest pseudoprime in base 3: given a number a > 1, a composite number n > aissaid to be pseudoprime in baseaifa"~' =1 (mod n); the ten smallest pseudoprimes in base 3 are 91, 121, 286, 671, 703, 949, 1105, 1541, 1729 and 1891;

the largest known number k such that the decimal expansion of 2° does not contain the digit 1; indeed, using a computer, one can verify that

2°! — 2475880078570760549798248448 does not contain the digit 1, while each number 2*, for k = 92,93,...,3000, contains it (see the number 71); the rank of the prime number which appears the most often as the tenth prime

factor of an integer : po; = 467 (see the number 199).

the number of integer zeros of the function M(z) := Ne y(n) located in the n1

THOSE

begins as follows:

FASCINATING

NUMBERS

Sil

1, 6, 92, 406, 1549, 5361, 12546, 41908, 141121, ...; it is

well known that the function M(«) changes sign infinitely often as x — oo (see E. Grosswald [98]), which implies in particular that the number m,; does indeed exist for each k > 1.

e the largest known number k such that the decimal expansion of 2° does not contain the digit 6; indeed, using a computer, one can verify that

2°3 — 9903520314283042199192993792 does not contain the digit 6, while each number 2*, for k = 94,95,...,3000, contains it (see the number 71);

e the smallest number > 1 whose sum of divisors is a seventh power: 7(93) = 2”.

e the seventh number whose sum of divisors is a perfect square: 7(94) = 11° the sequence of numbers satisfying this property begins as follows: 1, 3, 22, 66, 70, Sa lloes Oe 70, 210" 214s eee

e the smallest number

n such that each number

m

> n can be written as the

sum of distinct elements from the set {p,, : n = 1,2,...} = {3,5,11,17,...} (see R.E. Dressler & S.T. Packer [69]); e the smallest number m such that equation o(z2) = m has exactly four solutions, namely 42, 62, 69 and 77;

e the smallest number n > 1 such that y(n)?\a(n): the sequence of numbers satisfying this property begins as follows: 1, 96, 864, 1080, 1782, 6144, 7128, ...%°; if nz stands for?’ the smallest number n for which o(n)/y(n)* is an integer, then n1 = 6, nz = 96, n3 = 3538944 and n4 < 19698744770 118 549 504 (see

the number 1 782).

e the smallest prime number preceded by exactly seven consecutive composite numbers; indeed, there are no prime numbers between 89 and 97; if q, stands for the smallest prime number that is preceded by exactly k consecutive composite Dumberssthenagie=—so.egs — 11, gs; = 29, dy =89i.udo = 1149) ginn = 211, Chile) =

127, Chik = il847, Clie == 941, Cie =

907 and

Opal =

1 151;

e the largest two digit prime number.

361t is easy to see that there exist infinitely many numbers satisfying this property, namely all the numbers n = 2%3°, where a+ 1 is a multiple of 6 and 6 > 1 is odd. 387By examining the numbers of the form n = 2°38 with an appropriate choice of a and 8, one

can easily discover infinitely many numbers n for which y(n)*|o(n) for any fixed number k.

32

JEAN-MARIE

DE KONINCK

e the smallest solution of y(n + 1) — y(n) = 19: the only*® solutions n < 10° of this equation are 98, 135 and 11375; in fact, below is the table of all the

solutions n < 10° of equation y(n + 1) — y(n) = k for 1 < k < 100 (observe that this equation has no solution n < 10° for k = 5, 9, 25, 33, 35, 37, 51, 57, 61, 63, 65, 77, 81, 87 and 95): k

n < 10”

such that

y(n+1)-y(n) =k 3

4, 49

fe |e)ale 11 | 20, 27, 288, 675, 71199 13nis 1523024 15 | 16, 28 17 | 1681, 59535, 139239, 505925 19 | 98, 135, 11375 21 | 25, 2299, 18490 23 | 75, 1215, 1647, 2624 27 | 52, 39325 29 | 171, 847, 1616, 4374 31 | 32, 36, 40, 45, 60, 1375 39 | 76, 775 41 | 50, 63000 43 | 56,84 45 | 22747, 182182 47 | 92, 1444, 250624 49 | 54, 584, 21375, 23762, 71874, 177182720 53 | 147, 315, 9152, 52479 55 | 512, 9408, 12167, 129311 59 | 324, 4239 67 | 72, 88, 132, 5576255 69 | 82075, 656914 71 | 140, 3509, 114375 73 | 872, 1274, 3249 75 | 148, 105412, 843637 79 | 81, 104, 117, 156, 343, 375, 7100, 47375, 76895 83 | 164, 275, 5967, 33124, 89375, 7870625,38850559 85 | 126, 1016, 16128, 471968, 10028976 89 | 531, 11736 91 | 96, 100, 1050624 93 | 832, 201019, 1608574 97 | 3807, 4067, 12716, 73304 99 | 112, 1975, 8575 38It seems plausible that, for each odd number

k >

1, the number

of solutions of equation

y(n + 1) —7(n) = & is finite. In 2003, J.M. De Koninck & F. Luca [53] proved that if the abc Conjecture is true, then, for each odd integer k # +1, equation y(n +1) — y(n) = k has only a finite number of solutions.

THOSE

FASCINATING

NUMBERS

33

e the numerical representation adopted by the Greeks to denote the word AMEN

= apnv = 1+ 40+8+50 = 99 (see Ore [156], p. 28); e the third solution « of the Fermat-Pell equation x2? — 2y? = 1: it is well known that this equation has infinitely many solutions, the first nine being (x, y) = (3,2), (17,12), (99,70), (577,408), (3363,2378), (19601,13860), (114243,80782), (665857,470832) and (3880899,2744210).

100

e the only solution n < 10!” of o(n) = 2n + 17 (see the number 196); e the largest known solution y of the problem consisting in finding a right angle triangle whose sides x,y, z (all integers) represent respectively a triangular number, a perfect square and a pentagonal number: here the solution is

(x,y,z) = (105, 100, 145); the only other solution is (3,4,5) (R.K. Guy [101], D2):

101

e the largest known prime number of the form 10" +1; any other prime number, with more than three digits and of the form 10” + 1 must be such that n >

131071 (see the number 19841 for the factorization of those numbers of the

form 10? +1 for 1 2 such that n> +1 is*® prime, then ny = ne = ng = M4 = 2, (Vs =

30, ng =

OZ,

iy =

130, N=

278, (Oe) =

46, nio

=

824 and

ni

=

SO:

e the smallest solution of a(n) = 02(n +17).

103

e the smallest prime number p such that w(p+1) = 2 and w(p+2) = 3 (see also the number 64); e the largest prime number p < 2°? such that 43?~! = (mod p”): the only other prime number p < 2°” satisfying this congruence is p = 5 (see Ribenboim [169], p. 347).

104

e the smallest composite number n such that a(n + 6) = a(n) +6:

the only

numbers n < 10° satisfying this equation are 104, 147, 596, 1415, 4850, 5337, 370 047, 1630622, 35020303 and 120 221 396;

e the fourth solution of ¢(n) = ¢(n + 1) (see the number 15). 105

e the smallest number n such that $(n) < d(n +1) < ¢(n+ 2): here 48 < 52 < 106; n = 1484 is the smallest number n such that ¢(n) < ¢(n +1) < o(n + 2) < d(n +3); Nicolas Doyon claims (private communication) that it is possible to prove that, for all integers k > 2, if a1, a2,...,ax is any permutation of the integers 1,2,...,k, then there exist infinitely many integers n such that

o(n+a,) < d(n+ ag) 9 such that n+1,n4+2,...,n+k-—1 are also Niven numbers, then ng = 20, n3 = 110, ng = 510, ns = 131052, ng = n7 = 10000095 and ng = ng = 124324 220.

111

the smallest insolite number: a number > 1 is called insolite if it does not contain the digit 0 in its decimal expansion and if it is divisible by both the sum and the product of the squares of its digits: the sequence of insolite numbers is infinite and begins as follows: 111, 11112) 1122412, 1111111117 122 121216, TA 22a aa sb tO 22d G2 1 tio. PULEVIS1 216. PTI 122 2 ina 21h 232 i Lijo kee 10 273 111232541 SIP S220 22 is 312 2 Sites ol ior S21 III 232 22 232 S222 2 ie a2? 12. LP St2 32254 25d eho 22a iat Oe ee 1G is: 120 S220 22a eri hilt te oe 2ATA US 22 42 w2) A 2 22a 2S Od eS bee tee TUPI T1 103 S12, WEI 2 4 2 Od 313321 21651331611 S22 0b eee he following table provides, for each number k < 9, the list of the smallest insolite number n = nx containing the digit k: 431¢ has been known at least since 1997 (see Wilson [207]) that it is possible to construct a sequence of 20 consecutive Niven numbers, but that no strings of 21 consecutive Niven numbers exist. More recently, in 2008, De Koninck, Doyon & Katai [51] obtained, for each positive integer r < 20, an

asymptotic formula for the number of r-tuples (n,n +1,...,n+r-—1), where each n+i

is a Niven

number, with n < gz.

441¢ is easy to see that any number 11...1, where k is such that k|(10* — 1), is such a number, k

which is the case when k = 3, 9, 27, 81, 111, 243, 333, 729, 999, ... But this last sequence is infinite

because it contains all numbers of the form 3%, the reason being that 3% divides 10°” — 1 for each number a > 1, a result which can easily be proved by induction. J.M. De Koninck & N. Doyon [48]

proved that if I(x) stands for the number of insolite numbers < z, then al exp {= (log log a)? + O(log log z log log log x) K I(x)

K 29-462,

THOSE

FASCINATING

NUMBERS

ait

nm = Np insolite

Tu LZ, A) i oh a Ys 11121114112 WD oR |e Piri ael le

—

IL

al

yo)

Era! (pill Nf 19.0) We)

n = nx insolite IPP DAG W234 TPA? 121111216128 911131 213 824

ial

in this table, the star (*) next to the number n; indicates that it is the smallest known insolite number with this property;

e the value of the sum of the elements of a diagonal, of a line or of a column in

a 6 x 6 magic square (see the number 15).

[112] (=24-7) e the smallest number n having at least two distinct prime factors and which is such that p|n => p+ 8|n +8; the sequence of numbers satisfying this property begins as follows: 112, 135, 432, 532, 832, 847, 1372, 1792, 2632, 3072, 8092,

B70) ee

113 = n/logn reaches its maximal value e the number n at which the quotient Q(n) Saw) (see Rosser & Schoenfeld [{178]); in fact Q(113) ~ 1.25506; moreover, it is the

5 does not hold (since 4 logn

only number n > 2 for which the inequality m(n) < — —"

30 > 29.8791);

e the prime number which appears the most often as the seventh prime factor of

an integer (see the number 199).

114

e the smallest solution of a2(n) = g2(n + 19).

115

e the seventh number n such that n-2"—1 is prime: the only numbers n < 700000 satisfying this property, also at times called Woodall numbers, are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 882, 5312, 7755, 9531, 12379, 15 822, 18885, 22971, 23005, 98 726, 143018, 151023 and 667 071 (a result due

to Keller).

38

JEAN-MARIE

DE KONINCK

116

e the tenth number n such that n!+1 is prime: the only known numbers satisfying this property are 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380 and 26951. 117 e the tenth number n such that n- 10” — 1 is prime*®; the sequence of numbers satisfying this property begins as follows: 2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, ... (see the number 363).

118 e the smallest possible sum

common

to four triplets of numbers

having same

sum and same product, namely the triplets (14,50,54), (15,40,63), (18,30,70) and (21,25,72): (Problem E2872, Amer. Math. Monthly 89 (1982), p. 499); e the largest number which cannot be written as the sum of three powerful num-

erst iGal)

1 whose sum of divisors is a perfect square: o(119) = 12?;

e the third number n such that ¢(n)o(n) is a cube: the sequence of numbers satisfying this property begins as follows: 1, 3, 119, 357, 2522, 6305, 6596, 6604,

7566, 18915, 19788, 19812, 20520,... (see the number 170); e the smallest number n satisfying ¢(n) = 3¢(n + 1); the sequence of numbers satisfying this equation begins as follows: 119, 527, 545, 2849, 3689, 4487, 6649, 18619, 26771, 30377, 44659, 47585, 50507, 76997, 83021, ...(see the number

629);

e (probably) the largest number which cannot be written as the sum of two coprime numbers

933).

each having an index of composition

> 1.4 (see the number

451t is easy to prove that ifn =1 (mod 3), then the corresponding number n- 10" — 1 is not prime since it is a multiple of 3; see the number 363 for the idea behind the proof. 46Since any number which is not of the form 4°(8k + 7), with £ > 0 and k > 1, can be written

as the sum of three squares (see Grosswald [99]), it is clear that one only needs to verify that if n is of the form n = 8k + 7, then it can be written as the sum of three powerful numbers. The only numbers 8k + 7 < 127 which can be 63, 71, 79, 95 and 103. Moreover, if of three powerful numbers, then one the largest number n having exactly

written as the sum of three powerful numbers are 39, 47, 55, k(n) stands for the number of representations of n as the sum can prove that limn—oo k(n) = +00. Finally, if nj stands for j representations as the sum of three powerful numbers, then

my = 399, no = 1 263, n3 = 1335, na = 2103 and n5 = 1991. he

THOSE

FASCINATING

NUMBERS

39

120

e the smallest tri-perfect number

(n is tri-perfect if a(n) = 3n): only six tri-

perfect numbers are known, namely 120, 672, 523 776, 459 818 240, 1476 304 896

and 51001180160, and it seems that there are no others (see R.K. Guy [101], B2). e the only number n which can be joined with the numbers 1, 3 and 8 to form the set A = {1,3,8,n} so that ifz,y € A, x 4 y, then zy +1 is a perfect square; this result was obtained by Euler; for a thorough analysis of this problem, see

L. Jones [113] or the more recent paper*” of Dujella [73]; e one of the five numbers (the others being 1, 10, 1540 and 7140) which are both triangular and tetrahedral (see the number 10); e the smallest solution of o2(n) = a2(n +10): the only solutions n < 10° of this equation are 120, 942, 5395, 4737595, 98 600 035.

6811195,

11151355,

74699995

and

121

e the smallest number n > 1 which is both*® a star number and a perfect square: a star number is a number of the form 6n(n + 1) + 1; the sequence of numbers satisfying this property begins as follows: 121, 11881, 1164241, 114083761, 11179044361, 1095432263641, ...; e the only known perfect square of the form 1+p+p? +p? +p*, where p is prime: here with p = 3. 122 e the only known number whose square is the sum of a fourth power and a fifth

power:

here 122? = 114 + 3° (see H.Darmon & A.Granville [42] as well as

the number 21 063 928); in fact it is conjectured that the only co-prime integer solutions 2, y, z (non zero) of the equation z?+y!% = z", with ay < 1, where exactly*? one of the numbers p,q,r is equal to 2, are those appearing in the 47

set of m positive integers {a1,a2,...,@m}

perfect square for all

1 < i < j < m.

is called a diophantine

m-tuple if aja;

The first diophantine quadruplet,

+lisa

that is {1,3,8, 120},

was found by Fermat. In 1969, Baker & Davenport [11] proved that this quadruplet could not be extended to a diophantine quintuplet. Let us mention that in 1979, Arkin, Hoggatt & Strauss [7] proved that each diophantine triplet could be extended to a diophantine quadruplet: indeed, if {a,b,c} is such a triplet and if ab +1 = r?, ac +1 = s? and bc+1 = ??, where 1,s,t are positive integers, then one easily verifies that d = a+6b+ c+ 2abe + 2rst is such that {a, b,c, d} is a diophantine quadruplet. In 2004, Dujella [73] proved that no diophantine 6-tuple exists and that there can only exist a finite number of diophantine 5-tuples, and in fact that any element of a

diophantine 5-tuple must be smaller than 10107°. Let us add that it is easy to prove that there exist infinitely diophantine quadruplets; indeed, one only needs to prove that there exist infinitely many diophantine triplets and to use the result of Arkin, Hoggatt & Strauss mentioned above; one then only needs to verify that the triplets {1, r? —1,r? + 2r}, where r = 2,3,4,..., are all diophantine. 48Qne can establish the recurrence formula EL, = 98E,_1 — Ex—2 + 24, where Ey, stands for the kt? number which is both a star number and a perfect square.

49 According to the Beal Conjecture, there are no solutions with min(p, q,r) > 3.

4O

JEAN-MARIE

DE KONINCK

relations 1?+23 = 32, 25+72 = 34, 73413? = 29, 274178 = 71, 3°+114 = 122?, 177 + 762713 = 21063 9287, 14143 + 22134592 = 65’, 9 262% + 15312283? =

1137, 438+96 2223 = 30 042 907? and 33°+1 549 034? = 15 613° (see C. Levesque [124]).

123

e the eighth number n such that n- 2” — 1 is prime (see the number 115). 124

e the only number besides 188 which cannot be written as the sum of less than

five distinct squares (R.K. Guy [101], C20); e the second pseudoprime in base 5: the ten smallest pseudoprimes in base 5 are A 124,217, 50l.7S81.1 041, 1729 1801) 2821eand 4123:

125

e the smallest Canada perfect number®®, that is a number for which the sum of the squares of its digits is equal to the sum of its proper divisors > 1; thus

1? +2? +5? = 5425; the only numbers having this property are 125, 581, 8549 and 16999 (see J.M. De Koninck & A. Mercier [60]).

[126 |(=2- 3-7) e the smallest (and perhaps the only one) S-perfect number with three distinct

prime factors:

a number n is said to be S-perfect (or a Granville number)

if Dudindensdes@ = n, where S is the set of integers defined by 1 € S and 2 1 of ¢(a(n)) =n

is even, since ¢(n) is even for all n > 3.

521¢ would be interesting if one could prove that this sequence is infinite.

42

JEAN-MARIE

DE KONINCK

[is3] (=7-19) e the 100%” composite number; if we denote by ng, the k*” composite number, then we have the following®® table: Qa | N10

a

210%

fj) dks} Dy || exes

8 9

106 091 745 1053 422 339

By |) ine Al || Thay

10 | 10475 688 327 11 | 104287176419

OM

Sia

|

12 | 1039019 056 246

6 | 1084605 7 | 10708555

13 | 10358 018 863 853 14 | 103307 491 450 820

e the smallest solution of r(n + 11) = 7(n) + 11; the sequence of numbers satisfying this equation begins as follows: 13445, 16373, 21598, ...

133, 2489, 3958, 4613, 5765, 8453, 9593,

135 e the smallest number n such that n and n+ 1 each have four prime factors

counting their multiplicity: n, the smallest number

135 = 33-5 and 136 = 2° - 17; if we denote by

n such that n and n+ 1 each have k prime factors

counting their multiplicity (that is such that Q(n) = Q(n+1) =k), then ni = 2, no = 9, nz = 27, ng = 135, ns = 944) ne — 5 204, 17 — 29888, 1g — 00624, ng = 203391, nio = 3290624, ni, = 6082047 and ni2 = 32535 999 (see the number 230 for the similar question with the w(n) function); e the smallest solution of a(n) = o(n + 23); the sequence of numbers satisfying this equation begins as follows: 135, 231, 322, 682, 778, 1222, 1726, 1845, 5026,

92011, ...:

e the smallest odd number n > 1 such that y(n)|o(n); the sequence of numbers satisfying this property begins as follows: 11907, 41067, 43875," 33;

135, 891, 1521, 3375, 5733, 10935,

e the second solution of y(n + 1) — y(n) = 19 (see the number 98).

53We obtained the values appearing in this table in the following manner. It is clear that, for each k > 2, the number ng satisfies relation (*) nz = 1+ (np) +k. Using the Prime Number

Nk

Theorem in the form m(z) ~ z/logx+a/(log x)?, it follows from (*) that np ~ log nz

(as k —

oo), so that nz (1- cane - wah) ~ k. It follows in particular that logn, ~ logk.

il Combining these last two estimates, one obtains nz, ~ k/ (1= ek provides a starting point for the first approximation

i = lof (| L= can - rer

and setting s = s(n)

equal to 0, one replaces n by n+ a. &

log? Nk

of niga.

il

ae

Indeed,

= 1+7(n)+k—~n,

be -). Setting

Oe Pe eee

i

k = 10 =—

a

using the approximation

then, as long as a := s is not

THOSE

FASCINATING

NUMBERS

43

136

e the fifth number n such that ¢(n — 1)o(n — 1) =

@(n)o(n);

the sequence

of numbers satisfying this equation begins as follows: 6, 56, 57, 124, 136, 148, 176, 305, 352, 645, 1016, 2465, 19305, 61132, 162525, 476672, 567645,

712725, 801945, 2435 489, 3346 400, 3885057, 4556000, 8085561, 8369 361, 12516693, 22702 120, 29628 801, ... (see the number 55).

137 e the smallest possible value of the largest prime factor of n+ +1 for n > 4: this

lower bound is reached when n = 10 (see M. Mabkhout number 239);

[130] as well as the

the second Stern number: a number n is called a Stern number if it cannot be written as n = p+ 2a” for some prime number p and some number a: it seems that there exist only eight Stern numbers, namely 17, 137, 227, 977, 1187, 1493, 5777 and 5993 (only these last two are not prime numbers): see

L. Hodges [111]; e one of the only two prime numbers p (the other being 73) with the property®4

that any number of the form abcdabcd is divisible by p;°° e the smallest number n such that ¢7(n) = 2, where ¢7(n) stands for the seventh iteration of the ¢@ function;

if we consider

the sequence

(nz),>1

defined by

np = min{n : o*(n) = 2}, the first terms of that sequence are 3, 5, 11, 17, 41, 83, 137, 257, 641, 1097, 2329, 4369, 10537, ... (see Giblin [89], p.117 and R.K. Guy [101], B41);

e the fifth prime number p such that 19?-!=1

(mod p”) (see the number 43);

e the largest prime factor of 123456787654321.

139 e the smallest prime number p such that p+ 10 is prime and such that each number between p and p+ 10 is composite ; if q, = py stands for the smallest prime number such that p,;+1 — pr = 10k, we have the following table: 54This simply follows from the fact that 137 - 73 = 10001. 55Other interesting properties of the number 137 are mentioned in the book of Martin Gardner

[38].

44

JEAN-MARIE

10k

dk = Pr | Prti

10 139 20 887 30 4297 40 19 333 50 31907 60 43 331 70 173 359 80 542 603 90 404 851 100 || 396 733 110 || 1468277 | 10k

230 240 250 260 270 280 290 300 310

DE KONINCK

dk = Pr

149 907 4327 19 373 31957 43 391 173 429 542 683 404 941 396 833 1468387

dk = Pr

|) 607010093 || 391995 431 || 387096 133 || 944192 807 || 1391048047 || 1855047163 || 1948819133 || 4758958741 || 4024713661 2 300 942 549 6 291 356009

1 895 359 5 518 687 7621 259 13 626 257 33 803689 27915737 17051 707 142 414669 378 043979 20 831323 47 326693

Pr+i 1895 479 5518 817 7621 399 13 626 407 | 33803849 | 27915907 17051 887 | 142414859 | 378044179 | 20831533 | 47326913|

Pr+1

607 010 323 391 995 671 387 096 383 944 193 067 | 1391048317 | 1855047 443 | 1948819 423 | 4758959 041 | 40247113971 | 2300 942 869 | 6291 356 339

as for the smallest prime number p, such that p,11;—p, = 1000, see the number 22 439 962 446 379 651;

e the number of digits in the fourth prime number n whose digits are 1 and 2 in alternation, that is of the form n = 1212...121; the prime numbers of this form, with k digits, k < 2000, are those with k = 7, 11, 43, 139, 627, 1399, 1597 and 1979 digits respectively; e the largest known prime p such that 3? +2 is also prime; the other known prime

numbers p for which 3? + 2 is prime are 2 and 3.

140 e the only number n > 2 such that n? =

for a certain number m, here with

m = 50; it is interesting to observe that K.Gyory

the only solution of n’ =

[105] has established that

a other than the ones with k = @ = 2 is the one

k With 7 == 140.0 = 2° 7h = o0 and ik = 3:

e the smallest number > 1 which is not perfect or multi-perfect (we say that a

number n is multi-perfect if o(n)/n is an integer) but whose harmonic mean is

THOSE

FASCINATING

NUMBERS

45

an integer: the harmonic mean of a number n, denoted by H(n), is defined by =i!

d\n

the sequence of numbers satisfying this property begins as follows: 1, 140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600, 4713984, 4754880, 5772200, 6051500, 8506400, 8872200, ...; e the largest number which when raised to the square becomes a tetrahedral number (Sierpinski [185], p.87): the only numbers satisfying his property are 1, 2 and 140.

141 e the smallest number n > 1 such that n-2” + 1 is prime; numbers of the form n-2” +1 are called Cullen numbers®®; it is not known if infinitely many of these numbers are prime: the only known prime numbers of the form n-2" +1 are those with n equal to 141, 4713, 5795, 6611, 18496, 32 292, 32 469, 59 656,

90 825, 262 419, 361 275 and 481 899 (see W. Keller [{118]).

142

e the smallest solution of a(n) = o(n + 17); the sequence of numbers satisfying this equation begins as follows: 142, 238, 418, 429, 598, 622, 2985, 3502, ...;

e the only solution y of the diophantine equation x(a+1)(a+2)(x+3)(a+4)(a+

5) = y? —4, the solution (x, y) being (3, 142) (L.E. Mattics [132])°”. 143 e the smallest number r which has the property that each number can be written

as a{+a+...t+2/, where the z;’s are non negative integers (see the number 4); e the number of three digit prime numbers (see the number 21).

56Father Cullen first considered these numbers in 1905.

57Mattics studied the equation x(a + 1)(a + 2)(x + 3)(a@ + 4)(a@ +5) = y? —k, for |k| < 31, and obtained that this diophantine equation has solutions for k = 4 (with (x,y) = (3,142)), k = 9 (with (x,y) = (1,27)) and k = 25 (with (x,y) = (21, 12875)); he also proved that for the positive

solutions, we must have 1 < x < max(27, (40|k| + 1)1/3).

46

JEAN-MARIE

4a

(=e

DE KONINCK

37)

e the only Fibonacci number

> 1 which is a perfect square (see J.H.E. Cohn

[34])°*; e the smallest number whose fifth power can be written as the sum of four non

zero fifth powers: 1445 = 27° + 845 + 110° + 133° (R.K. Guy [101], D1); for a similar question with fourth powers, see the number 422 481;

e the only solution n < 10” of o(n) = 2n +115; e the smallest number

n having at least two distinct prime factors and such

that B,(n) = G(n)?: here 2* + 37 = (2 +3)?; the sequence of numbers satisfying this property begins as follows:

123 008, 380000,

144, 1568, 3159, 5346, 8064, 56000,

536544, 570752, 584064,

729088, 2267136,

8258048

...;

it is clear that if there exist infinitely many Mersenne primes, then equation

B,(n) = B(n)? has infinitely many solutions??; nevertheless, J.M.De Koninck & F. Luca [59] proved that, without any conditions, this sequence is infinite®°; e the smallest number > 2 which is equal to the product of the factorials of its digits in base 5: 144 = [1,0,3,4]; = 1!-0!-3!- 4!; the only known numbers satisfying this property are 1, 2, 144, 1728, 47775744 and 27 134 923 845 424 074 797 548 044 288 (see the number 17 280 for the table of the smallest numbers with this property in a given base).

145

e the smallest number > 2 which is equal to the sum of the factorials of its digits (145 = 1!+4!+5!); the only other number > 2 satisfying this property is 40585

(L. Janes, 1964)°. 147

e the number

of solutions

2 < x,

< aw

< Le

2 of Sos

5 1 of )7—=1

(see

i=1~?

RK Guy [f01)) Dll);

'8Recently (in 2006), Y. Bugeaud, M. Mignotte & S. Siksek [27] proved that 1, 8 and 144 are the only Fibonacci numbers which are powers. °°Indeed, one can easily check that if p = 2°-2 — 1 is prime, then n = 2% -p is a solution of

Bi(n) = B(n)?.

Tn fact they proved more, namely that if A(x) stands for the number of n < z such that equation

Bi (n) = B(n)? is verified, then the following bounds hold as x — oo:

a exp (2 / 34/3 + 0(1))\/logx log log z)

< A(z)
k, then qi = p3 = 5, PL 4k

go = Ps9 = 277 and q3 = p3eii39 = 5195977; one can prove” that q4 ~ 10'° (see the number 1307 for an analogue problem).

74Indeed, using inequalities

log

|

coe

B-

2 log? qa


1389.5 = (10!!)?/7.

More recently, G. Harman [108] proved that, for x large enough, C(x) > a ,

where 3 > 0.33. It is conjectured that, for any e > 0, C(x) > zi—€ if ¢ > zo(e).

100

JEAN-MARIE

DE KONINCK

671

e the fifth self contained number (see the number 293); e the value of the sum of the elements of a diagonal, of a line or of a column in

a 11 x 11 magic square (see the number 15).

(emai 2223) 21) e the second tri-perfect number (see the number 120). 674 e the smallest number which can be written as the sum of four fourth powers as well as five fourth powers: 674 = 34 + 34 + 44 4+ 44 = 144+ 244 244 24-45%.

675 e the third powerful number n such that n+1 is also powerful: here 675 = 33-5?

and 676 = 27 - 13? (see the number 288). 679 e the smallest number of persistence 5: we say that a number is of persistence k if the number of iterations required in the process of multiplying the digits to finally arrive at only one digit is k: 679

—> 378

—

168

—> 48 —> 32 —

6;

if nz, stands for the smallest number of persistence k, then n, = 11, no = 25, Ns = 39, na = 77, Ns. = 679,.ng = 6788, n7 = 68.889, 13. = 2677889, ng = 26 888 999 and nig = 3778 888 999.

e the smallest number n such that n?+1 is a powerful!?* number: here 6822+1 = 5°-617; the sequence of numbers satisfying this property begins as follows: 682, 1 268 860 318, 2 360 712 083 917 682, 4392 100 110 703 410 665 318, 8 171 493 471 761 113 423 918 890682, ..., and this sequence is infinite!°.

104T¢ is easy to see that such a number must be even, since otherwise 2|/n? +1 in which case n? +1 is not powerful. 5, Indeed, since each powerful number

is of me form xy? for certain numbers x and y (with u(y)= 1), we only need to show that equation n? +1 = x2y3 has infinitely many integer solutions (ip Gene But we ance know the solution (n,z,y)= (682,61,5), meaning that the Fermat-Pell equation n? — 125a? ==et has a solution (aamely (n, x) = (682,61)= (n1,21), say). Therefore, it follows that equation n? — 125”? = —1 has infinitely many solutions (n2441,22k41), given implicitly by N2k+1 + @2n41V125 =(m+21V

225)75t4,

k = 1,2,.

THOSE

FASCINATING

NUMBERS

e the fourth number n such that o3(n) is a perfect square:

101

indeed, o3(690) =

19656? (see the number 345). 693

e the second number n such that E,(n) := a(n+1) —o(n) satisfies B,(n+1) = E,(n): here the common value of E, is —204, since o(693) = 1248, o(694) = 1044 and 0 (695) = 840; the sequence of numbers satisfying this property begins as follows: 44, 693, 3768 373, 6 303 734, 15913 724, 20291270, ...

[697 |(= 17- 41) e the smallest 12-hyperperfect number:

ifm =1+4+12

a number n is said to be 12-hyperperfect

SS d, which is equivalent to 120(n) = 13n +11:

the smallest

d|n l 1 (see the number 78557 for a similar argument concerning the numbers of the form k - 2" + 1).

106

JEAN-MARIE

DE KONINCK

779

e the smallest number n such that ¢(n) = 6); if n, stands for the smallest number n such that ¢(n) = k!, then n; = 1, no = 3, ng = 7, ng = 35, n5 = 148, ng = 779, n7 = 5183, ng = 40723, ng = 364087, nip = 3632617, ni1 = 39916801

and n12 = 479045521; Erdés observed that equation ¢(n) = k! has a solution n for each number

k > 1, while this was less obvious for a(n) = k! for each

k > 3 (see the number 1 560). 780 e the smallest solution of a(n) = 3n + 12: this equation has only four solutions smaller than 10°, namely 780, 2352, 430272 and 184773 312. 782

e the second number n such that o(n) = o(n + 13) (see the number 182). 787

e one of the only two prime numbers with three digits (the other is 101) whose

digits are consecutive (see the number 67). 823

e the smallest number n such that ¢(n) > ¢(n+1) > (n+ 2) > d(n +3): here 822 > 408 > 400 > 348 (see the numbers 313 and 1484). 828 e the smallest number n which allows the sum

2 m 1, h(n) is either 0 or an even positive integer; if na, stands for the smallest number n such that h(n) = 2k, then we have the following!!! table:

fact | CE IL alteta te re Cea 3 Dae[wos [wen Be100|aTooo[araTOO Sonscoo|aR IE 1111f one could prove that for any positive integer /, there exists a number n such that h(n)= 2°, one would automatically obtain a proof of the existence of an elliptic curve of rank 2.

THOSE

FASCINATING

NUMBERS

107

834 the rank of the prime number which appears the most often as the 15*” prime

factor of an integer: pg34 = 6397 (see the number 199). 836

the second bizarre number (see the number 70); the fourth number which is not a palindrome, but whose square is a palindrome

(see the number 26). 839

the smallest prime factor of the Mersenne number 249 — 1.

[840] (= 23-3 -5-7) the largest number

‘| n such that Ss — 2, we denote by nz the smallest positive integer n such that

Gen

eat (nape hy.

we then have the following table:

rom (CUE

ne | 1 [4 | 843 |74848 | 671345 |8870024

108

JEAN-MARIE

DE KONINCK

and!!? most likely,

ng = 1770019 255 373 287 038 727 484 868 192 109 228 823, With} (is 4 1) — 0 for 7 — 1) 27

6.

844

e the smallest number n such that n,n+1,n+2,n+3,n+44 are all divisible by a square > 1: here 844 = 27-211, 845 = 5- 137, 846 = 2-3? - 47, 847 =7- 112,

848 = 24-53 (see the number 242).

854

e the largest number n such that if A and B stand respectively for the set of digits of n and of n?, then AUB = {1,2,3,...,9} and ANB = 9: here 854? = 729 316; the only other number satisfying this property is 567.

857 e the smallest prime number q such that 4 + 5 ot Ase ee 17

; > 1 (see the

number 347)!13,

858

e the second Giuga number (see the number 30).

860

e the smallest solution of o2(n) = o2(n + 8): the list of numbers satisfying this equation begins as follows: 860, 4316, 3790076, 5448956, 8921084, ... 114

112J.M.De Koninck & F. Luca [58] proved that the number n, exists for each k > 2 while also providing lower and upper bounds for the number np.

113 Given a large prime px, one can estimate the size of the smallest prime Qk such that

)

it

PkSPL 4k

is the nearest to 1. To do so, we use the formula De 2 ‘ = loglogx+c+O = ) Pk SPSL4k

1

1

(az), yielding ed

1

| = loglog a4 ~ loglog px + 0 ( = )wt, Pp log* pr

;

:

which occurs when log (ae) ~ loge, that is log q, © log py,, meaning that qx ~ pé. 14One can prove that if Hypothesis H is true (see its statement on page xvii), then this equation has infinitely many solutions.

THOSE

FASCINATING

NUMBERS

109

863

e the smallest prime factor of the Mersenne number 2*%! — 1, whose complete factorization is given by 2431 __1

=

863-3449 - 36238481 - 76859369 - 558062249

4642152737 - 142850312799017452169 - Pro: it is the smallest Mersenne number with exactly eight prime factors (see the number 223 for the list of the smallest Mersenne numbers which require a given

number of prime factors); e the second prime number p such that 13?~' = (mod p?): the only prime numbers p < 2° satisfying this congruence are 2, 863 and 1 747591 (see Ribenboim [169], p. 347).

864

e the smallest solution of o2(n) = a2(n + 12); the sequence of numbers satisfying this property begins as follows:

13:38.L626%

864, 1290, 6474, 5685114,

8173 434,

20422:

e the second number n > 2 such that

oO (n)

+ o(n)

is an integer (see the number

588). 870 e the smallest number which is not the square of a prime number, but which can be written as the sum of the squares of some of its prime factors!!®: here 870 = 2:3-5-29 = 27 + 5? + 297: the only numbers smaller than 10!° satisfying

this property number 5 209 if nz, for k > which can be

are 870, 188355, 298 995 972, 1152597606 and 1879 906 755; the 105 541 772 also satisfies this property (see also the number 378); 2, stands for the smallest number which is not a k‘” power, but written as the sum of the k‘” powers of some of its prime factors,

then

ny ig na

= = =

me

=

870=2-3-5-29=27+57 +4297, B18 28 SH oo Ee, 107827277891 825604 = 27-3-7-31-6718121 -34105993 = 3" 4 31° +677 + 18121, 178101 =3° -7-11-257=38°+7° + 11°,

1151T¢ is easy to generate these solutions using those of equation 72(n) = a2(n + 2) (see the number 1089) namely by examining the numbers n = 6m with (m,6) = 1, for which we have o2(6m) = o2(6m + 12), an equation which is equivalent to ¢2(m) = 2(m + 2). 116No one has yet been able to prove or disprove that such a number

n (that is with w(n) > 2

and such that n = Yoni p”) exists. For more on this matter, see the results of J.M. De Koninck & F. Luca [54].

110

JEAN-MARIE

ng n7 ng

= = =

no N19

= =

DE KONINCK

594839010 =2-3-5-17- 29-37-1087 = 2° + 5° + 29°, 275223438741 = 3-23-43 - 92761523 = 3” + 23” + 43”, 26584448 904822018 = 2-3-7-17-19- 113 - 912733109 SS a ee he 40373802 =2-3*-7-35603 = 2° + 3° + 7°, 420707 243066 850 = 2- 3? - 5? - 29 - 32238102917 =

910 Sf 510 a 9919.

e the fourth number which is equal to the sum of its digits added to the sum of the cubes of its digits: one can easily prove that the only numbers satisfying this property are 12, 30, 666, 870, 960 and 1998. 871 e the smallest number n which allows the sum

AS m 1, ng stands for the smallest number n which allows this sum to exceed k,

then the sequence (n,)x>1 begins as follows: 35, 871, 43217, 5296623, ...

872

e the 16" number n such that n!+ 1 is prime (see the number 116).

873

® the value of 14

2) -23lse

Ale Slee ol

877

e the seventh Bell number (see the number 52). 880 e the number of 4 x 4 magic squares (excluding rotations and reflections); if nz stands for the number of k x k magic squares, then nj = 1, no = 0, n3 = 1, n4 = 880 and ns = 275305 224: this last value was obtained by R. Schroeppel Ins LOVo:

881

e the largest known prime number p such that P(p?—1) = 11, where P(n) stands for the largest prime factor of n (see the number 4801).

‘THOSE FASCINATING

NUMBERS

111

882

e the 15% number n such that n- 2" — 1 is prime (see the number 115). 887 e the prime number which appears the most often as the 11'” prime factor of an integer (see the number 199);

e the smallest prime number p such that p+ 20 is prime and such that each

number between p and p + 20 is composite (see the number 139);

e the smallest prime factor of the Mersenne number 244% — 1, whose complete factorization is given by

2*43 _ 1 = 887 - 207818990653657 - P147; e the second number which does not produce a palindrome by the 196-algorithm

(see the number 196). 891

e the second odd number n > 1 such that y(n)|o(n) (see the number 135).

e the smallest number

n such that inequality log g(m) > /mlogm

holds for

all m > n: here g(m) = max (order of 7), where S,, stands for the group of TESm

permutations of m (J.P. Massias [131]). 907 e the smallest prime number which is preceded by 19 consecutive composite numbers; indeed, there is no prime number between 887 and 907. 911

e the first component of the fifth pair of prime numbers {p,q} such that

p?+=1

(modq*)

and

q?'=1

(mod p’);

here {p, gq} = {911, 318917} (see the number 2903). 915

e the smallest odd number which can be written as the sum of some powers of its prime factors: here 915 = 3-5-61 = 3°+5° +611.

Alay

117m 2005, J.M. De Koninck & F. Luca [54] studied the size of the set {n 2 andn= Pa is pp}, where each exponent ap can vary with the prime divisor p.

el

JEAN-MARIE

DE KONINCK

916

e the only composite number n < 10° such that o(n + 22) — a(n) = 22. 919

e the smallest number whose cube is the sum of two 3-powerful numbers:

indeed,

919° = 271° + 23 - 3° - 733 (see the number 776 151 559). 923

e the seventh Hamilton number: the sequence (hn)n>1 of Hamilton numbers can be generated in a recursive manner by setting

h,=1,

ho=2

- zie v—2

ee —j —

(n = 3);

k=0

this sequence begins as follows: 1, 2, 3, 5, 11, 47, 923, 409 619, 83 763 206 255.. .. 930

e the smallest number n such that gin) = =i the sequence of numbers satisn fying this equation begins as follows: 930, 1860, 2790, 3720, 4650, 5580, 7440, 8370, 9300, ... 933

e (probably) the largest number which cannot be written as the sum of two numbers whose index of composition is > 3/2: in other words, if 933 = a+ ),

then min(A(a), A(b)) < 3/2; if n, stands for the largest number n which cannot be written as n = a+ 6 with min(X(a), A(b)) > p, then we have the following table (all the values are based on numerical computations and are therefore only conjectured values)'!:

(en ea EN

Al 23 | 23 23, |

933

a[2351 |ee 8638 |62471

1109549 | ? ?

[935] (=5- 11-17) e the second Lucas-Carmichael number (see the number 399);

e the smallest square-free composite number n such that p|n => p+ 10|n + 10

(see the number 399).

118In a related matter, V. Blomer [22] established that the number of numbers < x which can be written as the sum of two powerful numbers

is > z/(log x)9-?53,

(thus in particular with an index of composition > 2)

THOSE

FASCINATING

NUMBERS

113

936

e the second number n such that (a(n) +7(n))/n is an integer: the only numbers n < 10° satisfying this property are 6, 936 and 1638.

937 e the third prime number q such that Ie ;P is a multiple of 100: here this sum

is equal to 67400 (see the number 563). 942

e the second solution of o2(n) = 72(n + 10) (see the number 120). 944 e the smallest number n such that n and n+ 1 each have five prime factors counting their multiplicity: 944 = 2+ -59 and 945 = 3°. 5-7 (see the number 135). 945

e the smallest odd abundant number: we have 0(945) = 0(3°-5-7) = 945 +975; the odd abundant

numbers!!9

< 104 are 945, 1575, 2205, 2835, 3465, 4095,

4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555 and 9765 (see also the number 5391 411 025); e the smallest of the eight existing primitive non deficient numbers: we say that a number n is non deficient if a(n) > 2n and we say that it is primitive non deficient if it is non deficient and if it is not a multiple of a smaller non deficient number (see Dickson [67], p.31): these eight numbers are 945, 1575, 2205,

7425, 78975, 131625, 342 225 and 570375. 946

e the fourth number n such that 2” = —2 (mod n) (Schinzel, see R.K. Guy [101], F10): the numbers n < 10° satisfying this property are 2, 6, 66, 946, 8 646, 180 246, 199 606, 265 826 and 383 846.

952 e the fourth number that is equal to the sum of the third power of its digits added

to the product of its digits: the only numbers satisfying this property'*° are 31, 370, 407 and 952.

119Qbserve that the first eight terms a, of this sequence

Oi

are given by a, =

945 + 630k, k =

ee

120One can argue, as in the footnote tied to the number 1324, that any such number can have at most five digits; using a computer, it is then easy to prove that 952 is the largest number with this property.

114

JEAN-MARIE

DE KONINCK

954 e the only three digit self replicating number: a number n all of whose digits are distinct and decreasing is said to be self replicating if the process of reversing its digits and subtracting this new number from the number n yields a number whose digits are the same as that of n: no numbers with 1, 2, 5, 6 or 7 digits satisfy this property; see M. Gardner [87]; the numbers 954, 7641, 98 754 210, 987 654 321 and 9876543 210 are the only ones satisfying this property.

957

e the third solution of o(n) = a(n +1) (see the number 14). 959

e the 13" number k such that k|(10**1 — 1) (see the number 303).

e the fourth number n divisible by a square > 1 and such that y(n+1)—y(n) = 1 (see the number 48); e the fifth number which is equal to the sum of its digits added to the sum of the cubes of its digits: the only numbers satisfying this property are 12, 30, 666, 870, 960 and 1998.

968

e the only solution n < 10! of o(n) = 2n + 59 (see the number 196); e the second powerful number which can be written as the sum of two co-prime

3-powerful numbers # 1: 968 = 343 + 625, that is 2? - 117 = 7° + 54 (see the number 841). 971 e the largest irregular prime smaller than 1000 (see the number 59). 974

e the 17" number n such that n! — 1 is prime (see the number 166). 975

e the tenth solution of ¢(n) = ¢(n + 1) (see the number 15).

THOSE

FASCINATING

NUMBERS

115

977 e the fourth Stern number (see the number 137). 983 e the smallest prime factor of the Mersenne number factorization is given by 9491 ae

||

Shs

249! — 1, whose complete

983 - 7707719 - 110097436327057 - 6976447052525718623 -19970905118623195851890562673 -3717542676439779473786876643915388439 14797326616665978116353515926860025681383;

it is the smallest Mersenne number with exactly seven prime factors (see the number 223 for the list of the smallest Mersenne numbers which require a given number of prime factors). 985

e the 13°" Markoff number (see the number 433). 987

e the smallest number n such that w(n) +w(n+1)+w(n+2)+w(n+3) = 12: here 987 = 3-7-47, 988 = 27-13-19, 989 = 23-43 and 990 = 2-3?-5-11; if we denote

by nx the smallest number n such that w(n)+w(n+1)+w(n+2)+w(n+3) =k, we have the following table:

13 803 18 444 134 043 282 489 1013 724

4 289 592 12582 633 57 495 513 260 628 717 801 621093

991 b

e the ninth prime number p, such that p;p2...pz — 1 is prime (see the number

317);

e the largest three digit number for which each permutation of its digits provides a prime number, namely in this case the prime numbers 991, 199 and 919; the only other known prime numbers satisfying this property are 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97 as well as all the primes of the form 11...1 (for these k

last primes, see the number 19).

—

16

JEAN-MARIE

DE KONINCK

997 e the largest three digit prixae number;

e the largest known value of p(k) whose last digit is 7 (see the number 389). 1001

e the 14*” number k such that k|(10**1 — 1) (see the number 303). 1008

e the smallest number n such that 7(n) = n/6 (see the number 330).

e the smallest four digit prime number;

e the smallest prime number which can be written in the following ten manners: x? + y?, 27 + 2y?, ..., 27 + 10y? (Problem stated by Gregory Wulcszyn and solved by A.M. Vaidya in Amer. Math. Monthly 75 (1968), p. 193); indeed, 1009

Se

15°98? = 197 49.187 = 91274 3-4? = 15 ed ek hed 2e O58 Gi Se eel oe ea Oe 9874.9). 57 = 37 +.10-107¢

e the smallest number which can be written as the sum of three distinct cubes

in two distinct ways: 1009 = 1° + 23 + 10° = 43 + 6° + 93; if we denote by nz the smallest number which can be written as the sum of three distinct cubes in k distinct ways, then

ng ng m

= = =

1009 BLO4) 13896

m5

=

161568

Me

=

1296378

n7

=

2016496

mg

=

2562624

=1?+2?+4 10° = 47 + 6? + 93, Sade ae? ai] oN 09 Sa? 1G? = 1% +12? + 233 = 23 + 43 + 248 =

(8° 0

Ome

Gaia

= 910

=2?+4 16° 454% — 9° + 153 4 54% = 17% 230° 4 463 =. 18° + 19° 53° = 26" 436° 4.462, =3° + 76° + 95° = 9° + 33° + 1087 D113 772 4949 = 319459? 102° = 33° + 81° + 90° = 603 + 75° + 872, =6?+4 72% + 118° = 10° + 66? + 1202 = 19° 219-4 1962 = 47° 97° 100 = 54° + 603 + 1183 = 663 + 90° + 100° = 83° 4+ 85° 494°, =8?+ 363 + 136° = 83 + 643 + 1323 = 123 + 1003 + 1163 = 173 + 463 + 1353 = 303 + 1033 + 113° = 363 + 603 + 1323 = 51> + 85° 122°) 69° 72 103°

THOSE

FASCINATING

NUMBERS

laure

no

=

14926249

= 2° + 342 + 946° = 12° + 186° + 2049 eee Sar 1G eg ee 114 87" = 72? + 90° 4: 2403 = 752? + 1902 41973 = 90? + 1863 + 1983 = 993 + 149% + 2203 106° 150° 2187.

mio

=

34012224

= 36% + 216° + 2883 = 39% + 1533 + 3123 = 41° + 1143 + 319% = 453 + 246% + 2679 = 723 + 1143 + 3183 = 100% + 192° + 2963 = 102? + 227° + 277% = 1189+ 162221652710 113

186° + 2963 2145+ 267%

1019 e the eighth prime number p, such that pyp2...p, +1 is prime (see the number 379).

1021 e the ninth prime number p, such that p)p2...pz +1 is prime (see the number 379).

1030

e the third number n such that a(n) = o(n + 5); the sequence of numbers satisfying this property begins as follows:

6, 46, 1030, 2673, 4738, 4785, 10437,

14025, 20038, 20326, 23914, 28702, ...

e the fifth number k such that ey 11... 1, is prime (H.C. Williams & H. Dubner [206]); k see the number

19.

e the smallest composite Phibonacci number; we say that a number n > 3 is a Phibonacci number if

o(n) = o(n — 1) + b(n — 2); the only composite Phibonacci numbers

< fO° rarest 037, 104) O27,

9 179,

55 387, 61133, 72581, 110177, 152651, 179297, 244967, 299651, 603461 and 619697; this notion was introduced by A. Bager [9] in 1981.

118

JEAN-MARIE

DE KONINCK

e the smallest prime number made up of four distinct digits; if q, stands for the smallest prime number made up of k distinct digits, then q1 = 2, g2 = 18, q3 = 103, g4 = 1039, gs = 10243, gg = 102359, g7 = 1023 467, gg = 10234589 and gg = 102345 689; it is clear that qio does not exist (see the number 9871 for the analogue question for the largest k digit prime number).

e the number of four digit prime numbers (see the number 21);

e the smallest irregular prime larger than 1000 (see the number 59).

e the smallest number n requiring eight iterations of the a;(n) function in order

to reach 1: indeed, 77(1069) = 1070, o7(1070) = 648, o7(648) = 121, o7(121) = 133, o7(133) = 160, 07 (160) = 6, o7(6) = 4 and o7(4) = 1 (see the number 193).

e the smallest number which can be written respectively as the sum of two, three

and four distinct cubes: 1.072 = 7° + 9° = 2° + 4° + 10° = 13 + 6° 4 7° + 8. the sequence of numbers satisfying this property begins as follows: 1072, 6 867, 6984, 8576, 9288, 9728, 10261, 10656, 10745, 10773, 10989, ...(see the

number 4802).

e the fourth solution of 02(n) = a2(n+2): the three smallest solutions are 24, 215 and 280; it is mentioned in the book of R.K. Guy [101], B13, that, according to P. Erdos, the above equation has only a finite number of solutions; nevertheless

one can prove (see J.M. De Koninck [45]) that if Hypothesis H is true (see its statement on page xvii), then the above equation has infinitely many solutions; there are 24 solutions < 10°, namely 24, 215, 280, 1079, 947519, 1362239, 2 230271, 14939999, 19720007, 32509 439, 45581759, 45841 247, 49 436 927, 78 436 511, 82 842911, 101014631, 166 828031, 225622 151, 225 757 799, 250 999 559, 377 129 087, 554998 751, 619 606 439 and 846 765 431; e the smallest number r which has the property that each number can be written

as 21° + 23° +... + 1°, where the z;’s are non negative integers (see the number 4).

THOSE

eal

e the smallest solution of ee

FASCINATING

NUMBERS

119

a)

(ee nae the only solutions n < 10° of this equation

are 1080, 6048, 6552, 435 708, 4713984 and 275 890 944.

e the only solution “MORE”

of the first known cryptogramme (or cryptarithm)

SEND +MORE

=

MONEY,

95674-1085

=

10652,

sent by the English ludologist Henri Ernest Dudiney (1857-1945) to its editor

(see Michel Criton [39]).

e the smallest number > 9 which is a proper divisor of the number obtained by reversing its digits: the numbers satisfying this property are 1089, 10989, 109 989, 1099 989, ... while their doubles are 2 178, 21 978, 219978, 2199978,...

e the seventh prime number p such that (3? — 1)/2 is itself a prime number; the prime numbers p < 10000 such that (3? — 1)/2 is also prime are 3, 7, 13, 71, 103A LOO M361 wl O2104177,.90l lLand 95oL,

e thesmallest!*! Wieferich prime: a prime number p is called a Wieferich prime if it satisfies the congruence 2?-' = 1

122

(mod p?); the only other known Wieferich

prime is 3511; R. Crandall, K. Dilcher & C.Pomerance [37] proved that there are no other Wieferich primes < 4-101, while computations done by volunteers through the Internet have established that there are no other Wieferich primes ody

ison hdgee

1211¢ is of some interest to display the complete factorization of 2199? — 1, a 329 digit number:

91082

= «34-577: 137 29: 43-53. 79-113 - 127- 157-313-337. 547-911 - 10937 1249 - 1429 - 1613 - 2731 - 3121 - 4733 - 5419 - 8191 - 14449 -21841 - 121369 - 224771 - 503413 - 1210483 - 1948129 - 22366891 -108749551 - 112901153 - 23140471537 - 25829691707 -105310750819

- 467811806281 - 4093204977277417 - 8861085190774909

-556338525912325157 - 275700717951546566946854497 -86977595801949844993 - 292653113147157205779127526827 -3194753987813988499397428643895659569. 122Tn 1909, Wieferich proved that if the first case of Fermat’s Last Theorem is false for a certain

prime number p (that is if 2? +y? = z?, where p does not divide ryz), then p satisfies the congruence 2P-1=1 (mod p?). However, Meissner was the first (in 1913) to observe that the prime number p = 1093 does indeed satisfy this property.

120

JEAN-MARIE

DE KONINCK

e the smallest number n such that ¢!°(n) = 2, where ¢/°(n) stands for the tenth iteration of the ¢ function (see the number 137).

e the ninth Keith number (see the number 197). 1105

(= 5 13-17)

e the second Carmichael number (see the number 561); e the smallest number which can be written as the sum of two squares in four distinct ways, namely 1105 = 4? + 33? = 92+ 32? = 12? + 31? = 23? + 24? (see

the number 50); e the value of the sum of the elements of a diagonal, of a line or of a column in

a 13 x 13 magic square (see the number 15).

e the 15‘" number k such that k|(10*++ — 1) (see the number 303).

e the smallest prime number that is preceded by 21 consecutive composite numbers; indeed, there are no prime numbers between 1129 and 1151.

(= 34?) e the smallest number which can be written as the sum of k perfect squares for each positive integer k < 1000 (Sierpinski [185], p.410); an example of a

representation of 34? as the sum of 1000 squares is 342 = 2-82 +2-42+4996-12.

e the largest number which cannot be written as the sum of five composite num-

bers (R.K. Guy [101], C20).

e the number which, when paired with the number 1 210, forms an amicable pair; this pair was discovered by Paganini when he was only 16 years old (see the

number 220).

THOSE

FASCINATING

NUMBERS

iDAL

1187

e the fifth Stern number (see the number 137).

1189 e the fourth solution w of the aligned houses problem (see the number 35).

1197 e the 1000

composite number (see the number 133)%

1201

e the fourth prime number of the form (x* + y*)/2: here 1201 = (14+7+)/2 (see the number 41).

1 210 e the number which, when paired with the number 1 184, forms an amicable pair

(see the number 1 184).

e the smallest number n such that n and n+1 are both divisible by a fifth power: 1215 = 3°-5, 1216 = 26 - 19; if n;, stands for the smallest number n such that n and n+ 1 are both divisible by a k*” power, then nz = 8, n3 = n4 = 80, ns = 1215, ng = 16 767, nz = 76544, ng = 636416, ng = 3 995 648, nyo = 24151 040 and nj, = 36315135 (for the analogue problem concerning three consecutive

integers, see the number 1 375).

IOGear)

p(n) =_ —; 8 the sequence of numbers satisfying this e the smallest solution of ——equation begins as follows:

n

1218, 2436, 3654, 4872, 7308, 8526, 9744, ...; it is

easy to establish that this sequence is infinite’*°.

123 Indeed, this follows from the fact that one can easily prove that n is a solution of this equation if and only if n = 2° - 38 .77 . 29° for certain positive integers a, 3, ¥, 6.

22

JEAN-MARIE

DE KONINCK

e the number of twin prime pairs < 10°; if 72(x) stands for the number of prime numbers p with p+ 2 prime, we have the following table:

gq

| waln)

108 | 440312

109 10! 101! 10!2

| 3424506 | 27412679 | 224376048 | 1870585 220

1013 | 15 834664 872 104 | 135 780 321 665

1225

e the third triangular number which is also a perfect square: here 1225 = °° 2

=

35” (see the number 36).

1229 e the number of prime numbers < 10000. 1 230

e the smallest number n such that the Liouville function 9 takes successively, starting with n, the values 1, —1, 1, —1, 1, —1, 1, —1, 1, —1 (see the number 6185). 1234

e the number of digits in the decimal expansion of the Fermat number gee ls 1253

e the third number n such that o(n + 1) = 20(n); the sequence of numbers satisfying this property begins as follows:

5, 125, 1253, 1673, 3127, 5191, 7615,

12035, 43817, 47795, 48559, 49955, 56975, 58373, 61721, 63545, 68033, 78395, 97411, ...; ifn, stands for the smallest number n such that o(n +1) = ka(n), then n1 = 206, no = 1253, ng = 1919 and ng = 37033919; as for the value of ns, one may at least say that ns < 14182439039, since 0(14182439040) = 50 (14182439039). 1257

e the fourth solution of o(¢(n)) = a(n) (see the number 87).

THOSE

FASCINATING

NUMBERS

IR

e the smallest number n such that ¢(n + 1) = 4¢(n); the sequence of numbers satisfying this property begins as follows: 1260, 13650, 17556, 18720, 24510, 42120, 113610, 244530, 266070, 712080, ... (see also the number 11 242770); if

nz stands for the smallest number n such that ¢(n + 1) = kd(n), then ny = 1, ng = 2, n3 = 6, nq = 1260 and ns = 11242770 (for the analogue problem with

equation ¢(n) = kf(n + 1), see the number 629); e the smallest vampire number: a number n with 2r digits, r > 2, is called a vampire number if it can be written as the product of two numbers a and b (each having r digits) and such that the set of numbers formed by joining the digits of a and b is the same set as that of the digits of n (in this case, 1260 = 21-60); there exist seven four digit vampire numbers, namely 1260, 1395, 1435, 1530, 1827, 2187 and 6880, while there are 155 six digit vampire numbers and 3382 eight digit vampire numbers (see E.W. Weisstein [201], p. 1894);

e the 16” highly composite number (see the number 180). 1 263 e the largest number of the form 8k + 7 which can be written as the sum of exactly three powerful numbers in exactly two distinct ways: here 1263 =

7? +53 + 33-11? = 22.7? + 23.7? + 3. 5? (see the number 118). 1270 e the rank of the prime number which appears the most often as the 16” prime factor of an integer: pi270 = 10343 (see the number 199).

1279

e the exponent of the 15” Mersenne prime 2'?79 — 1 (Robinson, 1952). 1 290

e the second solution of o2(n) = o2(n + 12) (see the number 864). 1291

e the largest solution n of n? —n+3 = 3m?: the only solutions (n,m) of n? —n+

3 = 3m? are (n,m) = (—1, 1), (1,1), (3,3), (11,21), (13,27) and (171,1291).124

124The interest for equation n? — n +3 = 3m? comes from the quest for the solutions (r,s) of

equation 1+2+...+r=12+2?+4+...+5?, that is of equation 3(r? +r) = s(s+1)(2s+1), which becomes n? —n +3 = 3m? after setting m = 2r+ 1 and n = 2s+1. R. Finkelstein & H. London

[81] proved that the only integer solutions of equation n° —n+3 = 3m? are those mentioned above,

which yields the solutions (r, s) = (1,1), (10,5), (13,6) and (645,85) of 3(r? +r) = s(s +. 1)(2s + 1).

124

JEAN-MARIE

DE KONINCK

e the smallest solution of r(n +1) —7(n) = 17: the six smallest solutions? 255 are 1295, 6399, 25599, 117649, 123903 and 173055 (see the number 399).

1297

e the largest known prime of the form 6” + 1 (here with n = 4); e the fourth prime number of the form n*++1: the smallest ten prime numbers of this form are 2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337 and 4477457; no one knows how to prove that there exist infinitely many prime

numbers of this form (see M. Lal [121])!*°, or even of the form m? + 1.

e the sixth number n > 9 such that n = )>’_,712? d', where dj,...,d, stand for the

digits of n: here 1306 = 11 + 3? + 0° + 6% (see the number 175).

e the smallest number n which allows the sum

1 Se - to exceed 3; the sequence m 2 which is equal to the sum of the factorials of its digits in base 15: here 1441 = (6,6, 1]15 = 6! + 6! + 1!; the only numbers with

this property are 1, 2,

1441 and 1442 (see the number 145).

e the largest number which is equal to the sum of the factorials of its digits in

base 15: here 1442 = [6, 6, 2]15 = 6! + 6! + 2! (see the number 1 441).

THOSE

FASCINATING

NUMBERS

129

(= 38”) e the smallest four digit perfect square which has only two distinct digits: Hitotumatu conjectured that, besides the numbers of the form 102", 4-10?” and 9-10”, there is only a finite number of perfect squares with only two distinct digits (R.K. Guy [101], F24);

e the second number n > 1 such that o(n) and o2(n) have the same prime factors, namely the primes 3, 7 and 127 (see the number 180); e the smallest perfect square whose last three digits are 444: the following!?° one is 213 444.

e the fourth Apéry number: fined by

~

n\

the sequence of Apéry numbers ao, a1, a2,... is de-

(nt+tk\\*

—

((n+k)!)?

aon ((::)Ce )) LELETCAE

Me ee

and begins as follows: 1, 5, 73, 1445, 33001, 819005, 21460825, 584307 365, 16 367 912 425, 468 690 849 005, 13 657 436 403 073, ...

e the third number n such that ¢(n), (n+ 1) and ¢(n +2) have the same prime factors, namely here 2 and 3: the sequence of numbers satisfying this property begins as follows: 35, 36, 1458, 3456, 16921, ...(see the numbers 266 401 and S1t)s

e the sixth number m such that >7,, 1 such that y(n + 18) =

4(n) + 18 (and there are no others < 10°): here we have y(n + 18) — y(n) = S103); 13 S18: e the third odd number n > 1 such that y(n)|o(n) (see the number 135). 1536

e the tenth Granville number (see the number 126).

1537

e the tenth Keith number (see the number 197). 1538

e the number of Niven numbers < 10000 (see the number 213).

1540

e one of the five numbers (the others are 1, 10, 120 and 7140) which are both triangular and tetrahedral (see the number 10).

132

JEAN-MARIE

DE KONINCK

e the number of Carmichael numbers < 10!° (see the number 646).

e the number of integer zeros of the function M(z) := SS y(n) located in the n 5G woe O j i

this last expression

representing the first three terms of the asymptotic expansion of Li(n): here

(1627) = 258 while [p= + pla + 235| anor © 257-832 (see the number 73);

a

e the ninth prime number p such that (3? — 1)/2 is itself a prime number (see the number 1091).

e the second number which can be written as the sum of the fourth powers of its

digits: 1634 = 14 + 64 + 34 + 4?; the others are 1, 8208 and 9474; e the sixth solution of a(n) = o(n+ 1) (see the number 206).

e the fourth number which is neither perfect nor multi-perfect but whose har-

monic mean is an integer (see the number 140); e the fourth solution of i

a s (see the number 1 488).

e the 1000 square-free number (see the number 165).

131The interest for this number

stems from a number

theory result according to which the k**

k

prime factor g,(n) of a number n is “usually” of the order of e© , in the sense that, for all « > 0 and any function €(n) which tends to +00 as n — oo,

sup

€(n) 1 and such that y(n+14) = y(n)+ 14: the smallest is 49 (and there are no others < 108); here y(n + 14) — y(n) =

(1715) — y(1701) = (73 -5) — 7(3°- 7) = 35 — 21 = 14.

THOSE

FASCINATING

NUMBERS

135

e the smallest composite number of the form 2n? + 29 (with n = 29).

e the smallest solution of o2(n) = a2(n + 16).

[i723] e the third Giuga number (see the number 30).

[i728 e the fourth number which is equal to the product of the factorials of its digits in base 5: 1728 = [2,3, 4,0, 3]5 = 2!-3!-4!-0!-3! (see the number 144).

[i729] (=7-13-19) e the smallest number which can be written as the sum of two cubes in two distinct ways: 1729 = 12°+1% = 10°+ 93 (an observation due to Bernard Frénicle de Bessy, 1657); a number which can be written as the sum of two cubes in two distinct ways is sometimes called a Ramanujan number!*; the sequence of numbers satisfying this property begins as follows: 1729, 4104, 13832, 20683,

32 832, 39312, 40033, 46683, 64232, 65728, 110656, 110 808, 134379, ...; if nz stands for the smallest number which can be written as the sum of two cubes in k distinct ways, then ng = 1729, n3 = 87539319, ng = 6963 472 309 248,

Ns = 48 988 659 276 962 496, while ng < 8 230 545 258 248 091 551 205 888; as for the sums of squares, see the number 50; for sums of fourth powers, see the number 635 318 657;

e the third Carmichael number (see the number 561); e the smallest pseudoprime in bases 2, 3 and 5.

[rea] (=27-37-72) e the smallest powerful number equidistant from the preceding (1728 = 2° - 3°) and following (1800 = 23 - 3? - 5%) powerful numbers; the sequence of numbers satisfying this property begins as follows: 1764, 7056, 729316, 1458 632, 2917 264, 11669056, 149 022 848 000, 260 102 223 752, 348 796 548 100, 697 593 096 200, 1040 408 895 008, 1 206 917 268 552, 1395 186 192 400, 2.413 834537 104, 4827 669074 208, 10 862 255 416 968, ... 1°8

132The reason for this name

is that one day, as G.H. Hardy visited Srinivasa Ramanujan

at the

hospital, the young Indian mathematician told his visitor that the number on the taxicab he had just stepped down from was highly interesting since it was the smallest number expressible as the sum of two cubes in two distinct ways: the taxicab number was 1729. This explains why Ramanujan numbers are also called taxicab numbers.

1331¢ would be interesting if one could prove that this sequence is infinite.

136

JEAN-MARIE

DE KONINCK

[1.782] (= 2-34-11) e the smallest number n > 1 such that o(n) = y(n)*, and the only one smaller than 10°: it is easy to see that such a number n > 1 must be even!*4 and cannot be square-free (compare with the number 96, as well as with the number 108); e the fourth positive solution x of the diophantine equation x? + 999 = y® (see

the number 251).

e the 11°” Granville number (see the number 126).

e the fifth solution of a(¢(n)) = a(n) (see the number 87).

e the fifth voracious number: consider the sequence (b,)~>1 defined by

Dy) =eok

bi asleep eas Or

Op Once

wa

leas

each term of this sequence is called a voracious number; thus the first voracious numbers are 2, 3, 7, 43, 1807, 3 263 443, 10 650 056 950 807, 113 423 713 055 421 844 361 000 443,

12 864 938 683 278 671 740 537 145 998 360 961 546 653 259 485 195 807, ...; it is interesting to mention that es 1/b; = 1; for more on this subject, see the recent papers of G. Myerson & J.W. Sander [146] and of J.W. Sander [179].

e the smallest number n such that >> m 1 is odd, then y(n)? = a(n) is also odd, so that n = m? for a certain m, in which case n < a(n) = y(n)? = y(m?)? = 7(m)? < m?, a contradiction. 135QOn the other hand, it is known that if one could find another one, there would be no others.

THOSE

FASCINATING

NUMBERS

iByE

e the smallest solution of (n+ 1)—7(n) = 9; the sequence of numbers satisfying this equation begins as follows: 1849, 11449, 23103, 28899, 38415, 63001, 66049,

195363, ... (see the number 399).

e the smallest number n such that P(n)

> P(n+1)

>...

> P(n+5):

here

617 > 463 > 109 > 103 > 53 > 29; if we denote by nz the smallest number n

such that P(n) > P(n+1) >... > P(n+k-—1), then we have the following table:

Fale o leon

aaa

0

6

c

8

[ng | [13 [13 [407 [851 |aero |12721 | jek 9 10 11 12 13 nx || 109453 | 586951 | 120797465 | 624141002 | 4044619541 (compare with the table displayed with the number 46189, this time for the

sequence of increasing P(n + 7%)).

(= 17- 109) e the number n which allows the sum

De

1

3 to exceed 3 (see the number 44).

m> m din, d 0, the set {n : max(P(n), P(n+

1)) < n*} is of positive density, implying in particular that the set of numbers n such that P(n) < Yn and P(n+1) < Yn-+1 is of positive density.

THOSE

FASCINATING

NUMBERS

147

e the eighth number n > 9 such that n = Dol d‘, where d;,...,d, stand for the

digits of n: here 2427 = 2! + 4? + 23 + 7+ (see the number 175).

e the third number

n > 2 such that eZ(n)

+ d(n) 5 7(n)

588).

is an integer (see the number

[2465] (= 5-17-29) e the fourth Carmichael number (see the number 561); e the value of the sum of the elements of a diagonal, of a line or of a column in a 17 x 17 magic square (see the number 15).

e the number of digits in the decimal expansion of the Fermat number Vem aA

e the second number n such that }> 1 such that ¢(n)o(n) is a fifth power (see the numbers 51 and 170).

e the fifth prime number of the form (x* + y*)/2 (see the number 41): here

3593 = (54 + 94) /2.

1451¢ is interesting to display the almost complete factorization of 23519 — 1, a 1057 digit number:

pee

tee

8-2 7d 19r Bln 73979. 131 15) e227) - 33) bal oll 2007 = tht ie o7al -35112 - 6553 - 8191 - 10531 - 15121 - 23311 - 65521 - 86113 - 87211 - 107251 -121369 - 262657 - 348031 - 409891 - 446473 - 1024921 - 1969111 - 4633201 -7623851 - 18837001 - 22366891 - 29121769 - 409251961 - 2400314671 -7830118297 - 26959262851 - 21497866557571 - 49971617830801 -145295143558111 - 385838642647891 - 571890896913727 - 93715008807883087 -194900834792501371 - 339175003117573351 - 4242734772486358591 -85488365519409100951 - 150832426800173710177 - 1439538040790707946401 -571403921126076957182161 - 5302306226370307681801 -255375215316698521591 - 4247713303224552237738169 -43725552532343303477113703251 - 134304196845099262572814573351 -27283345360345928653392998057 12535332071 -4897406518564079146139572699835240681611 -24841125429051585062538961751269988364169 - C1g3 - C209,

where C1g3 and C2099 stand for composite numbers made up respectively of 183 and 209 digits.

‘THOSE FASCINATING

NUMBERS

157

e the number of Carmichael numbers < 1011 (see the number 646).

e the 20" number n such that n! — 1 is prime (see the number 166).

e the smallest number n for which the Moebius function pz takes successively, starting with n, the values 1,0,1,0,1,0,1,0; if we denote by nz the smallest number n for which the Moebius function yz takes successively, starting with nm, the values 1,0,1,0,...,1 .0, then we have no = 15, n3 = 55, ng = 159, eee

PG

erent i

ipee oar ingen

OR

PE 15 245. nye 113.343,

13 = M14 = N15 = 1133759 and nig = 29149139: one can easily prove that there are no!*6 numbers nz with k > 17 (for the sequence —1,0,—1,0,..., see

the number 2749).

e the second number n such that n,

n+1, n+2 and n+3

have the same number

of divisors, namely eight (see the number 242).

3675)

(= 3)5427°)

e the smallest odd number n having exactly three prime factors and such that

y(n)| O(n).

e the 13" Keith number (see the number 197).

e the 15” solution of d(n) = 6(n + 1) (see the number 15).

e the seventh number such that 2” + n? is prime (see the number 2007).

146This simply follows from the fact that any sequence of 17 consecutive numbers, the first of which is square-free, must include two numbers r and s both divisible by 9 and therefore such that

p(r) = p(s) = 0 with r — s odd.

158

JEAN-MARIE

DE KONINCK

e the smallest number n such that >> m Ty = 7 < 8.13... = W4375 (see the number 2400). 4418

e the smallest solution of a(n + 13) = o(n) + 13.

e the exponent of the 20¢” Mersenne prime 24473 — 1 (Hurwitz, 1961).

e the fourth odd number k such that 2” + k is composite for all n < k (see the number 773): in fact, 2" + 4471 is composite for all n < 33548 and prime for n = 33 548.

62

JEAN-MARIE

_—

DE KONINCK

4481

e the sixth prime number of the form (x* + y*)/2: here 4481 = (74 + 9*)/2 (see the number 41).

3

4 503 e the third number n such that n,

n+1,n+2 and n+3

have the same number

of divisors, in this case eight (see the number 242). 4 547

e the 12°” prime number pz such that pip2...px +1 is prime (see the number 379). 4550

1 e the smallest number n which allows the sum yr ; to exceed 9 (see the number i 1 such that

Tee

get 2 ee he

for a certain number k: here k = 24 (see the number 70).

4913 | (= 17°) e the second number n whose sum of digits is equal to ~/n (see the number 512). 4933 e the number of digits in the decimal expansion of the Fermat number 2

41.

4991

e the fifth Lucas-Carmichael number (see the number 399). 5 002

e the sixth solution of o(¢(n)) = a(n) (see the number 87). 5 040

e the value of 7!;

e the 19’" highly composite number (see the number 180). 5041 e the smallest number n such that, if the Riemann Hypothesis (according to which all complex zeros of the Riemann Zeta function have their real part equal to +) is true, inequality o(m)/m < e7 loglogm (where ¥ is Euler’s constant) holds

for all m > n (G. Robin [176])148; e the fifth powerful number which can be written as the sum of two co-prime

3-powerful numbers # 1: 5041 = 128 + 4913, that is 712 = 27 + 173 (see the number 841).

148Tn that paper, Robin proves that the Riemann Hypothesis is equivalent to the fact that inequality a(n) < eYnloglogn is true for all n > 5041. :

THOSE

FASCINATING

NUMBERS

165

[5 044] (= 2? 13-97) e the smallest number n such that n,n+2,n+4,...,n+10

are all divisible by

a square > 1.

e the fifth odd number k such that 2” + k is composite for all n < k (see the number 773): in fact, 2” +5101 is composite for each n < 5759 and prime for a) (OU:

e the smallest number which can be written as the sum of three distinct cubes

in three distinct ways:

5104 = 1° + 12° + 15° = 2° + 10? + 16° = 9° + 10° + 15° (see the number 1 009).

e the largest number k such that inequality 0(p,) > k(log k+log log k —1) (where O(x) = >) p< logp) is false (G. Robin [175)).

[520](=2-5) e the smallest number n having at least two distinct prime factors and such that

B(n)3|Bi(n): here (2 + 5)3|(21° + 5); the only numbers < 10° satisfying this property are 5120, 419904, 885 735, 5315625 and 18003 384.

e the smallest number n such that ¢(n) = 7! (see the number 779).

e the only number n < 5-109 such that ¢(n) = ¢(n + 1) = ¢(n + 2): here the common value is 2592 = 2° - 34 (see R.K. Guy [101], B36)1*9;

e the 16%” solution of ¢(n) = 6(n +1) (see the number 15).

149Observe that 5186 = 2- 2593, 5187 = 3-7-13-19, 5188 = 27-1297 and that $(2593) = 2592 = 25 . 34 and ¢(1297) = 1296 = 24 - 3+, so that each odd prime number which shows up as a factor of the numbers n,

n+ 1 and n+ 2 is of the form 2° - 3° +1. One could ask the same question about

equations o(n) = o(n + 1) = o(n + 2); note that if there is a solution, it is larger than 5 - 109.

166

JEAN-MARIE

DE KONINCK

e the 17%” solution of 6(n) = ¢(n + 1) (see the number 15).

e the largest solution y of the diophantine equation 2? — 17 = y°?: T. Nagel [147] proved that this diophantine equation has exactly 16 solutions (2,y),

namely the solutions (+3, —2), (+4, —1), (+5, 2), (£9, 4), (+23, 8), (+282, 43), (+375, 52), (+378661, 5234) (see also Sierpinski [185], p.104). 5 264 e the smallest number n such that n and n+1 each have six prime factors counting their multiplicity: 5264 = 24-7-47 and 5 265 = 34-5-13 (see the number 135).

e the 16" number n such that n- 2” — 1 is prime (see the number 115);

e the second number n such that Eg(n) := ¢(n +1) — ¢(n) satisfies Eg(n+1) = E4(n): here the common value of Eg is 16, since (5312) = 2624, (5313) = 2640 and ¢$(5314) = 2656; the sequence of numbers satisfying this property begins as follows: 5186, 5312, 273524, ...

e the sixth composite number n such that o(n + 6) = o(n) + 6 (see the number 104).

[5346] (=2-3°- 11) e the fourth number n having at least two distinct prime factors and such that

By(n) = B(n)?: here 2+ 3° + 11 = (2+3-+11)? (see the number 144).

e the eighth odd abundant number (see the number 945).

e the number of integer zeros of the function M(z) := Ss u(n) located in the

interval [1, 10°] (see the number 92).

Sen

THOSE

FASCINATING

NUMBERS

167

e the third solution of 2(n) = o2(n + 10) (see the number 120).

e the sixth prime number q such that Ere p is a multiple of 100: here this sum

is equal to 1776 800 (see the number 563).

e the second dihedral 3-perfect number n, that is such that T(n) +o(n) = 3n: the only three numbers n < 10° satisfying this property are 60, 5472 and 2500 704.

(5525 (5? 13-17) e the smallest number which can be written as the sum of two squares in five

distinct ways (as well as six distinct ways), namely 5525 = 72 + 742 = 142 +

73? = 22? + 71% = 25? + 70? = 41? + 60? = 50? + 55? (see the number 50).

e the largest known!°? prime number p for which there exists an even number n (namely here n = 389 965 026 819 938) such that n — q is composite for each

prime number q < p (see J.Richstein [171]): here we have the “Goldbach representation” 389 965 026 819 938 = 5 569 + 389 965 026 814 369.

e the number of pseudoprimes in base 2 smaller than 10° (see the number 245).

e the smallest number n such that 6;(n) = 67(n+1) = B7(n+2), where G7(n) :=

Dpin,p>2 P: here 5694 = 2-3-13-73, 5695 = 5-17-67 and 5696 = 2°. 89, while the common value of 3;(n +7) is 89; the second number with this property is

2463 803 977 (see also the number 89 460 294).

e the sixth Lucas-Carmichael number(see the number 399).

150T¢ is possible to prove that there exist infinitely many prime numbers p satisfying this property.

JEAN-MARIE

168

DE KONINCK

e the fifth odd number n > 1 such that y(n)|o(n) (see the number 135). 5775 e the smallest abundant number n such that n+1 is also abundant; the sequence of numbers satisfying this property begins as follows: 5775, 5984, 7424, 11024, 21735, 21944, 26144, 27404, 39375, 43064, ...

5 776 e the sixth powerful number which can be written as the sum of two co-prime 3-powerful numbers 4 1: 5776 = 2401 + 3375, that is 24-19? = 12 8. 5?

4

(see the number 841).

5777

e the smallest composite Stern number (see the number 137). 5778

e the only triangular number > 3 which is also a Lucas number (L. Ming [137]). 5 795

e the third number n > 1 such that n-2"+ 1 is prime (see the number 141). 5 828

e the smallest number n such that P(n +7) < Vn+i for i = 0,1,2,3,4; the largest prime factors of these five numbers are respectively 47, 67, 53, 17 and 3, all smaller than 5828 ~ 76 (see the number 1518).

e the fourth bizarre number (see the number 70).

(=18°) e the third number n whose sum of digits is equal to {/n (see the number 512); e the smallest cube which can be written as the sum of three cubes in two distinct

ways: 5832 = 183 = 2°+123+16% = 93+123 415%: if nz stands for the smallest cube which can be written as the sum of three cubes in k distinct ways, we have my = 216 = 6°, no = 5832 = 183, ng = 157464 = 543, ng = 658503 = Sie. ms = 1259712 = 1083, ng =! n7 = 5268 024 = 1748, mg ='ng = nay = SA 022240

324",

952 763 904 = 9843.

ni

=

nj2

=

n3

=

119095488

=

4923,

44

=

25

=

THOSE

FASCINATING

NUMBERS

169

e the 30° Lucas prime number (see the number 613).

e the smallest number n such that the Liouville function 9 takes successively, starting with n, nine times in a row the value —1; if n, stands for the smallest

number n such that Ao(n + 7) = —1 for i = 0,1,2,...,k — 1, then we have the following table:

Em 1 ey, 3 4 5 6 G 8 9 10

2 lay eb Itsy

5879 || Waster“ |) tearey7fil | eye |) LCA Oe IAL TAO) 1004 646 1 004 646 1633 357 5 460 156

2 22 Was 24 25 26 27 28 29 30

O02 io0 | 21627159 || PAO SY, | 38821 328 | 41983 357 | 179376463 | 179376463 | 179376463 | 179376463 | 179376463

(see the number 1 934 for the list of the smallest numbers nx such that Ao(n~% +

Dl tore 082.

ek 91).

e the smallest number which can be written as the sum of the fourth powers of

two rational numbers (254 + 1494 = 5906-174), but not the sum of the fourth powers of two integers (A. Bremner & P. Morton [24]).

e the value of 1!+2!4+...+7!.

e the fourth number n such that n,

n+1,n+2 and n+3

of divisors, namely eight (see the number 242).

have the same number

170

JEAN-MARIE

DE KONINCK

e the smallest composite Wilson number; setting S(n) :=

I] 7, one can gen41

(i,n)=1

(mod n) if n = 2,4,p” eralize Wilson’s Theorem by showing that S(n) =—1 or 2p’, where p is an odd prime number, and S(n) = 1 (mod n) otherwise; Wilson numbers are therefore defined as the numbers n for which S(n) = +1 (mod n); the sequence of Wilson numbers begins as follows: 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326 713, 23025 711, 26921605, 341569 806,

399 292 158 (see T. Agoh, K. Dilcher & L. Skula [2]). 5 985

e the tenth odd abundant number (see the number 945). 5 993

e the largest known Stern number (see the number 137). 6 044

e the second number n such that o(n), o(n +1), o(n+ 2) and o(n +3) have the same prime factors, namely here 2, 3 and 7 (see the number 3777).

6 048 | (= 2° - 33 - 7) 1

e the second solution of a = < (see the number 1080).

6079

e the largest known prime number which divides a Mersenne number (here 21013 — 1) and is such that the prime number that follows, namely 6 089, is also a factor of a Mersenne number (here 27°! — 1); see the number 1 433. 6 099

e the only four digit Sastry number (see the number 183); if n, stands for the smallest r digit Sastry number, then n3 = 183, ng = 6099, n5 = 13224, ne = 106 755, ny = 2066115 and ng = 22 145 328.

e the 12" Granville number (see the number 126).

THOSE

FASCINATING

NUMBERS

ig

[e174] (=2-3?- 79) e the value of the Kaprekar constant for the four digit numbers (see the number 495);

e the fifth solution of ¢(n) = y(n)? (see the number 108).

e the smallest number n such that the Liouville function 9 takes successively, starting with n, the values 1, —1,1, —1,1,—1,1, —1,1, —1,1, —1; if ng stands for

the smallest number n such that Ag(n +7) = (—1)’, for i = 0,1,2,...,k —1, theienje—

_anse—

nio

=

1 230, ny

nig

=

hy9

n25

=

19 370 102,

=

N20

ween.

=n2

=

=

N21

ne

=

—

49.75

6 185, n43 =

N22

=

39 623 Ae

=

="

lg

=

4 9784, ny4

99 826,

n27

=

n23

=

Ty, = =

56, Ne

79 800, N15

7815614,

46 025 769, n28

=

= =

Ng N16

— =

342, N17

=

n24

=

11 435 684,

Nag

=

544 865 099

and n39 = 1075 790 572.

e the sixth number which is not perfect or multi-perfect but whose harmonic mean is an integer (see the number 140).

e the largest known number n such that n!4+2"—1 Oe, Oe tld, 167 and-2 609:

is prime; the others are n = 1,

e the first of the seven smallest consecutive numbers at which the Q(n) function takes distinct values, namely here the values 7, 2, 6, 4, 3, 1 and 5 (see the

number 726).

e the smallest prime factor of the Mersenne number 2°? — 1, whose complete factorization is given by

2°3 _ 1 = 6361 - 69 431 - 20394 401.

e the 18* number n such that n! +1 is prime (see the number 116); note that 6380! + 1 is a 21507 digit number.

=

QP

JEAN-MARIE

DE KONINCK

6 397 e the prime number which appears the most often as the 15” prime factor of an integer (see the number 199). 6 399

e the second solution of r(n + 1) — r(n) = 17 (see the number 1 295). 6 435

e the 11’ odd abundant number (see the number 945). 6451

e the =a third prime number p such that 31?-! = 1

(mod p?) (see the number 79).

6 474

e the third solution of o2(n) = o2(n + 12) (see the number 864). 6 489

ik e the smallest number n which allows the sum SS Hen) to exceed 17 (see the m 9 (besides 89) such that n = d, +d3+d$+d§+...+ r—1

d?, where d;, dz,...,d, stand for the digits of n: here 6603 = 6+62+04+38.

6611

e the fourth number n > 1 such that n-2"-+ 1 is prime (see the number 141). 6 694

e the third number n such that 5° P p 1 such that ¢(o(n)) =n (see the number 128). 6917

e 217* number n such that n! — 1 is prime (see the number 166).

Sue=5

6 930

gl 2:

e the fifth solution w of the aligned houses problem (see the number 35).

H

6977 e the seventh prime number

number 563).

q such that 5°)PS -, p is a multiple of 100 (see the

7 022

e the largest number n for which inequality p, < n(log n+log log n—0.9385) does not hold: here p7922 = 70919 > 7022(log 7022 + log log 7022 —0.9385) = 70918.6

(see G. Robin [175]). 7055

e the seventh Lucas-Carmichael number (see the number 399). 7140

e one of the five numbers (the others are 1, 10, 120 and 1540) which are both triangular and tetrahedral (see the number 10).

7192 e the fifth bizarre number (see the number 70). 7225 e the smallest perfect square m7 for which there exist numbers mj, m2,mg such that m? — (m; — 1)? = m?_, for i = 2,3,4: here 7225 = 85? = 842 + 132 —

84? + 12? + 5? = 84? + 12? + 4? + 3? (see the number 169).

THOSE

FASCINATING

NUMBERS

MD

e the smallest number n such that w(n) = w(n+1) = 4: here 7314 = 2-3-23-53 and 7315 = 5-7-11-19 (see the number 230).

e the seventh prime number of the form («* + y*)/2 (see the number 41): here

7321 = (14 + 114)/2.

e the 15" Keith number (see the number 197).

e the fourth of the existing eight primitive non deficient numbers (see the number 945).

e the fourth number which does not produce a palindrome by the 196-algorithm

(see the number 196).

e the smallest prime factor of the Mersenne number factorization is given by

22 C=

2!97 — 1, whose complete

(487, -268288039979128869297108670418919894904868938457 12448833.

e the 1000’” prime power, namely here a prime number (see the number 419).

e the largest known prime number p such that p—k? is composite for each number k < \/p; the other known prime numbers satisfying this property are 2, 5, 13, 31, 37, 61, 127, 379, 439, 571, 829, 991, 1549 and 3319 (see also the number 367).

—

76

JEAN-MARIE

DE KONINCK

7 560

e the smallest highly composite number

i i

which is not superabundant

(see the

number 110880): it is the 20°" highly composite number (see the number 180).

7641

e the only four digit self replicating number (see the number 954).

7 647

e the 16°" Keith number (see the number 197). 7 696 e the fourth number

whose square can be written as the sum

of three fourth

powers: 7 696% = 48* + 604 + 804 (see the number 481). 7 TAL

e the 31" Lucas prime number (see the number 613).

7 744 | (= 887) e the largest four digit perfect square which has only two distinct digits (see the

number 1 444). 7755

e the 17°" number n such that n- 2” — 1 is prime (see the number 115).

H

7776 e the fourth number n > 2 such that Z

(n) + 9(n)

is an integer (see the number

588).

e the fourth number n such that >>

17098? (see the number 2 474).

PSPn

p is a perfect square: here )>

e the 17" Keith number (see the number 197).

P 1 and such that 6(n+1) —d6(n) =1 (see the number 49). 18 496

e the fifth number n > 1 such that n-2"-+ 1 is prime (see the number 141). 18 523 e the 16’” prime number p,z such that pip2...pzx +1 is prime (see the number 379).

18 600

e the eighth number which is not perfect or multi-perfect but whose harmonic mean is an integer (see the number 140).

198

JEAN-MARIE DE KONINCK

18 620 e the ninth number which is not perfect or multi-perfect but whose harmonic mean is an integer (see the number 140).

18 737 e the smallest number which can be written as the sum of two squares, as the sum of two cubes and as the sum of two fourth powers and where each of the numbers in question is distinct from the others: here 18737 = 89? + 104? =

17? + 249 = 84 4 11°.

18 787 e the second component of the largest known pair of prime numbers (p,q) such that

pt '=1

(modg?)

and

gg? *=1

(mod 2);

here (p, qg) = (2903, 18787) (see the number 2903). 18 885

e the 21"** number n such that n- 2" — 1 is prime (see the number 115). . 19151

e the sixth self contained number (see the number 293). 19 279

e the number of Carmichael numbers < 10% (see the number 646). 19 333 e the smallest prime number p such that p+ 40 is prime and such that each

number between p and p + 40 is composite (see the number 139). 19 469

e the 37°" Lucas prime number (see the number 613). 19521

e the third 2-hyperperfect number (see the number 21).

THOSE

FASCINATING

NUMBERS

199

e the ninth number n such that n? — 1 is powerful: here 19601? —1 = 2°-34-5?.

7? 11? (see the number 485).

e the smallest prime number which is preceded by 51 consecutive composite numbers; indeed, there are no primes between 19609 and 19661.

(=27°) e the fifth (and largest) number n whose sum of digits is equal to ~/n (see the number 512).

[19 767] (=3- 11-599) e the number n which allows the sum

y mon

1 — to exceed 2 (see the number 402). m

Q(m)=3

e the smallest prime factor of 10°? + 1; if q, stands for the smallest prime factor

of 102° +1, then qg, = 101, go = 73, 93 = 17, qa = 353, gs = 19841,

ge = 1265011073, g7 = 257, g3 = 10753, gg = 1514497, gio =

1856 104 284 667 693 057, qi1 =

106907803649

and gig = 458 924033;

here are the complete factorizations!®’ of the numbers 102° +1 when2 2 such that oe

is an integer (see the number

588).

[27 625 |(= 5° - 13 - 17) e the smallest number which can be written as the sum of two squares in seven distinct ways (as well as in eight distinct ways), namely 27625 = 20? + 1652 =

277 +1642 = 457+ 160? = 60? +155? = 837+ 1442 = 8874141? = 10124132? = 115? + 120? (see the number 50).

[27730] (=2°.3?-5-7-11) e the smallest number n such that o(n) > 4n: in this case, Zin) = 4.05; the sequence of numbers satisfying this property begins as follows: 27720, 30240,

32 760, 50400, 55440, 60480, 65520, 75600, 83160, ...; moreover, if n = nz stands for the smallest number n such that o(n) > kn, then nz = 6, n3 = 120, ng = 27720, ns = 122522 400 and ng = 130 429 015 516 800; e the 25" highly composite number (see the number 180).

‘THOSE FASCINATING

NUMBERS

207

e the second number n such that each of the numbers n+i, i = 0,1,2,...,16, has

a factor in common with the product of the other 16 (see the number 2184).

e the tenth number which is not perfect or multi-perfect but whose harmonic mean is an integer (see the number 140).

e the smallest number n such that P(n +7) < /n +i for i = 0,1,2,3,4,5; the largest prime factors of these six numbers are respectively 73, 97, 131, 89, 163

and 53, and thus all smaller than /28032 ~ 167 (see the number 1518).

e the smallest number n such that 7(n) = r(n + 1) = T(n + 2) = T(n +38) = T(n+4) =7(n+5): the sequence of numbers satisfying this property begins as follows:

28374, 90181, 157493, 171893, 171.894, 180965, 180966, ... (see the

number 33).

e the eighth prime Fibonacci number (see the number 89).

e the smallest Niven number n such that n+ 30 is also a Niven number, but with no others in between; if nz, for k > 2, stands for the smallest Niven number n such that n + k is also a Niven number, but with no others in between, then AO)

==

90, 90,

=

7560,

n30

=

28 680,

n49

=

119 (GOR

n590

=

154 876,

né6éo

=

297 864, nz = 968 760, ngo = 7989 168, noo = 2879 865 and n109 = 87 699 842.

| e the ninth number

whose square can be written as the sum

of three fourth

powers: 28721? = 604 + 135+ + 1484 (see the number 481).

e the smallest number n such that r(n) > T(n +1) >... > T(n +4); it is also the smallest number n such that r(n) > rT(n +1) >... > T(n +5): here 16 >12>10>8>4> 2 (see the number 45).

208

JEAN-MARIE

DE KONINCK

e the smallest pseudoprime in bases 2, 3, 5 and 7; it is also the smallest pseudoprime in bases 2, 3, 5, 7 and 11; the sequence of pseudoprimes in bases 2, 3, 5 and 7 begins as follows: 29341, 46657, 75361, 115921, 162401, ...

e the eighth number such that 2” + n? is prime (see the number 2007).

e the second number n such that o(n), o(n +1), o(n+2), o(n+3) and o(n+4) have the same prime factors, namely here 2, 3, 5 and 7 (see the numbers 3777

and 20 154).

e the smallest number n such that n and n+ 1 each have seven prime factors counting their multiplicity: 29888 = 2° - 467 and 29889 = 3° - 41 (see the

number 135).

e the smallest number of the form p;p2 ...p~%-+1 which is not prime: here 30031 = 223-5240 911-13 -— 1 = 69-509:

e the smallest 4-perfect number, that is a number n such that o(n) = 4n; the list of 4-perfect numbers begins as follows: 30 240, 32 760, 2178540, 23 569 920, 45 532 800, 142 990 848, 1379 454720, 43 861 478 400, 66 433 720 320, 153 003 540 480, 403 031 236 608, ...; there seems to exist only 36 4-perfect num-

bers (see R.K. Guy [101], B2).

30 375 e the smallest number n which allows the sum De oe

fon Pm)

number 177).

to exceed 20 (see the

THOSE

FASCINATING

NUMBERS

209

e possibly the largest number n such that n(n + 1)(n + 2)...(n +6) and (n+ 1)(n+2)...(n +7) have the same prime factors: here

20618 230619) on. 230624 © 30619 -30620+...-30625

=

0b925395 57 11 290759..61 67 -113 - 173 - 251 - 271 - 457 -1531, 9°.32.55.77.11-99-.50.61 ‘67-113 - 173 - 251 - 271 - 457 - 1531;

if nz, for k > 2, stands for the largest number n such that n(n + 1)(n + 2)...(n+k —1) and (n+ 1)(n+ 2)...(n +k) have the same prime factors, then the conjectured values of the first n,’s (assuming the abc Conjecture’®°) are No = 2, ng = 24, ng = 32, ng = 400, ng = 480, n7 = 30618, ng = 34992, Ng =

39 366, nig

=

43740

and

ny1=

107 800.

30 693 e the ninth number which is not a palindrome, but whose square is a palindrome

(see the number 26). 30 784 e the tenth number

whose square can be written as the sum

of three fourth

powers: 30784? = 964 + 1204 + 1604 (see the number 481). 31 469 e the smallest prime number which is preceded by 71 composite numbers; indeed, there are no primes between 31397 and 31 469.

31 907 e the smallest prime number p such that p+ 50 is prime and such that each number between p and p+ 50 is composite (see the number 139). t

160Tndeed, it is easy to show that if the abc Conjecture is true, then each number nx is well defined.

Indeed, let k > 2 be fixed. First observe that P(n) < k, since otherwise p|n for some prime p > k, in which case p cannot divide any of the numbers n +7 for i = 1,2,...,k, thereby contradicting the

fact that n(n+1)...(n+k-—1) and (n+1)...(n+k) have the same prime divisors. By the same argument, we also have P(n +k) < k. Thus, applying the abc Conjecture (with a = n, b= k and c=n-+k), we have 1l+e

nthenomineny«

(TD) psk

an inequality which cannot hold if n is sufficiently large.

,

210

JEAN-MARIE

DE KONINCK

32 043

e the smallest number whose square uses each of the ten once: 87 numbers satisfy this property, namely: 32043, 35337, 35757, 35853, 37176, 37905, 38772, 39147, 39336, 44016, 45567, 45624, 46587, 48852, 49314, 49353, 50706, 55524, 55581, 55626, 56532, 57321, 58413, 58455, 58554, 61866, 62679, 62961, 63051, 63129, 65634, 65637, 66105, 68781, 69513, 71433, 72621, 75759, 76047, 76182, 77346, 80445, 81222, 81945, 83919, 84648, 85353, 85743, 85803, 89079, 89145, 89355, 89523, 90144, 90153, 90198, 91248, 95154, 96702, 97779, 98055, 98802 and 99066.1°

digits once and only 32286, 33144, 35172,

40545, 53976, 59403, 66276, 78072, 86073, 91605,

42744, 54918, 60984, 67677, 78453, 87639, 92214,

43902, 55446, 61575, 68763, 80361, 88623, 94695,

32 045 e the second number which can be written as the sum of two squares in eight

distinct ways: 32045 = 2? + 179? = 19? + 178? = 46? + 173? = 677 + 1667 = 74? + 163? = 862 + 157? = 109? + 142? = 122? + 131? (see the number 50). 32 214 e the third number n such that each of the numbers n +7, i = 0,1,2,...,16, has

a factor in common with the product of the other 16 (see the number 2 184). 32 292

e the sixth number n > 1 such that n-2"+ 1 is prime (see the number 141). 32 445

e the second number n such that o(n) = 2n + 6 (see the number 8 925). 32 469

e the seventh number n > 1 such that n- 2” + 1 is prime (see the number 141). 32 760

e the second number n such that a(n) = 4n (see the number 30 240). 32 768

e the eighth number n > 1 such that ¢(o(n)) =n (see the number 128). 161 Observe that none of these numbers is prime; indeed, each of them is a multiple of 3: this results

from the fact that to each such number n corresponds the number n? which is made up of the digits 0,1,2,...,9; since O+1+2+...+9 = 45 is divisible by 9, it follows that 3|n.

THOSE

FASCINATING

NUMBERS

PAUL

e the fifth Apéry number (see the number 1 445). 33 614

e the first of the three smallest consecutive numbers each divisible by a fourth power)33 6142477, 33615 = 3* -5..83, 33616 = 2*- 14-191 (see the number 1375). 33 617 e the smallest number n which allows the sum

1 y — to exceed 11 (see the number a

t 1. More precisely he showed

that

nm=1

(mod 3)

n=11

(mod 12)

n=0O

(mod 2)

n=1

(mod 4)

n=15

(mod 18)

n=27

(mod 36)

n=3

(mod 9)

7|78557 -2" +1 13/7857 2" 1 3|78557- 2" +1 5|78557-2" +1 19|78557-2" +1 37|78557-2" +1 73|78557 - 27 +1.

|

These congruences are respectively equivalent to

=

1,4,7,10,13, 16,19, 22, 25,28, 31,34 (mod 36), 11, 23,35

= Ill

Ss So S\S

(mod 36),

0,2,4,6,8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32,34 1,5,9, 13, 17,21, 25, 29,33 (mod 36),

=

15,33

=

27

=

3,12,21,30

(mod 36),

(mod 36),

(mod 36),

(mod 36).

Since these congruences cover all congruence classes modulo 36, this establishes that 78557 - 2" + 1 is composite for all integers n > 1. 167Tt is known today that the smallest number k which could possibly be a Sierpinski number is k = 10223; in fact, as of 2008, there remains six numbers smaller than 78557 which could possibly be Sierpinski numbers:

10223, 21181,

22699,

24737,

55 459 and 67607;

one of the last

“false Sierpinski numbers”, namely 19 249, was eliminated in 2007 by K. Agafonov, who proved that

19 249 . 213018586 4 1 is prime.

i)28

JEAN-MARIE

DE KONINCK

78975 e the fifth of the existing eight primitive non deficient numbers (see the number 945).

78 989

e the largest five digit prime number whose digits are consecutive (see the number I Nena

79 196 H

=

e the second number n > 1 such that n?+3

is a powerful number: here 79 196? +

3 = 7° - 19? - 37° (see the number 37). 79 427 e the fourth prime number gq such that Dine gp isa perfect square: here

> p< 79427 P= 292341604 = 17098* (see the number 22073). 79 800 e the smallest number n such that the Liouville function 9 takes successively, starting with n, the values 1, —1,1,—1,1,—1,1,—1,1,—1,1,—1,1,—1 (see the

number 6 185). 80518

e the only number of the form abcde such that abcde = a! — b! — c! — d! + e!; here

80518 = 8! — 0! — 5! — 1!4 8! (see the number 40585). 80 782 e the seventh solution y of the Fermat-Pell equation x? — 2y? = 1, namely that

given by (x, y) = (114243, 80782) (see the number 99).

[81.081 |(=34.7-11-13) e the smallest odd abundant number which is not divisible by 5: the sequence of numbers satisfying this property begins as follows: 81081, 153153, 171171, 189189, 207 207, 223 839, 243243, 261261, 279279, 297297, 351351 459 459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837 837, 891891, 908 523, 960 687, 999 999, ... (see the numbers 945 and 5 391 411 025).

THOSE

FASCINATING

NUMBERS

229

e the only five digit number (4 10000, 20 000, 30000) whose square contains only

two distinct digits: 81619? = 6 661661161 (see the number 109).

e the smallest number n which can be written as the sum of the squares of two prime numbers in 5 (as well as 6, 7 and 8) distinct ways: 81770 = 417 + 283? =

537

2811

ey

= 07 Se 2697137? E25

121572 + 2392 = 1792 +

2237 = 193? + 211? (see the number 338).

e the eighth number n > 2 such that ad(n)

+ o(n) 5 is an integer (see the number

y(n)

588).

e the 29%" highly composite number (see the number 180).

(= 174) e possibly the only fourth power b such that a+b = c, with (a,b) = 1, min(A(a), A(c)) = 3: here a = 857 375, b = 83521, c = 940 896 and

857 375 + 83521 Sor Ga

= tye re

940896 Og 117

with A(a) = 3, A(b) = 4 and A(c) & 3.283. 84 998 e the smallest number n which allows the sum

1 Ny ——

See

UE,

to exceed 22.

85 139

e the third number n such that ¢(n) = 4¢(n + 1) (see the number 629).

e the fifth number n such that ¢(n) + o(n) = 3n (see the number 312).

30

JEAN-MARIE

iw)

DE KONINCK

86 243

e the exponent of the 28°” Mersenne prime 2°° 743 — 1 (Slowinski, 1982). 86 453

e the seventh number k such that — 11...1 is prime (Baxter, 2000); see the numk

ber 19.

87 360

e one of the five known unitary perfect numbers (see the number 6). 87 750

e the second number n such that (o7(n) + y(n))/n

270).

is an integer (see the number

87 890 e the sixth number n such that each of the numbers n +7, 2 = 0,1,2,...,16, has

a factor in common with the product of the other 16 (see the number 2 184). 88 799

e the smallest prime factor of s;3 = 11+2?+33+...+131; in fact, s13 = 887993514531963 (see the number 3413 for the list of known numbers s,, which are prime); below is a table of the factorizations of the numbers 11+2?+3%+...+n” for 2 P(n+8): here 109453 > 54727 > 7297 > 6841 > 4759 > 2027 > 823 > 421 > 107 (see the number 1 851).

e the fifth number which is not a palindrome, but which divides the number obtained by reversing its digits (see the number 1089).

|110 487| (= 3 - 13 - 2833) e the 100000" composite number (see the number 133).

e the exponent of the 29” Mersenne prime 2!19°°3 — 1 (Colquitt and Welsch,

1988).

e the 25°” superabundant number: we say that n is superabundant if o(n)/n > a(m)/m for each number m < n; the sequence of numbers satisfying this property begins as follows: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166 320, 277 200, 332 640, 554 400, 665 280, 720720, 1441440, ...

e the smallest number n for which the Moebius function py takes successively, starting with n, the values 1,0,1,0,1,0,1,0,1,0,1,0 (see the number 3 647).

e the 12" number n such that n?—1 (see the number 485).

is powerful: here 114 243?—1 = 23-134-239?

e the 17’” Granville number (see the number 126).

i)38

JEAN-MARIE

DE KONINCK

114 689 1, which in particular is the smallest e the smallest prime factor of Fiz = TE Fermat number for which the complete factorization is not yet known (see the

number 70525 124 609). 115 921)

(= 13%37 - 241)

e the fourth pseudoprime in bases 2, 3, 5 and 7 (see the number 29 341).

115 975

e the tenth Bell number (see the number 52). 119 972 e the smallest Niven number n such that n+40 is also a Niven number, but with

no others in between (see the number 28 680).

120120)

(=27 23-577 -11-13)

e the smallest number n such that Q(n)’\™ > 2n: here Q(n)’\™/n & 2.18235; the sequence of numbers satisfying this inequality begins as follows: 120120, 240 240, 480480, 1021020, 2042040, ...(to discover the smallest number n such that Q(n)“™ > n, see the number 60; as for the maximal value of the

i

quantity Q(n)*“™ /n, see the number 3569 485 920);

e the smallest number n such that a(n) = 21- d(n) (see the number 416 640).

121 056

e the fifth number n such that ¢(n)o(n) is a fourth power: (121 056)o(121 056) =

3364 (see the number 170).

e the largest known prime number p such that 12?-' = 1 (mod p?): the only other prime number p < 2°? satisfying this property is 2693 (see Ribenboim [169], p. 347).

e the second number which quadruples when its last digit is moved in first position

(see the number 102564).

THOSE

FASCINATING

NUMBERS

239

130 304

e the sixth solution of a(n) = 2n + 2 (see the number 464).

e the smallest Niven number n > 6 such that n+1,n+2,n+3 also Niven numbers (see the number 110).

and n+4

are

131070

e the fifth number n such that o(¢(n)) = n/2: the first four are 2, 6, 30 and 510; the sixth one is 8589 934 590. 131 071

e the sixth Mersenne prime: 131071 = 2!” — 1: the following table provides the list of all Mersenne primes 2? — 1 known as of May 2009: Rank

Pp

(ZA NS Ze 315 ial eam Be Cy pal Be lh ies Fa |19 Se uieot 9

61

10 | 89 107 Lag, 9127, Ly)

ODE

2 7 31 127 8191 131071 524 287 2147 483 647

OAT

24 ss 26 27 28 29 30 31

Dp

| 19937 21701 | 23209 | 44497 | 86243 | 110503 | 132049 | 216091 756 839

| ry a

|

De ee g21701 _ 4 223209 _ 4 g44497 _ 4 986243 __ 4 3110005 "1 2132049 _ 4 9216091 _ 4 9756 839 _ 4

DE = i

39

289 _ 1 2107 J 127 _ 4

D399 433° 33 | 859433 34 ial, U2 Sidleet POR 1 35. S898 260qal5 26928269 1 20) 2 20g 22 i a20 ean Steel Of moles ODS T Maes poe en INCOR PALATE fom a eres Zickes ew OO S466 97a 2t2 eet AO 71220 996 01 .1"220298 Ot Stee AUSG S83 abo 5 eal Ales 25 064 9501 2700s ee AD AS N30 402457 0|-2°9 02 se 1 N32.082 Goie\ede 7 AAD

Vissi erialeee irae al 2607 _ 1 14 | 607 15 Gl 270 met 27? 1 2.203 lee — 1 162 yom? 8 eS e228 17 iReosoe SIPING! wyWE a 1) |ngSET hpiacees is 4423) pees 200 21 Bla9 G89 | 42? 29 1 229

Rank

ieeeet t= *]

Anes

STL bO GOT | 2° te

a

09 0 1| |208720 46_|1126 ag_|ii2is |a78—1 | |43

e the sixth of the existing eight primitive non deficient numbers (see the number 945).

240

JEAN-MARIE

DE KONINCK

e the exponent of the 30°” Mersenne prime 213?°49 — 1 (Slowinski, 1983).

e the number of possible arrangements of the integers 1,2,...,9 with the restriction that the j*” integer must not be in the j"” position for each 7, 1 p+6|n +6 the number 399).

(see

148 349

e the only known number n which is equal to the sum of the sub-factorials of its digits: 148 349 =!1+!4+!8+!3+!4+19; the sub-factorial function !n is defined by

nant

1

i

(1-45

i}

—g

173Qne can prove that there exist infinitely many

te

numbers

ji

FUG)

satisfying this property by simply

considering the sequence of numbers 142 857, 142857 142857, 142 857 142 857 142 857, and so on.

242

JEAN-MARIE

DE KONINCK

150 287

e the smallest prime factor of the Mersenne number factorization is given by

: :

2163 _ 1, whose complete

9163 _ 1 — 150287 - 704161 - 110211473 - 27669118297 - 36230454570129675721.

151 023

e the 26%" number n such that n- 2” — 1 is prime (see the number 115).

153 720

e the third number n such that ¢(n) + o(n) = 4n (see the number 23 760 as well as the number 312). 153 846

:

e the third number

which quadruples when its last digit is moved

to the first

position (see the number 102 564).

154 876 e the smallest Niven number n such that n+ 50 is also a Niven number, but with no others in between (see the number 28 680). 155 863

e the smallest prime number gq for which the value of the corresponding sum ap ,,—,,¢(m) is a multiple of 100.000; here the sum is equal to 7697 600 000. ’

(=

1321)

e the smallest 10-hyperperfect number: we say that a number n is 10-hyperperfect if it can be written as n = 1+ 10 ay) d (which is equivalent to the

condition 100(n) = 11n + 9).

d|n l 1 and such that (see the number 49).

6(n+1)—46(n) = 1

e the first term of the smallest sequence of ten consecutive prime numbers all of

the form 4n + 3 (see the number 463). 185 527

e the smallest number which can be written as the sum of the cubes of three distinct prime numbers in two distinct ways: here

185527 = 19° 4+ 31° + 53° = 13° + 43? 4. 473; if n, stands for the smallest number which can be written as the sum of the cubes of three distinct prime numbers in k distinct ways, then ng = 185527, n3 = 8627527 and n4 = 999 979 163.

246

JEAN-MARIE

DE KONINCK

e the second number which is not the square of a prime number, but which can be written as the sum of the squares of some of its prime factors: here 188 355 = 3-5- 29-433 = 5? + 29? + 4332 (see the number 870).

e the first of the smallest eight consecutive numbers at which the Q(n) function takes distinct values, namely here the values 6, 8, 1, 7, 3, 5, 2 and 4 (see the

number 726).

194 979 e the largest number which can be written as the sum of the fifth powers of its

digits: 194979 = 1° +9° + 4° + 9° + 7° + 9° (see the number 4150). 195 556

i

e the only composite number n < 10° such that o(n + 10) = o(n) + 10.

197 210 e the second number which can be written as the sum of the squares of two prime

numbers in 5 distinct ways (as well as 6 and 7): 197210 = 31? + 443? = 677 +

439? = 107? + 431% = 173? + 409? = 199? + 397? = 241? + 373? = 311? + 317? (see the number 338). 199 584

a(n) e the second solution of oes

= (see the number 35 640).

199 606

e the seventh number n such that 2” =—2

(mod n) (see the number 946).

203 391 e the smallest number

n such that n and n +1 each have nine prime factors

(counting their multiplicity); indeed, 203 391 = 38-31 and 203 392 = 2” -7-227 (see the number 135).

THOSE

FASCINATING

NUMBERS

247

203 433 e the smallest number n such that n, n+ 1 and n + 2 are square-free and each have four prime factors: 203433 = 3-19-43 -83, 203434 = 2-7-11- 1321, 203435 = 5 - 23-29-61 and 203436 = 5 - 23-29-61; the sequence of numbers satisfying this property begins as follows: 203433, 214489, 225069, 258013, 294593, 313053, 315721, 352885, 389389, 409353, ... (see the number 1309).

205 097

e the smallest number n requiring 12 iterations of the o;(n) function in order to reach 1, which gives rise to the sequence 205098, 136736, 4274, 2138, 1070, 648, 121, 133, 160, 6, 4, 1 (see the number 193). 205 128 e the fifth number which quadruples when its last digit is moved in first position

(see the number 102 564). 205 206

e the third solution of a(n) = a(n + 69) (see the number 8 786). 208 012

e the 12°’ Catalan number (see the number 14). 208 335 e the largest number which is both triangular and square pyramidal: a number is said to be square pyramidal if it can be written as the sum of the first m squares for some number m: here

908335 = 1e- 0 3

4645 = 17 07a 32-85:

the only numbers which are both triangular and square pyramidal are 1, 55, 91

and 208 335 (for an important reference, see the number 645). 210101

e the smallest six digit prime number whose digits are consecutive (see the num-

ber 67). 211673

e the smallest number n such that Q(n) = O(n +1) =... = O(n +6): here the common value is 3 (see the number 602).

248

JEAN-MARIE

DE KONINCK

(= 3022) e the second perfect square whose last three digits are 444 (see the number 1 444).

e the third number n such that a(n), o(n +1), a(n +2), a(n +38) and o(n + 4) have the same prime factors, namely here 2, 3, 5, 7 and 17 (see the numbers

3777 and 20154).

(= 477-97) e the smallest 31-hyperperfect number (see the number 21).

(= 463" e the smallest powerful number n such that n + 6 is also powerful: here n + 6 = 54 . 73: the sequence of numbers satisfying this property begins as follows: 214369, 744769, 715819215721, ...1”° (see the number 463).

e the second even number n such that 2” = 2

e the ninth number n > 2 such that =

(mod n) (see the number 161 038).

is an integer (see the number

588). 216091

e the exponent of the 31"** Mersenne prime 276°! — 1 (Slowinski, 1985).

e the smallest number n such that n,n+1,n+2,n+3,n+4,n+5 and n+6 are all divisible by a square > 1: here 217070 = 2-5-77-443, 217071 = 32-89-271, 217012 = 27- 13567, 217 O73 11 113) 210014 = cs L1-= 13503001 WOKS

5? . 19-457 and 217076 = 2? - 54269 (see the number 242).

175 This sequence is infinite, since in 1982, W.L. Daniel [41] proved that each positive integer can be written as the difference of two powerful numbers in infinitely many ways. Moreover, a few

years later, R.A. Mollin and P.G. Walsh [141] proved that each integer

m > 0 has infinitely many

representations as m = P—Q, where P and @ are powerful numbers neither of them being a perfect

square.

THOSE

FASCINATING

NUMBERS

249

e the third number n such that (o7(n) + y(n))/n is an integer (see the number 270).

e the smallest number

n which allows the sum

aS m, where d;,d2,...,d, stand for the digits of n. 230 387

e the smallest number n such that 1(n) >

logn

n

2n

6n

log*n

log’n

log'n

een

eee

eis

last expression representing the first four terms of the asymptotic expansion of

Li(n): here we have (230387) = 20474 while E= aE en tices 2n

BEL 6

soem ~ 20473.9 (see the number 73). law)

230 578

e the second even number n such that o;(n) = o7(n+2) (see the number 54178).

e the sixth number which quadruples when its last digit is moved in first position

(see the number 102 564).

234 256 | (= 227) e the second number n > 1 whose sum of digits is equal to ~/n (see the number 2401).

e the smallest number n such that t(n) < T(n+1)

< ... < rT(n +8):

here

4 1 such that n- 2” + 1 is prime (see the number 141).

obtains

V/1234567898765432112345678987654321 = 32 - 37 -V11133366688900000111333666889,

V 123456789876543211234567898765432112345678987654321 = 3? . 37- V/1113336668890000011133366688900000111333666889, and so on.

i)56

JEAN-MARIE

DB KONINCK

362 880 e the value of 9!.

364 087

e the smallest number n such that ¢(n) = 9! (see the number 779).

366 439 e the 217% prime number px, such that pip2...p~ +1 is prime (see the number 379 wa

369 119 e the third prime number q which divides the sum of all the prime numbers < q

(that is q| }°, 5: 371549 = 284+ 135 =

310 + 2? .57, each of these last four numbers having as index of composition 8, 5, 10 and 5.49485 respectively; the sequence of numbers satisfying this property begins as follows: 371549, 1016807, 2657333, 19592147, 143123 843,

257 400 763, 1586 054 707, 2 461 855 097, 2579 381 441, 4110 348 907, 4755 143 089, 6 866 235 461, 8 382 781 919, 9 803 235 961, ... (see the number 34 525 900 789 931). 373 457

e the fourth composite number n such that 2"~-? =1 20 737.

(mod n); see the number

373 649

e the first term of the smallest sequence of ten consecutive prime numbers all of

the form 4n + 1 (see the number 2593). 378 661 e the largest solution x of the diophantine equation x? — 17 = y?, namely the one

given by (2, y) = (378661, 5234) (see the number 5 234). 383 846

e the ninth number n such that 2” = —2

(mod n) (see the number 946).

388 961

e the largest known prime number p such that P(p?—1) < 17 and P(p*—1) = 17;

here p? — 1 = 2. 3+.5-7+-11-13-17 (see the number 4801). 389 052 e the smallest number which can be written as the sum of the cubes of three distinct prime numbers and as the sum of the cubes of four distinct prime

numbers: here 389 052 = 2? + 3° + 733 = 73 + 433 + 473 + 593; the sequence of numbers satisfying this property begins as follows: 389052, 493191, 1 442 932, 19 927 270, 31867 188, 36 265069, 49 431241, 95444028, 110621861, ...

180The interest for this number comes from the fact that no one knows if there exists a number which can be written as the sum of two fifth powers in two distinct ways. This question is however

settled for the fourth powers (see the footnote attached to the number 635 318657).

258

JEAN-MARIE

DE KONINCK

(= 58) e the third number n > 1 whose sum of digits is equal to “/n (see the number 2401).

e the 22Ӣ (and the largest known) prime number p,; such that pip2...pz +1 is prime (see the number 379).

[sonaas] (=2°-3.7.73) e the sixth Erdés-Nicolas number (see the number 2016).

e the 18‘" Granville number (see the number 126).

e the smallest prime number p such that p+100 is prime and such that each number between p and p+ 100 is composite (see the numbers 370 261, 378 043 979, 4758 958 741 and 22 439 962 446 379 651, as well as the table given at the number 139 ee

e the largest known number n = (dj, d2,...,d,] such that n = dé + tees +...+

d%: here 397612 = 3 + 91 + 7° + 67 + 19 + 23: the only other known number satisfying this property is 48625, an observation due to Patrick De Geest. 404 851 e the smallest prime number p such that p+ 90 is prime and such that each

number between p and p + 90 is composite (see the number 139). 409 113 ® the value of b=?)

2.

- 2 9Ol

e the eighth Hamilton number (see the number 923).

‘THOSE FASCINATING

NUMBERS

259

e the fifth solution of equation o(n) = a(n + 69) (see the number 8 786).

e the smallest number n such that o(n) = 17-¢(n): here 416 640 = 2’ -3-5-7-31, o(416640) = 1566720 and ¢(416640) = 92160; if nz stands for the smallest number n such that o(n) = k- d(n), then nz = 3, n3 = 2, na = 14, n5 = 56, Vea

0,07

ni3

=

nig

=

—

630, ny4

see =

oe ge

420, n15

291 060, n20

=

=

O0, Dig =

840, N16

83 160, na

=

=

120120

1108, 4

20 790, n17 and

no

=

=

p92 OSUe191 416 640, nig

=e 210, =

9 240,

5 165 160.

e the only number n < 10° such that B(n) = B(n +1) = B(n +2), where (Gy) =

eon ap.

here 41162

— 2-3. 251- 277, 417163

=

17 - 53 - 463.

417164 = 2?-11-19-499, and the common value of B(n +7) is 533 (see the number 89 460 294).

[419904](= 2°.3° e the second number n having at least two distinct prime factors and such that

B(n)3|Bi(n): here (2 + 3)3|(2° + 3°) (see the number 5 120).

[421590] (= 2-3-5-13-23- 47) e the ninth ideal number (see the number 390).

e the smallest number whose fourth power can be written as the sum of three non zero fourth powers:

422.4814 = 95 8004 + 2175194 + 414560?; this is an observation made by R. Frye around 1988; Euler believed that there

were no such numbers!*!,

e the third solution of o(n) = 3n + 12 (see the number 780).

181fyler wrote: Just as there are no non trivial solutions of equation x* + y* = z3, there are none for equations x* + y* + 24 = ut and a> +y°+ 2° +u° =v", and so on for higher powers. For the fifth powers, a counter example was found (see the number 144).

260

JEAN-MARIE

e the tenth number n > 2 such that

DE KONINCK

(n) + o(n) is an integer (see the number 5 7(n)

588).

435 708 | (= 2? - 32 - 77-13 - 19)

a(n) e the fourth solution of erreur

10

(see the number 1080).

e the number of twin prime pairs < 10® (see the number 1 224).

e the 13°" and largest prime number q for which the value of the corresponding sum Dene qP uses each of the digits 0,1,2,...,9 once and only once: in this case,

>» p 1 and such that 6(n+1)—6(n) = 1 (see the number 49).

e the smallest number which is equal to the sum of the seventh powers of its digits added to the product of its digits: the only numbers satisfying this property are 455 226, 3653 786, 4210818, 7774369 and 9 800 817.

e the smallest integer n such that w(n),w(n+1),...,w(n+5)

are all distinct,

namely in this case with the values 6, 1, 4, 2, 3 and 5 (see the number 417).

e the eighth powerful number n such that n+ 1 is also powerful: here 465 124 =

2? -11?- 31? and 465 125 = 5° - 61? (see the number 288).

THOSE

FASCINATING

NUMBERS

261

465 125 |(= 5° - 617) e the smallest powerful number n such that n — 1 is a perfect square (namely 6827): the second number satisfying this property is

1610 006 506 595 061 125 = 1 268 860 3187 + 1 = 5° - 17? - 612 - 109441? (see the number 682). 467 458

e the third even number n such that o7(n) = o;(n + 2) (see the number 54178). 470 449 e the 13" number n such that n? — 1 is powerful: here

ATOAA97 — Ve 2°? 3°07

11 977

(see the number 485). 470 832

e the eighth solution y of the Fermat-Pell equation x” — 2y? = 1: here (zr, y) = (665857, 470832); see the number 99.

472 601

e the second composite number n such that o(n+56) = o(n)+56 (see the number 76571). 480 441

e the sixth number n such that ¢(n)o(n) is a fourth power: here

(480441)o (480441) = 6724 (see the number 170).

480 852

e the smallest number n such that 7(n) = n/12 (see the number 330). 481 824

e the 13°” number n such that o(n) and o2(n) have the same prime factors, namely the primes 2, 3, 5, 7 and 13 (see the number 180).

i)62

JEAN-MARIE

DE KONINCK

481 899

e the 12¢” and largest known number n > 1 such that n-2” +1 is prime (see the

number 141). 485 475

e the smallest number n such that (n+ j)4+(n+j+1)4, for j = 0,1,2,3,4, are all primes: here these primes are 111096215892626372264401, 111097131255474983589617, 111098046623980093284097, 111098961998141724650737 and 111099877377959900992481. 491 531

e the largest prime number p < 2%? such that 7?-! = 1

(mod p”): the only

other known prime number p satisfying this property is p = 5. 505 925

e the largest known solution of y(n + 1) — y(n) = 17 (see the number 1681). 507 904 @

GasBa(@) —

9** Granville number (see the number 126).

509 203 e the largest known Riesel number:

an odd number k is called a Riesel number

if k - 2” — 1 is composite for each number n > 1: H.Riesel [172] proved that k = 509 203 satisfies this property!®? (see also the number 78557). 512 000

e the smallest number n such that 10! divides 224

1+ 2-+...+n

(see the number

514 229 =

e the ninth prime Fibonacci number (see the number 89). 522 752

e the seventh solution of a(n) = 2n + 2 (see the number 464).

182Riesel also proved that there exist infinitely many Riesel numbers: he did this by showing that the numbers 509203 + 11184810r, r = 0,1, 2,..., are all Riesel numbers. As of May 2009, we know that the smallest number k which could possibly be a Riesel number is k = 2293.

THOSE

FASCINATING

NUMBERS

263

[523 776 |(= 29-3- 11-31) e the third tri-perfect number (see the number 120).

524 160 | (= 2’ .32-5-7-13) e the smallest solution of zl) = a n

524 287 |(= 219 — 1) e the seventh Mersenne prime.

529 620

e the seventh number n such that ¢(n)o(n) is a fourth power: here

(529620)

(529620) = 672+ (see the number 170).

531 434

e the second solution of o(n + 7) = a(n) +7

(see the number 74).

540 857

e the fifth composite number n such that 2"~-? = 1 20 737.

(mod n); see the number

542 603 e the smallest prime number p such that p + 80 is prime and such that each number between p and p+ 80 is composite (see the number 139).

545 835

e the eighth horse number (see the number 13). 548 834 e the only number > 1 which can be written as the sum of the sixth powers of

its digits: 548 834 = 5° + 46 + 86 + 86 + 36 + 4°. 548 856

e the eighth number n such that o(¢(n)) = n (see the number 744).

264

JEAN-MARIE

DE KONINCK

556 160

e the seventh number n such that ¢(n) + a(n) = 3n (see the number 312).

e the smallest number

n such that the decimal expansion of 2” contains ten

consecutive zeros, namely from the 94170*" decimal of 2” to the 94179*" (see the number 53).

e the eighth number n such that ¢(n) + a(n) = 3n (see the number 312).

e the largest of the existing eight primitive non deficient numbers (see the number 945).

575 119 e the smallest prime number q; such that each number q; = 3q;_1 + 2 is prime for 1 = 2,3,...,7: such a sequence of prime numbers is in some sense similar

to the Cunningham chains (see the number 1 122 659).

e the number of seven digit prime numbers (see the number 21).

e the smallest number n such that P(n)

> P(n+1)

>... > P(n +9): here

086951 > 73369 > 21739 > 9467 >°1319 > 1193 > 1181 > 1091 S677

(see the number 1851).

[590 226] (= 2-3-7- 13-23-47) e the tenth ideal number (see the number 390).

+29

THOSE

FASCINATING

NUMBERS

265

e the smallest number n > 1 for which €(n) is an integer, where €(n) stands for

vo of numbers satisfying satisfying thithis Da ee ged(in)' here’®3 €(n) = 486 361; the sequence q property begins as follows: 614341, 618233, 1854699, 11746427, 26584019, 35 239 281, 79752057, 85393 399, 118082503, 345592 247, 354 247 509, 505 096 361, 802 597 537, 1036 776 741, 1062 742 527, 1515 289 083, 2149579 159, 2 243 567 557, 3695 178 641, 5077547629, ... 154

(= 288) e the fourth number n > 1 whose sum of digits is equal to ~/n (see the number

2.401).

e the eighth prime number of the form n* +1, here with n = 28 (see the number

1297).

e the smallest number which can be written as the sum of two and three distinct

fourth powers: 617057 = 74 + 284 = 34 + 204 + 264; the sequence of numbers satisfying this property begins as follows: 617057, 1957682, 3502 322, 3959 297, 6959682, 9872912, 31322912, 40127377, 46712801, 48355 137,

49 981 617,...(see the number 4802).

e the smallest number n such that G(n) < B(n+1)

< ... < B(n+ 8): here

423 < 589 < 811 < 1487 < 1616 < 7026 < 41151 < 308575 < 617147 (see the

number 714).

e the second number n for which €(n) is an integer (see the number 614341).

(= 2 78401) e the number n which allows the sum

183Qne can easily show that €(n) = + lias

Sy

i!

— to exceed 5 (see the number 44). m

Zatlig

pti

1841+ would be interesting if one could show that this sequence is infinite.

i)66

JEAN-MARIE

DE KONINCK

629 693

e the smallest number n such that min(A(n), A(n + 1), A(n + 2), A(n + 3)) >

& OOo]

here

min(A(n), A(n + 1), A(n + 2), A(n + 3)) ~ min(1.54162, 1.78225, 1.53833, 1.35053) = 1.35053; the sequence of numbers satisfying this property begins as follows: 629693, 11 121 381, 16176510, 20 188 925, 26 315 199, 82 564 351, 148 629 247, 185 966 873, 283 760 125, 1791 156975, 1972524741, 3047548 776, ...; most likely there exist infinitely many numbers n satisfying this inequality (see for that matter the

number 14018750 and its footnote). 632 501 e the smallest prime number q such that Seer gpisa multiple of 100000:

here

>» p 1 such that ¢(a(n)) =n (see the number 128).

[74340] (=2°-3?-5-31) e the seventh Erdés-Nicolas number (see the number 2016).

e the fourth number n such that n? + 2 is powerful: here

716 0357 + 2 = 3° - 41? - 33617 (see the number 265).

e the largest number n that is divisible by each of the numbers < n!/°, namely here the numbers 1, 2, 3, ..., 14 (see N. Ozeki [157]);

e the 11° colossally abundant number (see the number 55 440).

e the smallest number n such that 6(n) < B(n+1) /, 5: 217 4 311.5 — 7° 4+ 96 . 5° each of these last four numbers having composition 17, 5.05684, 5 and 6 respectively (see the number 371

the sum of 1016807 = as index of 549).

1 023 467 e the smallest prime number made up of seven distinct digits (see the number 1039).

(= 17-19 - 3169) : e the number n which allows the sum

se

1

to exceed 3 (see the number 402).

m +. 97° + 100° = 54° + 60° + 118° = 66° + 90? + 1002 63° 485-194"

(see the number 1 009). 2 025 797

e the eighth self contained number (see the number 293). 2066 115

e the smallest seven digit Sastry number (see the number 6099). 2118 656

e the 11% dihedral perfect number (see the number 130). 2124679

e the second (and largest known) Wolstenholme prime (see the number 16 843). 2 142 720 iN

e the third solution of a te

278,540)

(see the number 35 640).

(2229725 77 413-19)

e the third 4-perfect number (see the number 30 240).

i)82

JEAN-MARIE

DE KONINCK

2199 978

e the eighth number which is not a palindrome, but which divides the number obtained by reversing its digits (see the number 1 089).

2 230 271

e the seventh solution of o2(n) = o2(n + 2) (see the number 1079). 2 424 833 e the smallest prime factor of the Fermat number fy = oe factorization is given by

1, whose complete

Fo = 2424833 - 7455602825647884208337395736200454918783366342657 - Pog.

(= 197-6841) e the third 18-hyperperfect number (see the number 1 333).

2481 757

e the second prime number p such that 23?-! = 1 13.

(mod p”); see the number

e the third and largest known dihedral 3-perfect number (see the number 5 472).

e the smallest number which can be written as the sum of three distinct cubes

in eight distinct ways:

2562624

= =

8° +36? +136 = 8° + 64? + 1327 = 12° + 100° + 116° 173+ 46? + 135° = 30° + 103? + 113° = 36° + 603 + 132° Bil? 85° 129° == 60" 2 70> 193°

(see the number 1009).

e the third even number n for which 2” =2

(mod n); see the number 161038.

THOSE

FASCINATING

NUMBERS

283

2 590 623 e the first of the three smallest consecutive

by a fifth power:

numbers

each of which is divisible

here 2590623 = 3° -7- 1523, 2590624 = 2°- 73-1109 and

2 590 625 = 5° - 829 (see the number 1375). 2 646 798

e the ninth (and possibly the largest) number n > 9 such that n = )7/_, d!, where d,,...,d,. stand for the digits of n: here 2646 798 = 2! + 62443 464475496 +87

(see the number 175). 2 657 333 e the third number which can be written in two distinct ways as the sum of two co-prime numbers each with an index of composition > 5: 2657333 = 274 3!%.5 = 5° +2" . 34, each of these last four numbers having as index of composition 7, 5.46252, 5 and 8.25538 respectively (see the number 371549).

2674 440

e the 14°” Catalan number (see the number 14).

2677 889

e the smallest number of persistence 8 (see the number 679).

2 702 765

e the 12°” Kuler number (see the number 272).

2 704 900 e the fourth multiple of 100 such that the following 100 numbers include exactly 17 prime numbers, namely 2704901, 2704903, 2704907, 2704909, 2704927, 2704931, 2704937, 2704939, 2704943, 2704957, 2704963, 2704969, 2704979, 2704981, 2704987, 2704993, 2704997 (see the number 400).

2718 281 e the third prime number built from the first digits of the decimal expansion of

the Euler number e (see the number 271).

284

JEAN-MARIE

DE KONINCK

e the ninth solution y of the Fermat-Pell equation 2? — 2y? = 1: here (a,y) = (3880899, 2744210); see the number 99.

2 759 640

e one of the three numbers n such that the polynomial 2° — x+n can be factored: the other two are n = 15 and n = 22440: here 2° —2+2759640 = (a7 +1224

377) (x3 + 12x? — 2332 + 7320) (see the number 22 440).

e the fourth prime number p such that 31?-' = 1 79).

(mod p?) (see the number

e the third number which can be written as the sum of the cubes of its prime

factors: 2836295 = 5-7-11-53-139 = 5° + 7° + 113 + 53? + 139° (see the number 378). 2 879 865 e the smallest Niven number n such that n + 90 is also a Niven number, while

no others are located in between (see the number 28 680).

e the smallest seven digit automorphic number:

2890625?

= 8355712890625

(see the number 76). 2914 393

e the second and largest prime number p < 2°” such that 97?-' = 1

(mod p?)

(see Ribenboim [169], p. 347): the smallest is p = 7.

e the exponent of the 36’” Mersenne prime 2297622! — 1 discovered by Spencer in 1997 using the programme developed by G.F.Woltman (see the number 1398 269).

THOSE

FASCINATING

NUMBERS

285

2 999 999

e the largest number n such that fe(n) > n, where fe(n) = fe([di, do,...,d,]) = d§ + d§+...+d®, where d;,d2,...,d, stand for the digits of n.

3 020 626

e the fourth even number n such that 2" = 2

(mod 7); see the number 161 038.

3021 377

e the exponent of the 37” Mersenne prime 2? 97137” —1 discovered by R. Clarkson (a 19 year old student) in 1998 using the programme developed by G. Woltman

(see the number 1 398 269). 3 263 443

e the sixth voracious number (see the number 1 807).

3 290 624

e the smallest number n such that n and n+ 1 each have ten prime factors (counting their multiplicity); indeed, 3290624 = 2° - 6427 and 3290625 =

3+ - 5° - 13 (see the number 135). 3 343 776

e the 11°" number n such that 4(n) + a(n) = 3n (see the number 312).

3 345 408

e the 13" number n > 1 such that ¢(o(n)) = n (see the number 128). 3 358 169 e the first term of the smallest sequence of 12 consecutive prime numbers all of

the form 4n + 1 (see the number 2593). 3370 501

e the sixth and largest solution x of the Bachet equation x? + 999 = y® (see the number 251).

286

JEAN-MARIE

DE KONINCK

3 405 122

e the smallest number n such that Q(n) = Q(n+ 1) =... = Q(n +8): here the common value is 4; it is also the smallest number n such that Q(n) = Q(n+1) = ... =Q(n+4+ 9) (see the number 602). 3 424 006

e the number of twin prime pairs < 10° (see the number 1224). 3 485 664 | (= 2° -3?- 77-13-19)

15

e the third solution of we) aris (see the number 293 760). n 3 523 884

e the smallest number n such that 7(n) = n/14 (see the number 330).

(= 27-39) e the smallest number n such that y(n)?|a(n) (see the number 96). 3 565 979 e the smallest prime number equally distant, by a distance of 48, from the preceding and the following prime numbers: p254473 = 3565931, pos4a79 = 3565979 and P254480

=

3 066 027.

3612 703 e the only prime number p such that p|A, for all n > p, where A, := n! — (n —

1)!4+ (n—2)!—...—(—1)"1], a result established by M. Zivkovié [209]. 3 628 800 e the value of 10! .

e the fourth prime number

of the form n! + n+ 1, here with n = 10 (see the

number 52).

e the smallest number n such that ¢(n) = 10! (see the number 779).

THOSE

FASCINATING

NUMBERS

287

e the second number which is equal to the sum of the seventh powers of its digits

added to the product of its digits (see the number 455 226).

e the smallest number n such that P(n +7) < W/n+i, for i = 0,1,2,3:

here

P(3678723) = P(3?-19- 71-101) = 101 < 3678723 ~ 154, P(3678724) = P(2?.72-137) = 137 < */3678724 ~ 154, P(3678725) = P(5?-37-41-97) = 97
p+ 12|n+ 12

(see the number 399). 4695 456

e the 12°" number n such that ¢(n) + o(n) = 3n (see the number 312).

4713 984 | (= 29- 33-11-31) e the fifth solution of ae = * (see the number 1080).

4729 494 e the number appearing in the famous “cattle problem” of Archimedes, namely

in the Fermat-Pell equation 2? — 4729 494 y? = 1 (see J. Stillwell [191]). 4737 595

e the fourth solution of 72(n) = a(n + 10) (see the number 120). 4989191 iL

e the smallest number n which allows the sum ys 7 to exceed 16 (see the number

83).

t

p= 3672424151449 = (463 - 4139)?

p 2, the probability that there is only a finite number of numbers n such that :

min (A(n), A(n + 1),...,A(n +k is equal to zero.

k

—1)) > aT

300

JEAN-MARIE

DE KONINCK

14684570 e the number of possible arrangements of the integers 1,2,...,11 with the restriction that the integer 7 must not be in the j-th position for each 7, 1 1 whose sum of digits is equal to ~/n: the only

n > 1 satisfying this property are 52521875,

60466176

and

205 962 976.

e the second solution of

a(n) jae ;(see the number 8 910720).

e the value of 11+ 2? +...+ 8% (see the number 3 413).

e the smallest prime number p such that w(p+1) = 2, w(p+2) = 3, w(p+3) = 4, w(p + 4) = 5 and w(p+ 5) = 6 (see the number 103).

302

JEAN-MARIE

DE KONINCK

e the smallest number n satisfying ¢(n) = 5¢(n + 1); the sequence of numbers satisfying this property begins as follows: 17907119, 18828809, 31692569, 73421039, 179467469, ... (see the number 629).

18 003 384 | (= 2° - 3° - 7°) e the fifth number n having at least two distinct prime factors and such that

G(n)3|Bi(n): here (2+ 3 + 7)3|(23 + 38 + 73) (see the number 5120).

e the smallest of the first four consecutive numbers being each divisible by a eube > 1; 18035622 =2-3*-11-29-349, 18035623 = 17° - 3671, 18035624 = 23 . 163 - 13831, 18035625 = 3-54-9619 and 18035626 = 2-7? - 61 - 431 (see

the number 844).

18 506 880 |(= 27 -3°-5- 7-17) e the smallest solution of

a(n)

13

meee Ce

é

the only solutions n < 10° of this equation

are 18506 880, 36 197 280 and 299 980 800. 18 673 201

e the second number n such that ¢(n + 1) = 5¢(n) (see the number 11242770). 19 099 919

e the smallest prime number q; such that each number q; = 2q;-1 + 1 is prime for 1 = 2,3,...,8: such a sequence of prime numbers is called a Cunningham

chain (see the number 1 122659).

e the fourth number which can be written in two distinct ways as the sum of two co-prime numbers each with an index of composition > 5: 19592147 =

318 + 2'8 . 13° = 24.38 +117, these last four numbers having as index of composition 13, 5.12746, 6.45259 and 7 respectively (see the number 371549).

e the ninth solution of o2(n) = o2(n + 2) (see the number 1079).

THOSE

FASCINATING

NUMBERS

303

20 291 270

e the sixth number n such that E,(n) := o(n +1) — o(n) satisfies E,(n +1) = E,(n): here the common value of E, is 365040, since (20291270) = 37374480, o (20291271) = 37739520 and o (20291272) = 38104560 (see the number 693). 20 427 264 | (= 2° -3?- 11-13-31) Gi

e the fourth solution of ee =F (see the number 4320).

20 511 392 e the smallest number which can be written as the sum of two fifth powers and

as the sum of four fifth powers: 20511392 = 3° + 29° = 45 + 10° + 205 + 28°. 20 831 323 e the smallest prime number which is followed by at least 200 consecutive composite numbers (in fact here by exactly 209 composite numbers); see the number 370 261.

20 840 574

e the third solution of o(n) = a(n + 15) (see the number 26). 20 916 224

e the smallest number n such that 12! divides 224):

1+ 2+...+ 7 (see the number

20 996 011

e the exponent of the 40°” Mersenne prime 27°99°°!! — 1 (a 6320430 digit number) discovered by Michael Shafer on November 17, 2003 using the programme

developed by G. Woltman (see the number 1 398 269). 21 063 928 e the third number whose square can be written as the sum of a cube and a

seventh power (see B. Poonen, E. Schaefer & M. Stoll [165]): here 21 063 928? = 76271° + 17°; it is indeed possible to prove that the diophantine equation x? =

y> +z" has only three solutions!?! in positive integers x, y, z with (2, y, z) = 1; here is the table of these solutions: 1917, 1995, Henri Darmon and Andrew Granville [42] proved that if p, q, r are three positive integers such that : + : + 4 < 1, then equation Ax? + By? = Cz", where A, B,C are non zero integers,

has only a finite number of solutions in positive integers x,y, z with (x,y,z) = 1.

w 04

JEAN-MARIE

DE KONINCK

21621 600

e the 14*” colossally abundant number (see the number 55 440). 21 772 800

e the smallest number > 2 which is equal to the product of the factorials of its digits in base 12: 21772800 = [7, 3,6, 0,0, 0, O]12 = 7! - 3! - 6!-0!-0!- 0! -O!; the only known numbers satisfying this property are 1, 2, 21772800, 2090188800, 14 497 650 943 439 560 735 142 707 200 000 000, 75 445 311 584 829 283 999 739 123 702 169 600 000 000 000,

962 493 562 543 459 590 626 671 870 630 428 672 000 000 000 000 and 45 883 517 654 351 824 863 158 584 663 538 863 253 527 461 888 000 000 000 000 000

(see

the number 17 280 for the table of the smallest numbers with this property in

a given base).

e the 15’ number n such that ¢(n) + o(n) = 3n (see the number 312).

e the smallest eight digit Sastry number (see the number 6099).

e the 13°” Euler number (see the number 272). 23 002 083 | (= 3° - 13 - 717) e the 10000 powerful number (see the number 3 136).

e the 16" number n such that ¢(n) + a(n) = 3n (see the number 312).

23 569 920 | (= 2° -3°-5- 11-31) e the fourth 4-perfect number (see the number 30 240).

THOSE

FASCINATING

NUMBERS

305

23 592 593

e the smallest prime number q such that Yip 1 which is equal to the sum of the squares of the factorials of its digits in base 8: here 25417732 = [1,4,0,7,5,4,0,0,4]g =

112 +412 +01? + 71? +51? 4+4!? + 01? + 0!2 +.4!? (see the numbers 145 and 40585). 25 430 981 e the second number > 1 which is equal to the sum of the squares of the factorials of its digits in base 8: here 25 430981 = [1, 4, 1,0,0,5,7,0,5]g = (eee char te

Ol? + 01? + 5!2 + 71? + Ol? + 5!? (see the number 25 417 732). 25 457 760

e the eighth number n such that 3(n)|G(n+1) and B(n+1)|G(n+2): here 164|984 and 984|1157184 (see the number 225 504). 25 658 441

e the first component p of the third 8-tuple (p,p+2,p+6,p+8,p+12,p+18,p+ 20,p + 26) made up entirely of prime numbers: the smallest such 8-tuple is (11, 13, 17, 19, 23, 29, 31,37), while the second is the one whose first component is 15 760091. 25 741 470

e the fourth solution of o(n) = a(n + 15) (see the number 26). 25 964 951

e the exponent of the 42" Mersenne prime 27° 96491 — 1 (a 7816 230 digit number) discovered by Martin Nowak (an eye surgeon) on February 18, 2005, using the programme developed by G. Woltman (see the number 1 398 269).

26 888 999

e the smallest number of persistence 9 (see the number 679). 26 890 623 e the first of the smallest three consecutive numbers each divisible by a sixth power: 26 890 623 = 3° -36887, 26 890 624 = 27-19-11057, 26 890625 = 5®-1721

(see the number 1375).

THOSE

FASCINATING

NUMBERS

307

27 412 679

e the number of twin prime pairs < 101° (see the number 1 224). 27 644 437

e the 13°” Bell number (see the number 52). 28119418

e the sixth even number n such that o7(n) = 07(n + 2) (see the number 54178).

28 600 321 | (= 31? - 29761) e the second 30-hyperperfect number (see the number 3901).

29 149 139 e the smallest number n for which the Moebius function p takes on successively, starting with n, the values 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0 (see the number 3 647).

30 002 960

e the 17’" number n such that ¢(n) + a(n) = 3n (see the number 312). 30 042 907 e the largest known number whose square is the sum of a cube and an eighth

power: here 30042907? = 96 222% + 43° (see the number 122). 30 402 457

e the exponent of the 43"¢ Mersenne prime 29° 497 457 — 1 (a 9152052 digit number) discovered by Curtis Cooper and Steven Boone on December 15, 2005, using the programme developed by G. Woltman (see the number 1 398 269).

30 459 361 | (= 55197) e the smallest powerful number n such that n+14 is also powerful: heren+14 =

33 . 5° - 192: the sequence of numbers satisfying this property begins as follows: 30 459 361, 717498 661, 1090122275, 185 344 887 289,... 19.

192This sequence is infinite (see the number 214369).

oo08

JEAN-MARIE

DE KONINCK

32 509 439

e the tenth solution of o2(n) = a2(n + 2) (see the number 1079). 32 535 999

e the smallest number n such that n and n+1 each have 12 prime factors counting

their multiplicity: here 32535999 = 3!° - 19-29 and 32536000 = 2° - 5° - 7? - 83 (see the number 135). 32 582 657

e the exponent of the 44*” Mersenne prime 29°87’—1 (a 9 808 358 digit number) discovered by Curtis Cooper and Steven Boone on September 4, 2006, using the programme developed by G. Woltman (see the number 1 398 269).

32 694 619 e the smallest prime number q; such that each number q; = 3q;—1 + 2 is prime for i = 2,3,...,8 (see the number 575119): such a sequence of prime numbers is somewhat similar to the Cunningham chains (see the number 1 122659).

33 550 336 |(= 2!2(213 — 1)) e the fifth perfect number (the smallest four are 6, 28, 496 and 8 128). 33 721 216

e the 12‘ dihedral perfect number (see the number 130). 33 757 004

e the number n which allows the sum

SS m 1 whose sum of digits is equal to ¢/n: the only other numbers n > 1 satisfying this property are 8303765625, 24794911296 and 68 719 476 736; e the smallest number which can be written as the sum of three distinct cubes

in ten distinct ways:

34012224

(see the number 1009).

= 36° + 216° +.288° = 397 + 153° = = 41° + 114° + 319° = 45° 4 246° 4 $e 12s 1149 318° 100 1 1097 291027 297", HO77> = 118° 201 86> 162° + 216° + 270° = 173° + 2148

312° 267° 00063 4 296? + 2673

THOSE

FASCINATING

NUMBERS

309

34 385 818

e the seventh even number n such that o7(n) = o;(n+2) (see the number 54178).

35 020 303

e the ninth composite number n such that a(n + 6) = a(n) + 6 (see the number

104).

35 080 502

e the eighth even number n such that o7(n) = o7(n+2) (see the number 54178).

35 357 670

e the 16°” Catalan number (see the number 14).

35 515 634

e the smallest number n such that w(n) + w(n +1) +u(n +2) =

17:

here

35515634 = 2-7-11-23- 37-271, 355156385 = 3-5-17- 41-43-79 35 515636 = 2? - 13-19 - 103 - 349 (see the number 2210).

and

36 171 409

e the largest known prime number p such that P(p*—1) < 29 and P(p*—1) = 29:

here p? —1 = 2°-3-5-73-11- 13-17-23 - 29? (see the number 4801).

36 197 180 | (= 2° -3°-5- 72-19) 13 e the second solution of ae mers (see the number 18 506 880).

36 315 135

e the smallest number n such that n and n+ 1 are both divisible by an eleventh power: here 36315135 = 31! -5- 41 and 36315136 = 213 - 11-13-31 (see the number 1 215).

310

JEAN-MARIE

DE KONINCK

36 721 681

e the smallest number n such that a(n) < o(n+1) < o(n+2) < o(n+3) 1 which is equal to the sum of the ninth powers of its

digits (see the number 146511 208).

537 346 048

e the 21"** dihedral perfect number (see the number 130).

565 062 156

e the smallest number n such that P(n) < P(n+1)

2 which can be written as the sum of two co-prime numbers each with an index of composition > 6: we have

607 323 321 = 37 11° - 419 = 2° - 5° 47990°. where

MBs

419) 2 19198,

d(2° - 5°) = 7.09691,

A(29°) =6.

| 608 892 570 | (= 2-37-5-119- 13-17-23) e the smallest number which is not a prime power, but which is divisible by the sum of the fourth powers of its prime factors, a sum which here is equal to

407286 = 2-3?-11°-17 (see the number 378). 612 220 032 | (= 2” - 314) e the smallest number n > 1 whose sum of digits is equal to W/n: the only other numbers n > 1 satisfying this property are 10 460 353 203, 27512614111, 52 523 350 144, 271 818611 107, 1174711 139 837, 2207984167552 and 6 722 988 818 432.

THOSE

FASCINATING

NUMBERS

333

624 041 002

e the smallest number here 312070501

89163001

>

n such that P(n) 208047001

>

> P(n+1)

156035251

>

>...

124828201

> P(n +11): >

104023501

>

> 39008813 > 4079353 > 48647 > 3079 > 2719 > 431 (see the

number 1 851). 635 318 657 e the smallest number which can be written as!?® the sum of two co-prime fourth powers in two distinct ways: 635318657 = 594 + 1584 = 1334 + 1344: the sequence of numbers satisfying this property begins as follows: 635318657, 3 262 811042, 8657 437 697, 68 899 596 497, 86 409 838577, 160 961 094577, 2094 447 251 857, 4231525 221 377, ...; it is known that this sequence is infinite!®’; on the other hand, the sequence of numbers with two representations

as the sum of two fifth powers is most likely finite!®®. 683 441 871

e the first of the smallest ten consecutive numbers at which the Q(n) function takes distinct values, namely here the values 6, 8, 1, 11, 7, 5, 4, 3, 2, 10 (see

the number 726). 699 117 024

e the 22” number n such that $(n) + a(n) = 3n (see the number 312).

196 Ruler was the first to make this observation, that is some 150 years before Ramanujan (see the number 1729) informed G.H. Hardy that the number 1729 was a very interesting number since it was expressible as the sum of two cubes in two distinct ways. Hardy claims he then asked Ramanujan if he knew of any number which could be written as the sum of two fourth powers in two distinct ways, a problem for which Ramanujan had no solution. 197Indeed, consider the expression 2(7, a,b) := a” + a°b* — 2a%b* + 3na*b° + ab®. Checking that x*(1,a,b) + 24(—1,b,a) = 2*(—1,a,b) + x4(1,b, a), and setting 1 < a < b, one can then generate infinitely many such representations. This result is essentially due to Euler, as was pointed out by Jean-Marc Deshouillers. 198Tn fact, no one knows of any number which can be written in two distinct ways as the sum of two fifth powers. By a heuristic argument, one can show that such a representation is unlikely. Indeed, let x be a large number. It is clear that the number of numbers n = a° + b° < z is approximately (x1/5)2 = x7/5_ Hence, the probability that a number n chosen at random is of the form n = a°+b®

n2/5

is of the order of

n

= ——\. n3/5

Therefore, it follows that

Danie 1 mia Prob[n = a° +b? ete=c ome +05] ee~ (2) a 76/8 This is why the expected number of numbers n € [r,2z] having two distinct representations as the sum of two fifth powers is of the order of

1 ye an”

x

Lap

| ior

r P(n+1) >... > P(n+ 12): here 4044619541 > 674103257 > 19168813 > 10317907 > 5737049 > 1447609 > 465809 > 451207 > 199429 > 162109 > 115399 > 82021 > 1913 (see the

number 1 851).

4118 054 813

e the number of prime numbers < 10!!.

2031¢ is easy to establish that this number n is of the form n = 2°11 p2...pr for certain positive a= 5 andk=9. integers a and k; one can then prove that the optimal choice is obtained by setting

346

JEAN-MARIE

DE KONINCK

4 294 967 295

e the largest known number n such that o(¢(n)) = y(n); the known numbers satisfying this equation are

3 15 955 OA 65535 LORS 4294967295

= 3, 96 5; ("93-5 17, eo es SS SMM Te D57, = 2 ee 119-127 1003, = 3-5-17-257- 65537

(see also the number 744).

4 294 967 297 |(= 2?’ + 1 = 641-6 700417) e the sixth Fermat number, and the smallest composite one.

4 428 914 688 |(= 2! . 32-11-43 - 127) 1 e the fourth solution of ae = * (see the number 360).

4473 671 462 e the fourth number which can be written as the sum of the cubes of its prime factors:

WATB6T1AG2 = 213-179 - 5903" 1621 = 2°" 18°

179° 4 503°

16212

(see the number 378). 4679 307 774

e the smallest number which is equal to the sum of the tenth powers of its digits. 4 700 063 497 | (= 19 - 47 - 5 263 229)

e the smallest solution of 2” =3

(mod n) (see R.K. Guy [101], F10).

4758 958 741

e the smallest prime number p such that p + 300 is prime and such that each number between p and p + 300 is composite (see the number 396 733).

THOSE

FASCINATING

NUMBERS

347

4 931 691 075

e the 13" powerful number n such that n+1 is also powerful: here 4931691075 = 3° . 5%. 17? . 53? and 4931691076 = 2? - 13? - 37? - 73? (see the number 288).

15991 411 025 |(=5?-7-1113-17-19 -23- 29) e the smallest odd abundant number which is not a multiple of 3.

5 394 826 801 | (= 7- 13-17-23 -31- 67-73) e the smallest Carmichael number which is the product of seven prime numbers

(see the number 41 041). 5 425 069 447 e the 14°” powerful number n such that n+1 is also powerful: here 5 425 069 447 =

7° - 41? -97? and 5425 069 448 = 23 - 26041? (see the number 288).

[5471 312310] 1 e the smallest number n which allows the sum Ss — to exceed 23 (see the number v i

orl &

here

min(A(n), A(n + 1), A(n + 2), A(n + 8), A(n + 4), A(n + 5)) ~ min(1.29744, 1.27631, 1.32039, 1.21002, 1.30973, 1,21811)=1.21002; see the number 51 767 910.

27512614111

e the third number n > 1 whose sum of digits is equal to (/n (see the number 612 220 032).

28 091 567 595

e the 12" horse number (see the number 13).

31111 221 312

e the 20°” insolite number (see the number 111).

[31 998 395 520 |(= 27. 3° .5-72-13-17-19) e the second 5-perfect number (see the number 14 182 439 040). 32111111 232

e the 21"* insolite number (see the number 111); it is the smallest insolite number whose first two digits are different from 1. 33 489 857 205

e the number of 12 digit prime numbers (see the number 21).

37 607 912 018 e the number of prime numbers < 10!”.

THOSE

FASCINATING

NUMBERS

S00

38 358 837 677 e the smallest known prime number p for which inequality

ro) < 2x ()

does not hold; Ramanujan proved that this inequality holds for p sufficiently

large (see B.C. Berndt [21], as well as the number 2418).

43 861 478 400 | (= 21° . 33 - 5? - 23 - 31 - 89) e the eighth 4-perfect number (see the number 30 240).

44 496 177 152 e the smallest number with an index of composition > 2 which can be written as the sum of two co-prime numbers each with an index of composition > 7: we have

44 496 177 152 = 2"! .7* 9049 = 198 + 31’, where

KO

9049) =~ 2.08679,

A 19°) =8,

ABI= 7%

(see the number 607 323 321).

51 001 180 160 | (=2'4-5-7-19-31- 151) e the sixth tri-perfect number and the largest one known (see the number 120). 52 523 350 144 | (= 34”)

e the fourth number n > 1 whose sum of digits is equal to ¥/n (see the number 612 220 032).

55 420 693 056 e the seventh number which is both triangular and a perfect square: 55 420 693 056 =

Bae

= 235416? (see the number 36).

61 917 364 224

e the smallest fifth power which can be written as the sum of four fifth powers: 61917 364224 = 1445 = 27° + 84° + 110° + 133° (Lander & Parkin [122]).

356

JEAN-MARIE

DE KONINCK

63927 525 575)(=3 05 1 23) A(n + 1)) reaches its e most likely the number n at which Q2(n) := min(A(n), maximum value, namely 2.65551: we have

n = 63927525375 = 3°-5°-77-23 n+ 1 = 63927525376 = 28 .114-13-41

= with = A(n) & 3.19419 with A(n+1) © 2.65551

66.433 720 320) (= 2)*- 3% 5-11-43 - 127) e the ninth 4-perfect number (see the number 30 240).

68 719 476 736 e the largest number 34012 224).

n whose sum of digits is equal to ¢/n (see the number

68 899 596 497 e the fourth number having two representations as the sum of two co-prime fourth powers:

68 899 596 497 = 5024 + 2714 = 4974 + 2984

(see the number 635 318 657).

70 525 124 609 e the smallest prime factor of the Fermat number Fg = yas

1; the table below

reveals the status of the factorizations?°* of the Fermat numbers F,, = 22” + 1 for 0 1 whose sum of digits is equal to */n (see the number 20 047 612 231 936).

99 999 999 999 973 e the largest 14 digit prime number.

100 000 000 000 031 e the smallest 15 digit prime number.

103 307 491 450 820 | (= 2? -5- 31-61-97 - 28160383)

e the 100000000 000000" composite number (see the number 133).

374

JEAN-MARIE

DE KONINCK

119 429 556 097 859

e the eighth number (and the largest one known) which can be written as the sum of the cubes of its prime factors:

119 429556097859

=

7-53-937- 6983-49199 7? + 53° + 9373 + 6983° + 49199°

(see the number 378).

130 429 015 516 800 | (= 2” -3°.5?-7?-11-13-17-19- 23-29) e the smallest number n such that a(n) > 6n: here o(n)/n = 6.017... (see the number 27 720). 170 824 677 031 250

e possibly the number n at which the quantity Q4(n) := min(A(n), A(n+1), A(n+ 2), \(n+3)) reaches its maximal value, namely approximately 1.41419: indeed, we have

Q4(170824677031250)

*

min(1.41776, 1.41419, 1.44547, 1.42225) 1.41419;

if, for each number k > 2, nz stands for the number n at which the quantity

Qz(n) := min(A(n), An + 1),...,A(m +k —-1)) reaches its maximal value, the following table gives the conjectured values of Ve for 2k

8/7 is known, although common sense indicates that there are infinitely many such numbers (see the footnote tied to the number 14018 750).

171 160 044 505 600 | (= 21! . 34.5? e the smallest solution of A n

77. 11-13-19-31)

= =

THOSE

FASCINATING

NUMBERS

aD

230 283 190 977 853

e the 15*” horse number (see the number 13).

[232 630 479 398 401 |(= 78 - 40353601) e a 6-hyperperfect number, possibly the sixth one (see the number 301).

| 248 155 780 267 521 | e the largest number 20 047 612 231 936).

n whose sum of digits is equal to ~/n (see the number

249 393 770 611 256

e the number of 16 digit prime numbers. 250 058 907 189 001 e the smallest sixth power which can be written as the sum of eight sixth powers: 250 058 907 189 001 = 251° = 8° + 12° + 30° + 788 + 102° + 138° + 165° + 246°.

274 859 381 237 761 | (= 31 -61- 73-109 - 991 - 18433)

e the smallest value of X such that C(X) > X1/3, where C(X) number of Carmichael numbers

stands for the

< X:

here C'(274 859 381 237 761) = 65019 >

274 859 381 237 7611/3 = 65018.48... number 646).

(see R.G.E. Pinch [162] as well as the

279 238 341 033 925

e the number of prime numbers < 10!°. 323 113 121114112

e the 69* insolite number (see the number 111); it is the smallest insolite number whose first three digits are different from 1. 432 749 205 173 838 | (= 2-3-7-59- 163 - 1381 - 775 807)

e the seventh Giuga number (see the number 30). 443 372 888 629 441 | (= 17- 31-41-43 - 89-97 - 167 - 331)

e the smallest known Carmichael number (R. Pinch; see R.K. Guy [101], A13).

n such that pln =>

p? — 1\n —-1

376

JEAN-MARIE

DE KONINCK

511 643 454 094 368

e the 25" powerful number 7 such that n+ 1 is also powerful: here

511 643454094368 511643 454094369

= =

2°.32 192.297 414.592 177-2417.5521

(see the number 288). 999 999 999 999 989 e the largest 15 digit prime number.

1 000 000 000 000 037 e the smallest 16 digit prime number. 1111111111114112

e the 38*” insolite number (see the number 111). 1 238 926 361 552 897 e the smallest prime factor of the Fermat number Fg = g2° 4 1, whose complete factorization is given by Fg = 1238926361552897 -93461639715357977769163558199606896584051237541638188580280321.

[1 436 697 831 295 441 |(= 11-13-19-29-31-37-41-43-71127) e the smallest Carmichael number which is the product of ten prime numbers

(see the number 41041). 2 056 364 173 794 800 e the smallest number which can be written as the sum of four seventh powers in two distinct ways, namely

2 056 364 173 794 800 = 107 + 147 + 123” + 1497 = 157 +. 907 + 1297 + 1467.

2 125 390 162 618 116 |(= 2? - 3? - 491? - 156497) e the 100000000 powerful number (see the number 3 136).

‘THOSE FASCINATING

NUMBERS

a7

2 346 318 816 620 308 | e the number of 17 digit prime numbers.

|2 360 712 083 917 682| e the third number n such that n? + 1 is powerful: here

2 360 712 083 917 6827 + 1 = 5° - 612 - 3001? - 2306865012 (see the number 682).

| 2623 557 157 654 233 e the number of prime numbers < 101”.

3 904 305 912 313 344 | (= 54°)

e the smallest number n > 1 whose sum of digits is equal to ~/n: the only other numbers n > 1 satisfying this property are 45848 500718 449031 and 150 094 635 296 999 121.

5 056 584 744 960 000 e the value of 1!-2!-...-8!.

5 315 654 681 981 355

e the 16’” horse number (see the number 13).

6 992 962 672 132 095 | (= 3-5- 17-353 - 929 - 83 623 937)

e the largest known number n for which ¢(n)|(n + 1) (see the number 65 535).

8 314 460 009 856 000 e the smallest number n = [d;, d2,...,d,] such that (dj +7) -(d2+7)-...-(d-+

7) =n; Patrick Letendre established, for 1 < ¢ < 9, the following solutions of

(dy +t)- (dp +t)-...-(dp +t) =n:

378

JEAN-MARIE

DE KONINCK

n 18 (one can prove that it is the only solution)

12, 24, 35, 56

(no known solution) 120, 315, 4752, 7744, 24960, 57915, 3386 880 50, 210, 450, 780, 1500, 3920, 16500, 91728, 269 500, 493 920, 1 293 600, 266 378 112, 317 447 424, 1277337600, 14 948 388 000, 48 697 248 600, 379 748 636 467 200 6 | 90, 840, 4320, 59 400, 60 480, 917 280, 2 419 200, 34 992 000, 3714984000, 460 522 782 720, 896 168 448 000, 2 194 698 240 000, 39 109 522 636 800, 229 419 122 688 000, 239 446 056 960 000, 650 997 662 515 200, 3 954 407 288 832 000, 182 279 345 504 256 000, 883 270 791 696 384 000 7 | 8314460 009 856 000, 31 746 120 037 632 000, 92 632 873 013 093 597 184000 000, 1 108 240 107 492 643 314 063 114 240 000

whnd or Ft

8 | (no known solution) 9 | (no known solution) 9 077 457 159 999 998

e possibly the second number n such that min(A(n), A(n + 1), A(n + 2)) > 1.7; here we have

min(A(n), \(n + 1), A(n + 2)) © min(1.83944, 1.73736, 1.70566) = 1.70566; see the number 85016574 for the smallest number n satisfying the above inequality.

|9 999 999 999 999 937 | e the largest 16 digit prime number. 10 000 000 000 000 061 e the smallest 17 digit prime number. 11111 731111111113

e the smallest odd insolite number whose digits are not only 1’s; the smallest five

numbers satisfying this property?! are 11111731 111111113, LE DISTT AIL SEL 110, UP US1127 TALIS 4111, IST LAL TIS ISL iat i.

a ian

211One can prove that, except for these five numbers, any other number satisfying this property is larger than 10?!.

‘THOSE

FASCINATING

NUMBERS

379

[11117 311111311111] e the second odd insolite number which is not of the form 11...1.

RIS Plies ety e the third odd insolite number which is not of the form 11...1.

|11 721 060 349 748 875|(= 5° - 1813 - 2513) e the 1000000" 3-powerful number (see the number 216).

12 345 678 987 654 321| e the square of 111111111.

[13 111 131117111111 | e the fourth odd insolite number which is not of the form 11... 1.

14 737 133 470 010 574 | (= 2-3-7-71- 103 - 67213 - 713 863)

e the eighth Giuga number (see the number 30). 17 111 113131111111 e the fifth odd insolite number which is not of the form 11... 1.

22 114 397 130 086 627 e the number of 18 digit prime numbers.

22 439 962 446 379 651 e the smallest prime number p such that p+ 1000 is prime and such that each

number between p and p+ 1000 is composite (see the number 396733): this prime number p was discovered in 2003 by Nyman & Nicely [154]. 26 584 448 904 822 018 e the smallest number which is not an eighth power, but which can be written as the sum of the eighth powers of some of its prime factors: here

26 584 448 904 822018 = 2-3-7-17-19- 113 - 912733109 = 2° + 178 + 113° (see the number 870); at least one more number satisfies this property, namely 210913096528905026899530575850386805453832507856329770499303938.

380

JEAN-MARIE

DE KONINCK

|45 848 500 718 449 031 |(= 71°) e the second number n > 1 whose sum of digits is equal to ~/n (see the number 3 904 305 912 313 344).

48 988 659 276 962 496 | e the smallest number

n which can be written as the sum

of two cubes in five

distinct ways:

48 988 659 276962496

= —

387879 + 365757? = 107839° + 362753° 2052923 + 342952° = 2214243 + 3365883 931518° 4331954"

(D.W. Wilson [208]; see the number 1729). 59 649 589 127 497 217 e the smallest prime factor of the Fermat number F7 = gz" 4 1, whose complete factorization is given by

F, = 59649589127497217 - 57044689200685129054721. 60 977 817 398 996 785 | (=5-7-17-19-23-37- 53-73-79 - 89 - 233) e the smallest Carmichael number which is the product of 11 prime numbers (see

the number 41 041). 99 194 853 094 755 497

e the 12’" prime Fibonacci number (see the number 89). 99 999 999 999 999 997 e the largest 17 digit prime number. 100 000 000 000 000 003

e the smallest 18 digit prime number. 107 827 277 891 825 604 e the smallest number which is not a fourth power, but which can be written as the sum of the fourth powers of some of its prime factors: here

107 827 277 891 825 604

2? .3-7-31-67- 18121 - 34105993

3* + 314 +67 + 181214; at least three other numbers satisfy this property, namely 48 698 490 414 981 559 698, 93 310 754811 505 006 990 350 670 730 and 3137 163 227 263 018 301 981 160 710 533 087 044 (see the number 870).

THOSE

FASCINATING

NUMBERS

381

130 370 767 029 135 901 | e the 17‘" horse number (see the number 13).

| 150 094 635 296 999 121 |(= 81°) e the largest number n whose sum of digits is equal to ~/n (see the number 3 904 305 912 313 344).

[201 446 503 145 165 177] e the first Sierpinski number to have been discovered (by Sierpinski in 1960): a number

k such that k - 2” + 1 is composite for each number n > 1 is called a

Sierpinski number; see the number 78 557. 209 317 712 988 603 747

e the number of 19 digit prime numbers.

212 104 218 976 916 644 | (= 2? - 7? - 1109? - 296637) e the 1000000000" powerful number (see the number 3 136). 234 057 667 276 344 607

e the number of prime numbers < 1019 (an estimate due to Marc Deléglise).

539 501 733 634 012 578 | (= 2-3" - 11-13-19? -23-31-37-41- 477) e the smallest number which is not a prime power, but which is divisible by the sum of the sixth powers of its prime factors, namely by 19184230593 =

37-11-19. 47? (see the number 378); at least three other numbers satisfy?!” this property, namely 280 128 388 470 016 293 362 568 270, 560 320 704 841 008 416 047 743 000 and 20 497 203 366 427 937 245 153 868 828 160. 550 843 391 309 130 318 | (= 2-3-7-71- 103 - 61559 - 29 133 437)

e the ninth Giuga number (see the number 30).

212T> obtain these numbers, one can proceed as follows.

of a combination of the first 25 prime numbers, set

For each number n1 which is the product

s = y(m), where m =

Ae, p®, and examine

if s|n1; if such is the case, one can conclude that the number n = n1 - m/s is such that oie. divides n.

382

JEAN-MARIE

DE KONINCK

|711 813 411 914 121 216 | e the smallest insolite number containing the maximum possible number of distinct digits, namely the digits 1, 2, 3, 4, 6, 7, 8 and 9 (see the number 111).

999 999 999 999 999 989 e the largest 18 digit prime number.

1 000 000 000 000 000 003 e the smallest 19 digit prime number.

1111111111111111111

e the second prime number of the form 11...1 (see the number 19).

1 610 006 506 595 061 125 | (= 5° - 17? - 61? - 1094417) e the second powerful number

n such that n — 1 is a perfect square,

namely

1 268 860 318” (see the number 682).

[1 986 761 935 284 574 233 | e the number of 20 digit prime numbers.

2 206 550 475 483 180 841

e the smallest sixth power which can be written as the sum of seven sixth powers:

2 206 550 475 483 180 841 = 1141° = 74°+234°+402°+474° +702° +894°+1077°. 2 220 819 602 560 918 840

e the number of prime numbers < 107° (an estimate due to Marc Deléglise).

2 305 843 008 139 952 128 | (= 2°°(231 — 1)) e the eight perfect number. 2 305 843 009 213 693 951

e the ninth Mersenne prime, namely 2°! — 1.

THOSE

FASCINATING

NUMBERS

383

|2393 703 338 691 891 312| e the smallest known

number n for which @,(n) = Pinel) B.(n +3) = B.(n +4) = B.(n +5), where 6,(n) = 6B

p, (PZ)

=

(n) = P(n) => one: p 1 whose sum of digits is equal to */n: the only other numbers satisfying this property are 19 687 440 434072 265 625, 53 861 511 409 489 970 176, 73 742 412 689 492 826 049, 179 084 769 654 285 362 176 and 480 682 838 924 478 847 449. 15 407 021 574 586 368 000 e the fifth number which is equal to the product of the factorials of its digits in base 7:

15407021 574 586.3608 000:— {3,6,4,0,4)2,4,0,3.3,0,3,2,451,6,0,3,2)2,0, lati

(see the number 248 832000). 18 446 744 073 709 551 617 e the smallest composite Fermat number; its factorization is

2?° +1 = 18 446 744 073 709 551617 = 274177 - 67280421310721.

19 687 440 434 072 265 625 | (= 851°) e the second number n > 1 whose sum of digits is equal to /n (see the number 13 744 803 133 596 058 624).

19 698 744 770 118 549 504 | (= 2°3 - 37) e perhaps the smallest number n such that y(n)*|o(n) (see the number 96). 21 127 269 486 018 731 928 e the number of prime numbers < 10?!. 32 032 215 596 496 435 569

e the smallest prime factor of the Mersenne number 2!%” — 1, whose complete factorization is given by

2137 _ 1 — 32032215596496435569 - 5439042183600204290159. 43 252 003 274 489 856 000 | (= a

)

e the number of possible permutations of the Rubik cube (3 x 3 x 3), a result

obtained by E.C. Turner & K.F. Gold [196].

THOSE

FASCINATING

NUMBERS

385

|48 698 490 414 981 559 698 | e the second number which is not a fourth power, but which can be written as the sum of the fourth powers of some of its prime factors: here

48 698 490 414 981 559 698

Ne

ecb

Dade Via

oalyfe May pctes RY eo iE Sale pS

ooo 3le:

the smallest number satisfying this property is 107 827 277 891 825 604 (see the

number 870).

53 861 511 409 489 970 176 | (= 941°) e the third number n > 1 whose sum of digits is equal to \/n (see the number 13 744 803 133 596 058 624).

73 742 412 689 492 826 049 | (= 971°) e the fourth number n > 1 whose sum of digits is equal to \/n (see the number 13 744 803 133 596 058 624).

89 726 156 799 336 363 541 e the largest left truncatable prime number whose last digit is 1 (Gerry Myerson, West Coast Number Theory Problems, 1999); see the number 73 939 133.

|

92 801 587 319 328 411 133

e the 19%" horse number (see the number 13).

99 999 999 999 999 999 989

e the largest 20 digit prime number.

100 000 000 000 000 000 039

e the smallest 21 digit prime number.

154 345 556 085 770 649 600 |(= 2)° .3°-5?-72-11-13-17-19-31- 43 - 257) e the smallest 6-perfect number, that is a number n such that a(n) = 6n (see the number 6); it is believed that there are no more than 245 such numbers (see R.K. Guy [104]).

386

JEAN-MARIE

DE KONINCK

179 084 769 654 285 362 176 |(= 106'°) e the fifth number n > 1 whose sum of digits is equal to ‘V/n (see the number 13 744 803 133 596 058 624).

[186 264 514 898 681 640 625 |(= 54 - 30517578 121) e a 4-hyperperfect number, possibly the third one (see the number 1 950 625).

|201 467 286 689 315 906 290 | e the number of prime numbers < 1072.

295 147 905 179 352 825 856|(= 2°) e the smallest power of 2 which uses all the digits from 0 to 9.

328 256 967 373 616 371 221 | (= 37! - 31381059607) e a 2-hyperperfect number, possibly the sixth one (see the number 21).

|449 177 399 146 038 697 307 | e the second narcissistic prime with more than one digit (see the number 28 116 440 335 967).

480 682 838 924 478 847 449 | (= 117!°) e the largest number n whose sum of digits is equal to 13 744 803 133 596 058 624).

/n (see the number

632 127 050 601 113 666 430 |

e possibly the second number n such that P(n)*|n and P(n + 1)4|\n + 1: here

632 127050601 113666430 632 127 050601 113666431 (see the number 11859 210). 999 999 999 999 999 999 899 e the largest 21 digit prime number.

= =

2-3°-5- 13-41-71 -2lel 7-193-2521-3691%:

THOSE

FASCINATING

NUMBERS

387

| 1000 000 000 000 000 000 117 | e the smallest 22 digit prime number.

|1 180 591 620 717 411 303 424 | e the largest known number n of the form n = 2* for which the sum of the digits is equal to k: here 27° = 1180591620717 411 303 424; the only other known

number of this type is 32 = 2°.

1357 913 579 135 791 357 913 | e the second prime number of the form 1357913579135..., that is whose digits are the odd numbers 1, 3, 5, 7 and 9, repeated: the smallest is 13. [1791 562 810 662 585 767 521 |(= 11-13-17-19-31-37-43-71-73-97-109-113-127)

e the smallest Carmichael number which is the product of 13 prime numbers (see the number 41041). |1 834 933 472 251 084 800 000 e the value of 1!-2!-...-9!.

[2 361 183 241 434 822 606 848 | (= 27!) e the largest known power of 2 which does not contain the digit 7 in its decimal

expansion (see David Gale [86]; see also the footnote tied to the number 71).

2 677 687 796 244 384 203 115 | e the 20°” horse number (see the number 13).

[3 508 125 906 290 858 798 171 | e the tenth Hamilton number (see the number 923). E 551 349 655 007 944 406 147

e the smallest prime number of the form n® + 1091, here with n = 3906 (see the number 3906). 9 999 999 999 999 999 999 973

e the largest 22 digit prime number.

388

JEAN-MARIE

DE KONINCK

[10 000 000 000 000 000 000 009| e the smallest 23 digit prime number. 11111 111111111111111111

|

e the third prime number of the form 11...1 (see the number 19). 11111111111 111111122112

e an insolite number (see the number 111). 21 000 000 000 000 000 000 001

e the fourth prime number of the form k- 10* + 1 (see the numbers 3001 and 201).

35 452 590 104 031 691 935 943| e the largest narcissistic prime number:

there exist only three narcissistic prime

numbers (see the number 28 116 440 335 967). 87 674 969 936 234 821 377 601 | (= 7-13-17-19-23-31-37-41-61-67-89-163-193-241)

e the smallest Carmichael number which is the product of 14 prime numbers (see the number 41041). 99 999 999 999 999 999 999 977 e the largest 23 digit prime number. 100 000 000 000 000 000 000 117 e the smallest 24 digit prime number. 146 361 946 186 458 562 560 000 | (= 218 .3-54-7-11-13-19-37-79- 109-157-313)

e the largest known unitary perfect number (see the number 6); the other four are 6, 60, 90 and 87 360.

244 294 249 389 674 423 328 124 e possibly the smallest number n such that

Qr(n) = min(A(n), (m+ 1),...,A(n +6) > : here

Q7(n)

2

min(1.32585, 1.17568, 1.20547, 1.19621, 1.23233, 1.16671, 1.16896) 1.66671.

THOSE

FASCINATING

NUMBERS

389

|357 686 312 646 216 567 629 137 | e the largest known left truncatable prime number (see the number 73 939 133).

|514 843 556 263 457 213 182 265 | e the integer part of ee, that is the expected size of the fourth prime factor of

an integer (see the number 1618).

[1 337 735 048 956 150 266 042 387 | e possibly the third number n for which P(n)*|n and P(n + 1)4|(n +1): here 1337 735 048 956 150 266 042387

=

3-13-109- 157-359-619 - 17334,

1 337 735 048 956 150 266042388

=

27-37-53-97-193-557- 20117;

see the number

11 859 210.

2 264 832 171 464 196 096 000 000 e the smallest number > 2 which is equal to the product of the factorials of its digits in base 9: 2 264 832 171 464 196 096 000 000 Loot OnoOwom NOamen. 5.1. 4G). 0,0)0, 0, 0,019; the only other number satisfying this property is 57101730555178822329718244269801471229938910753095152775987 20000000000000000000

(see the number 17 280 for the list of the smallest numbers with this property

in a given base). 2 512 088 574 784 743 818 066 896

e possibly the fourth number n for which P(n)*|n and P(n + 1)*|n + 1: here

2512 088 574 784 743 818 066 896 2512088 574 784 743 818066897 see the number

Detigael 1 = 119% 107e Did 193-197-269 -389 -397 -11237;

=

11 859 210.

8 230 545 258 248 091 551 205 888

e the smallest known number (found by D.W. Wilson [{208]) which can be written as the sum of two cubes in six distinct ways: 8 230 545 258 248 091 551 205 888

= = = (see the number 1729).

11 239 317° + 201 891 435° 17 781 264? + 201 857 064? 63 273 192° + 199 810 080° 85970916? + 196567 548? 125436328° + 184269296° 159363450° + 161127942°

390

JEAN-MARIE

DE KONINCK

113 423 713 055 421 844 361 000 443

e the eighth voracious number (see the number 1 807). 244 197 000 982 499 715 087 866 346 |(= 2-311

2331547137228 282147)

e the tenth Giuga number (see the number 30).

|453 694 852 221 687 377 444 001 769 | e the smallest perfect square which can be written as the sum of 2, 3, 4, 5, 6 and 7 squares:

453 694 852 221 687 377 444 001 769 = 21300 113 901 6137 = 21300 113 901 612? + 6526 8857 = 21300 113901 612? + 6 526 884” = 21300113 901 612? + 6526 884? = 21300 113 901 6127 + 6526 884? = 21 300 113 901 612? + 6526 8847 = 21300 143.901.0612" + 6526 884~

+ + + + +

3613? 3612? + 857 36127 + 84? + 13? 36127 + 84? 4+ 127 + 5? 36124 S47 1 27 as

ae

618 970 019 642 690 137 449 562 111 e the tenth Mersenne prime, namely 2°° — 1.

6 658 606 584 104 736 522 240 000 000 e the value of 1!-2!-...-10!.

[27 134 923 845 424 074 797 548 044 288 | e the seventh (and possibly the largest) number which is equal to the product of the factorials of its digits in base 5: 27 134 923 845 424 074 797 548 044 288 =| 2A 2A 68) 2525 1, 0)2, SyOpoyayAnSetls 0/8 53, 485 N24 aeOnos)

4,2,4,4,4,0,4,1,2,3]5 (see the number 144).

34111 227 434 420 791 224 041 472 000| (= 2?” .3°.53.7-11-132-19-29-31AB 6 lia e127) e the second 6-perfect number: n is 6-perfect if o(n) = 6n (see R.K. Guy [104], B2).

THOSE

FASCINATING

NUMBERS

391

|93 310 754 811 505 006 990 350 670 730| e the third number which is not a fourth power, but which can be written as the sum of the fourth powers of some of its prime factors: here

93 310 754 811 505 006 990 350 670 730 ee

Ole

3,- le Of LOS =.37311 93,2.8609

-2045107145539 - 2218209705651794191 = 2* + 1034 + 3734 + 11934 + 2045107145539; the smallest number satisfying this property is 107 827 277 891 825 604 (see the

number 870). 241 573 142 393 627 673 576 957 439 049 e the smallest prime factor of the number SS 11...1; indeed, we have 71

107-1 7

=

times

241573142393627673576957439049 -45994811347886846310221728895223034301839,

a factorization first obtained by Davis and Holdridge in 1984, using the quadratic sieve due to Pomerance.

554 079 914 617 070 801 288 578 559 178 |(= 2-3-11-23-31-47 059-2 259 696 349110 725 121051) e the 11%” (and largest known) Guiga number (see the number 30). 162 259 276 829 213 363 391 578 010 288 127 e the 11°” Mersenne prime, namely 2!9 — 1.

14 497 650 943 439 560 735 142 707 200 000 000

e the fifth number which is equal to the product of the factorials of its digits in base 12: ’ 14 497 650 943 439 560 735 142 707 200 000 000 = [5, 1,0, 10, 8, 9,9, 9,6, 4,0, 0,0, 0,0, 0,0,0,0,0,0,0,0,0,0,0,0, 0,0, 0,0, OJ12

(see the number 21772 800). 130 547 383 608 518 581 304 037 589 860 381 057

e the smallest known number n such that Sry 378).

divides n (see the number

392

JEAN-MARIE

DE KONINCK

[2 658 455 991 569 831 744 654 692 615 953 842 176 |(= 2°(2' — 1)) e the ninth even perfect number. |3 137 163 227 263 018 301 981 160 710 533 087 044|

e the fourth number which is not a fourth power but which can be written as the sum of the fourth powers of some of its prime factors: here

3 137 163 227 263 018 301 981 160 710 533 087 044 — 2?.32.7.11-191 - 283 - 7541 - 1330865843 - 2086223663996743 = 344+ 74 +1914 + 1330865843+: the smallest number satisfying this property is 107 827 277 891 825 604. |115 132 219 018 763 992 565 095 597 973 971 522 401

e the largest narcissistic number (D. Winter, 1985); see the number 88. |170 141 183 460 469 231 731 687 303 715 884 105 727 |

e the 12*” Mersenne prime, namely 2!27 — 1.

[340 282 366 920 938 463 463 374 607 431 768 211 457 |(=2? +1) e the eighth Fermat number: and Brillhart:

its factorization was obtained in 1974 by Morrison

2?" + 1 = 59649589 127 497 217 - 5 704 689 200 685 129 054721.

437 489 361 912 143 559 513 287 483 711 091 603 378

e the smallest known number n such that P(n)°|n and P(n + 1)°|(n +1): here

437 489 361 912 143 559 513 287 = 2-39.77. 11? - 67-1151 437 489 361 912 143 559 513 287 = 43 - 61-107 - 269 - 421 -

483 711 091 603 - 1439 - 1609 483 711 091 603 617 - 653 - 2689

378 2557 - 4957°, 379 - 6619°:

the numbers 4402074374845013694517762402276831087215, 227489926 15102631934745928628382078239867, 2954120615478394653060385065220308608058260, 8250351204235843413274102593592289950249874, 506676284977787252656476827071452300208916227 and 4556139057327711835147225814446398763593460517 also satisfy this property

(see the number 6 859).

THOSE

FASCINATING

NUMBERS

393

[1770 019 255 373 287 038 727 484 868 192 109 228 823 e possibly the smallest number n such that

f(n +1) = f(n+2) =...=f(n +8), where f(n) stands for the product of the exponents in the factorization of n: here the common value of f(n +7) is 6 (see the number 843). |1 831 607 359 566 125 048 834 492 989 440 000 000 000 |

e the sixth number which is equal to the product of the factorials of its digits in base 7: 1831 607 359 566 125 048 834 492 989 440 000 000 000 =A OOM oe wl eO. lyase dua03,0,2, dn0,0: 10,2, 1.2, 1. 253;-55 25 4,0, 4, 4, 5, 6,0, 6, 2,0, 0,2, 2,3, 4]7 (see the number 248 832 000). 6 153 473 687 096 578 758 448 522 809 275 077 520 433 167

e the 11’” Hamilton number (see the number 923).

20 988 936 657 440 586 486 151 264 256 610 222 593 863 921 |(= (2!48 + 1)/17) e the largest known prime found before the computer era (some times called the

Ferrier number), namely in 1952 by A. Ferrier (see H.C. Williams [204]). 75 445 311 584 829 283 999 739 123 702 169 600 000 000 000 |

e the sixth number which is equal to the product of the factorials of its digits in base 12: 75 445 311 584 829 283 999 739 123 702 169 600 000 000 000 = [Bibel 2, Oslo OneS,0,4,11, 11.4.1, 2.2.8,0,0,0,.0; 0,0,0,,0,.0,.0,0,0,0,0, 0,0,0,0,0,0,0,0,0]12 (see the number 21772800). 208 492 413 443 704 093 346 554 910 065 262 730 566 475 781

e the fifth (and largest known) prime of the form 11 + 2? +...+n”, here with

n = 30 (see the number 3413). 273 457 513 497 334 816 890 950 735 729 000 448 000 000 000 000

e the seventh number which is equal to the product of the factorials of its digits in base 7: 273 457 513 497 334 816 890 950 735 729 000 448 000 000 000 000 = (1, 2, 0, 2, 2, 0,3, 3, 3, 4, 2, 3, 5, 5, 6, 5, 6, 2, 5, 1,0, 1,0, 5, 1, 1, 4, 2,6, 5,6, 3, 4, 4, 6024620, 0n0etl non0920223, 004,050 Oro. 1, 4|7

(see the number 248 832000).

394

JEAN-MARIE

DE KONINCK

962 493 562 543 459 590 626 671 870 630 428 672 000 000 000 000 |

e the seventh number which is equal to the product of the factorials of its digits in base 12: 962 493 562 543 459 590 626 671 870 630 428 672 000 000 000 000 0,0,0705.0; =o ie OvOnte ll. ls: ti. 7, 6,1,.0) 902,070,050, 0,0; 0,.0;,0, 0,0,0,0,0,0,0,0,0,0,0,0, 0,0, 0]12 (see the number 21772800). 12 864 938 683 278 671 740 537 145 998 360 961 546 653 259 485 195 807 |

e the ninth voracious number (see the number 1 807). [191 561 942 608 236 107 294 793 378 084

(= 25(27 — 1)

303 638 130 997 321 548 169 216 |

e the tenth perfect number. 27 418 521 963 671 501 273 905 190 135 082 692 041 730 405 303 870 249 023 209

(23

ee hls de lye ale Ag A

Ads

00"

a)

e the smallest cube whose sum of divisors is also a cube (Rubin): 0(27418521963671501273905190135082692041730405303870249023209)

= 65400948817364742403487616930512213536407552000000000000000 = 40289760243532800000°. 45 883 517 654 351 824 863 158 584 663 538 863 253 527 461 888 000 000 000 000 000 e the eighth number which is equal to the product of the factorials of its digits in base 12: 45 883 517 654 351 824 863 158 584 663 538 863 253 527 461 888 000 000 000 000 000 ==) LA 10s 70.651, O14, 6. 4,0, t,o, Sls UL, Or ort, o.6,0 ONOgOs 0, Unni. 0.

0;,0;:0;0; 0;,0,0,,,0;/0,.0;.0;, 0, 0,,0,,0, 0,0; 0,'0;,0;,0, 0,0,,0;.0,, 0)15

(see the number 21772800). 210 913 096 528 905 026 899 530 575 850 386 805 453 832 507 856 329 770 499 303 938

(=2-3-7-11-17-19-37- 243871 - 61732369 -2537372468554462665091597215251362804089751) e the largest known number which is not an eighth power, but which can be written as the sum of the eighth powers of some of its prime factors: indeed,

210913096528905026899530575850386805453832507 8563297 70499303938

= 2° + 17° + 61732369° (see the number 870).

THOSE

eeu

FASCINATING

NUMBERS

395

1) |

e the 11" perfect number, a 65 digit number.

69 113 789 582 492 712 943 486 800 506 462 734 562 847 413 501 952 000 000 000 000 001

e the largest known prime of the form 1!-2!-3!-...-r!+1, here with r = 14 (see

the number 125 411328001).

52!4+52+4+1 e the largest known prime of the form n!+n-+ 1, in this case a 68 digit number; the only known numbers n such that n!+n-+1 is prime are n = 2, 4, 6, 10 and

52 (see the number 52). 37 032 592 805 942 775 592 027 297 064 629 098 681 015 432 812 873 314 981 288 646 788 880 382 401

e the largest prime factor of 7979 + 1, whose complete factorization?! is given by 797° 41

=

2*.5.34919188021 - 45780868646549 -918150595356645610443476284621975471619075398615249 - Pra.

6 086 555 670 238 378 989 670 371 734 243 169 622 657 830 773 351 885 970 528 324 860 512 791 691 264

e the largest known sublime number: this number is sublime because n = 2!2°(2°!— 1)(29+ — 1)(219 — 1)(2" — 1)(2° — 1)(23 — 1), and therefore one can easily check that t(n) = 26(27 — 1) and o(n) = 2176(2127 — 1) are both perfect: the only other known sublime number is 12. Dize (Qtet —

1)

e the 12%” perfect number, a 77 digit number.

213T 9 obtain this factorization, one can use a technique that allows to factor numbers of the form

n”™ +1 when n = 3 (mod 4), which is indeed the case for n = 79. Indeed, using MATHEMATICA and the feature Factor[((n * x7)” + 1)/(n * 2? + 1)], one obtains two polynomials (in x) both of degree n — 1. Then, setting 2 = 1, two non trivial factors of n™ + 1 appear. In the case n = 79, the second of these factors is precisely the largest prime factor of 7979 +1. When n=2 (mod 4), the

same approach works, but this time by writing Factor [((n* x2)” + 1)/(n? x4 +1)], in which case one gets two polynomials of degree n — 2.

396

JEAN-MARIE

DE KONINCK

gq 44 e the ninth Fermat number, a 78 digit number; it is a composite number, whose factorization was obtained in 1980 by Brent and Pollard:

2° 44

=

1238926361552897 .9346163971535797776916355819960689658405 1237541638188580280321.

398 075 086 424 064 937 397 125 500 550 386 491 199 064 362 342 526 708 406 385 189 575 946 388 957 261 768 583 317

e the smallest prime factor of the RSA-576 number, a 174 digit number which no one could factor, until December 2003; this number was finally factored by Jens Franke, who obtained that the number 188198812920607963838697239461650439807 1635633794173827007633 564229888597 152346654853 190606065047430453173880113033967161 99692321205734031879550656996221305168759307650257059 is the product of the two prime numbers 398075086424064937397125500550386491199064362342526708406 38518957594638895 7261768583317 and 472772146107435302536223071973048224632914695302097116459 852171130520711256363590397527.

e the tenth Fermat number, a 155 digit number:

it is a composite number, and

its smallest prime factor (found by Western in 1903) is

2.424 833 = 37-216 +1.

e the 13” Mersenne prime, a 157 digit number.

e the 14” Mersenne prime, a 183 digit number.

g10

208 eA e the 11°" Fermat number, a 309 digit number: smallest prime factor, namely

45592577 — hls

it is a composite number, whose

etale

THOSE

FASCINATING

NUMBERS

397

was obtained by Selfridge in 1953, while the factor

6 487 031 809 = 395 937-2144 1, was obtained by Brillhart in 1962 (see the number 70525 124 609).

| 2520 (2521 _ 4) | e the 13”” even perfect number, a 314 digit number.

e the fourth prime number all of whose digits are 1 (see the number 19).

2606 Caw

2, 1)

e the 14°” even perfect number, a 366 digit number.

e the 15” Mersenne prime, a 386 digit number.

e the 12” Fermat number, a 617 digit number; which two of its prime factors, namely

319489 = 39-241

and

it is a composite number,

974849=119-2'3+41,

were obtained by Cunningham in 1899 (see the number 70 525 124609).

e the 16°” Mersenne prime, a 664 digit number.

e the 17°” Mersenne prime, a 687 digit number. 91 ih)

pte 279

_-

1)

e the 15*” even perfect number, a 770 digit number.

for

398

JEAN-MARIE

DE KONINCK

e the 18°” Mersenne prime, a 969 digit number. ues ah

Ss

1031

e the fifth prime number all of whose digits are 1 (see the number 19).

e the 13°” Fermat number, a 1234 digit number; which four of its prime factors are

q, = 114689 = 7-214 +1, g3 = 63766529 = 973-21 +1

and

it is a composite number for

g2 = 26017793 = 397 - 2°° + 1, q4 = 190274191361 = 11613415 -2'4 +1;

q1 was obtained by Pervouchine and Lucas in 1877, gz and q3 were obtained by Western in 1903, while qs was discovered by Hallyburton and Brillhart in 1974, so that nz,

27° +1=q1- 42°93" 94° C1202, where C} 292 is a composite 1 202 digit number (see the number 70525 124 609).

e the 19%” Mersenne prime, a 1 281 digit number.

22 202 (22 203

_

1)

e the 16” even perfect number, a 1327 digit number.

e the 20’” Mersenne prime, a 1 332 digit number.

22 280 (Oe Pfeil

1)

e the 17°” even perfect number, a 1373 digit number. 23 216 ey 217 _

1)

e the 18” even perfect number, a 1937 digit number.

THOSE

FASCINATING

NUMBERS

399

gis

? ae aa e the 14°” Fermat number, a 2467 digit number; it is a composite number, whose smallest prime factor, namely

2710 954 639 361 = 41 365 885 - 276 + 1, was obtained by Hallyburton and Brillhart in 1963

(see the number 70 525 124 609). 24 252 (26 253

1)

e the 19” even perfect number, a 2561 digit number.

24 A22 (23 423

__ 1)

e the 20” even perfect number, a 2663 digit number.

29 689

1

e the 21"%* Mersenne prime, a 2917 digit number. 99941 _ 4

e the 22Ӣ Mersenne prime, a 2993 digit number.

gil 213

_

1

e the 23” Mersenne prime, a 3376 digit number. 22°

as 1

e the 15*” Fermat number, a 4933 digit number; it is a composite number (Self-

ridge and Hurwitz, 1963), of which no prime factor is known as of today. 29 688 (29689 _ 1) |

ns

e the 217** even perfect number, a 5834 digit number. 29940 (99941 _ 1)| e the 22”% even perfect number, a 5985 digit number.

400

JEAN-MARIE

DE KONINCK

e the 24*” Mersenne prime, a 6002 digit number. 921701 _ 4

e the 25’” Mersenne prime, a 6533 digit number.

gil 212 (2*1 213

_

1)

e the 237¢ even perfect number, a 6751 digit number.

e the 26" Mersenne prime, a 6 987 digit number.

gis

Pid

ere A

e the 16" Fermat number, a 9 865 digit number; it is a composite number, whose smallest prime factor, namely

1214251009 = 579-271 +1, was obtained by Kraitchik in 1925 (see the number 70525 124609). 219936 (919937 _ 4) e the 24°” even perfect number, a 12003 digit number.

221 700(221701 _ 4) e the 25’” even perfect number, a 13066 digit number.

e the 27°" Mersenne prime, a 13395 digit number. 228 208 (923 209 _ 1)

e the 26" even perfect number, a 13973 digit number.

THOSE

7

216

FASCINATING

NUMBERS

401

ae

e the 17’" Fermat number, a 19729 digit number; whose smallest prime factor

it is a composite number,

825 753601 = 1575-219 +1 was obtained by Selfridge in 1953;

e the smallest counter example to the conjecture of F. Eisenstein (1823-1852) according to which each number of the form 2

oe is prime:

ta

observe that the four smallest numbers of this form are 2+ 1 = 3, 2

2741=5, 27 4+1=17 and 2” +1 = 65537 (see the number 70525 124 609).

e the 28°" Mersenne prime, a 25962 digit number.

244 496 (244497 _ 4) e the 27°" even perfect number, a 26790 digit number.

318 032 361 - 2107001 _ 4 e the first member of the largest known twin prime pair (Underbakke & Carmody,

2001): it is a 32220 digit number. gii0 503 __ 1

e the 29°” Mersenne prime, a 33 265 digit number.

e the 18%? Fermat number, a 39457 digit number; it is composite and its smallest prime factor,

31 065 037 602 817 = 59251 857- 2'9 + 1, was found by Gostin in 1980 (see the number 70 525 124 609). 9132 049 _

1

e the 30¢” Mersenne prime, a 39 751 digit number.

402

JEAN-MARIE

DE KONINCK

lL §sol

SS)

49 081

e the sixth prime number whose digits are only 1’s (see the number 19).

286 242 (ee 243

1)

e the 28'” even perfect number, a 51924 digit number. 9216091 _ 4

e the 31”%* Mersenne prime, a 65050 digit number.

9110 502 (2ai0 503

_

)

1)

e the 29% even perfect number, a 66530 digit number.

18 22 i

a

e the 19” Fermat number, a 78914 digit number; it is composite and its smallest

prime factor, discovered by Western in 1903, is 5242881 = 5-279 +1 (see the number 70525 124609).

26241927024

ed

e the largest known Cullen prime number, a 79002 digit number (Darren Smith,

1998) (see the number 141). 2132 Des (2132 049 _

1)

e the 30%" even perfect number, a 79502 digit number. i Gara

Ss

86 453

e the seventh prime number whose digits are only 1’s (see the number 19). Ove 2303 093 ao

e the largest known prime of the form k - 2” + 1, a 91241 digit number (Jeffrey

Young, 1998).?14

214These numbers are very useful in the search for prime factors of Fermat numbers, because it is

well known (since Euler) that each prime factor of 22” + 1 is of the form k- 2'+2 + 1.

THOSE

FASCINATING

NUMBERS

403

(lybecome

—— 109

297

e the eighth prime number whose digits are only 1’s (see the number 19). 2216 0902216091 _ 1)| e the 31%’ even perfect number, a 130100 digit number.

22°

ve i

e the 20°" Fermat number, a 157827 digit number; it is a composite number, two of its prime factors are known, namely 70525124609 = 33629-27441

(discovered by Riesel in 1962) and 646 730 219521 = 308 385-27! +1 (discovered by Wrathall in 1963); see the number 70 525 124 609. 9756839 _ 7 e the 32”4 Mersenne prime, a 227 832 digit number.

9859 433

__

1

e the 33% Mersenne prime, a 258 716 digit number.

1G eH!

SSS

270 343

e the ninth prime number whose digits are only 1’s (see the number 19).

92

20 a

=

e the 217% Fermat number,

a 315653 digit number;

it is a composite number

(see the number 70525 124 609). Qi 257 787 __ :

e the 34¢” Mersenne prime, a 378 632 digit number. pa 398 269 _ E

e the 35¢” Mersenne prime, a 420921 digit number (see the number 1 398 269).

oS04

JEAN-MARIE

DE KONINCK

eae 838 (2756 839 _ 4)

e the 322 even perfect number, a 455663 digit number.

9859 A32 (eee A433 __ 1)

e the 337% even perfect number, a 517430 digit number.

227?

A: 1

e the 22Ӣ Fermat number, a 631 306 digit number; it is composite and its smallest prime factor, discovered by Wrathall in 1963, is

4 485 296 422 913 = 534689 - 27341 (see the number 70525 124609).

a91 257 786

(21 257787 __ 1) |

e the 34°” even perfect number, a 757 263 digit number.

Qi 398 283 (24 398 269

_

i)

e the 35*” even perfect number, a 841 842 digit number.

22 976221

_

1

e the 36%” Mersenne prime, a 895 932 digit number (see the number 2976 221).

93 021377

_

1

e the 37°" Mersenne prime, a 909525 digit number (see the number 3021 3a) 22”? ‘i 1

e the 2374 Fermat number, a 1 262612 digit number; it was proven to be composite by Crandall, Doenias, Norrie and Young in 1995 (see the number 70525 124 609)

22 976 220 (22 976221

_

1)

e the 36” even perfect number, a 1791864 digit number.

THOSE

93021 376 (23021 S 7a

FASCINATING

NUMBERS

405

1) |

e the 37" even perfect number, a 1819050 digit number. 96972593 _ 4

e the 38°" Mersenne prime, a 2098 960 digit number (see the number 6 972593). Dan 44

e the 24'" Fermat number, a 2525 223 digit number; it is composite and its largest prime factor, discovered by Pervouchine in 1878, is 167772 161 = 5-27°+1 (see

the number 70525 124609). 913466917 _

e the 39” Mersenne prime, a 4053 946 digit number (see the number 13 466 917). 26 972 592 (oe 972593

_

1)

e the 38” even perfect number, a 4197919 digit number.

224

2am

aL

e the 25” Fermat number, a 5050446 digit number; it was established in 1999

that it was a composite number (see the number 70525 124609). 220996011 _

e the 40°” Mersenne prime, a 6320 430 digit number (see the number 20 996 011). 924036183 _

e the 417% Mersenne prime, a 7 235 733 digit number (see the number 24 036 183). 225 964951

_

1

e the 42”4¢ Mersenne prime, a 7 816 230 digit number (see the number 25 964 951). gis 466 216(248 466917

_

1)

e the 39°” even perfect number, a 8107891 digit number.

aN06

JEAN-MARIE

930 402457

_

DE KONINCK

1

e the 43™¢ Mersenne prime, a 9 152052 digit number (see the number 30 402 457).

232 582657

_

1

e the 44*” Mersenne prime, a 9 808 358 digit number (see the number 32 582657).

237 156667

__ 1

e the 45%” Mersenne prime, a 11 185 272 digit number (see the number 37 156 667).

220 996 SEES

996011

_

1)

e the 40°” even perfect number, a 12640858 digit number.

243 112609

_

1

e the 46°” Mersenne prime, a 12 978 189 digit number (see the number 43 112 609).

924 036 2821224 036183

_

1)

e the 417%’ even perfect number, a 14471 465 digit number.

225 964 950 (22 964951

_

1)

e the 42Ӣ even perfect number, a 15632458 digit number.

930 402 456 (220 402457

_

1)

e the 43"¢ even perfect number, a 18304 103 digit number.

932 582 656 (222 582657

_

1)

e the 44‘” even perfect number, a 19616714 digit number. 937 156 666 (238 156667

_

1)

e the 45" even perfect number, a 22370543 digit number.

THOSE

FASCINATING

NUMBERS

407

Be 112.608 (43112609 _ 4) | e the 46" even perfect number, a 25956 377 digit number. 3 . 2402653211 _ 3

e the number of steps required for the Goodstein sequence starting at 4 to reach 0: this is a 121210695 digit number; in 1944, the English logician R.L. Goodstein introduced an algorithm to generate sequences of positive integers which, contrary to what we may think, converge to 0; to describe the process put forth by Goodstein, we introduce the notion of “complete representation” of a positive integer in base 6: first write this number as a sum of multiples of powers of b, and then do the same with the exponents found in this sum, then the exponents of these exponents, and so on, until the representation becomes stable;

for example, the complete representation in base 2 of 266 (= 28 + 2° + 2!) is g2rt 4 9241 +21; thus, the Goodstein process for the number 3 is the following:

a)

ok a pe a eee eae 1

—

1-44-1=3=3.4°

=

3.-5°9-1=2=2-5°

—

2-6°9-1=1=1-6°

==S

lls

Sap

Goodstein [94] proved in 1944 that every Goodstein sequence converges to 0; this fact is somewhat counter intuitive. Indeed, although the sequence obtained

by beginning with 3 converges rapidly to 0 (in only five steps), a very different situation occurs by starting at 4, since the number k of steps required by the

Goodstein process to bring 4 down to 0 is k = 3 - 240763211 _ 3 (L. Kirby and Jo Paris (119))2*")

101°

1034

e the Skewes number;

this number

occupies an important place in the history

of the function 7(a#): indeed, in 1933, assuming the Riemann Hypothesis, Skewes proved that the smallest number zo for which m(x%9) > Li(xo) satisfies Zp < 191. this result was at that time very significant, since many great mathematicians, Gauss and Riemann being two of them, believed that

n(x) < Li(z) for all x > 2, an inequality which can be verified for all x < 10*°, but which is not always true; indeed, Littlewood proved in 1923 that the difference 7(x) —Li(a) changes signs infinitely often; in fact, he proved more, namely that there exists an increasing sequence of real numbers 20,71, %2,... tending 215Kirby and Paris established that the very slow convergence of the Goodstein sequences to 0 is related to the Goodstein Theorem which cannot be proved within the setup of elementary arithmetic.

408

JEAN-MARIE

DE KONINCK

to™--oovand' such that for 1 = 0,1, 2, 2... :

m(2n41) —Li(tang1) :

haces (Xan)oe — Li(tan)

>


Li(a) is not known; interestingly, these past years, the size of x, has been gradually narrowed down: il in 1933, Skewes shows that, assuming the Riemann Hypothesis, Ge, SA

910

1034

;

ST,

in 1955, Skewes shows that, without any hypothesis, z, Li(z); . in 2000, C. Carter & R.H. Hudson [30] show that 2(ax) > Li(z) for a certain number z close to 1.39 - 10?!®.

THOSE

FASCINATING

409

NUMBERS

Appendix The prime numbers

2 31 73 127 179 233 283 353 419 467 547 607 661 739 811 877 947 1019 1087 1153 1229 1297 1381 1453 1523 1597 1663 1741 1823 1901 1993 2063 2131 2221 2293 2371 2437 2539 2621

3 37 io 131 181 239 293 359 421 479 507 613 673 743 821 881 953 1021 1091 1163 1231 1301 1399 1459 1531 1601 1667 1747 1831 1907 1997 2069 2137 2237 2297 2377 2441 2543 2633

5 Al 83 137 191 241 307 367 431 A487 563 617 677 751 823 883 967 1031 1093 rel 1237 1303 1409 1471 1543 1607 1669 1753 1847 1913 1999 2081 2141 2239 2309 2381 2447 2549 2647

ic 43 89 139 193 251 311 373 433 49] 569 619 683 757 827 887 971 1033 1097 1181 1249 1307 1423 1481 1549 1609 1693 1759 1861 1931 2003 2083 2143 2243 2311 2383 2459 2551 2657

11 AT o7 149 LO 257 313 379 439 499 571 631 691 761 829 907 LE 1039 1103 1187 1259 1319 1427 1483 1553 1613 1697 ALArarg 1867 1933 2011 2087 2153 2251 2333 2389 2467 2057 2659

13 53 101 151 199 263 317 383 443 503 577 641 701 769 839 911 983 1049 1109 1193 1277 1321 1429 1487 1559 1619 1699 1783 1871 1949 2017 2089 2161 2267 2339 2393 2473 2579 2663

< 10000

17 59 103 157 211 269 331 389 449 509 587 643 709 773 853 919 991 1051 ey 1201 1279 1327 1433 1489 1567 1621 1709 1787 1873 1951 2027 2099 2179 2269 2341 2399 2477 2591 2671

19 61 107 163 223 2A 337 397 457 521 593 647 (aes) 787 857 929 997 1061 1123 1213 1283 1361 1439 1493 1571 1627 1721 1789 1877 1973 2029 PaaS 2203 2273 2347 2411 2503 2593 2677

23 67 109 167 221 200 347 AQ] 461 523 599 653 727 797 859 937 1009 1063 1129 1217 1289 1367 1447 1499 1579 1637 1723 1801 1879 1979 2039 2113 2207 2281 2351 2417 2521 2609 2683

29 71 113 173 229 281 349 409 463 541 601 659 733 809 863 941 1013 1069 1151 1223 1291 1373 1451 1511 1583 1657 1733 1811 1889 1987 2053 2129 2213

2287, 2357 2423 2531 2617 2687

JEAN-MARIE

410

2689 2749 2833 2909 3001 3083 3187 3259 3343 3433 3017 3581 3659 3733 3823 3911 4001 4073 4153 A241 4327 4421 4507 4591 4663 A759 4861 4943 5009 5099 5189 5281 5393 5449 5027 5641 5701 5801 5861 5953 6067

2693 2753 2837 297 3011 3089 3191 3271 3347 3449 3027 3083 3671 3739 3833 3917 4003 4079 4157 A243 4337 4423 4513 4597 4673 4783 A871 4951 5011 5101 5197 5297 5399 5471 5031 5647 OTL 5807 5867 5981 6073

2699 2767 2843 2927 3019 3109 3203 3299 3359 3457 3529 3093 3673 3761 3847 3919 4007 4091 4159 4253 4339 4441 4517 4603 4679 A787 A877 4957 5021 5107 5209 5303 5407 5477 5057 5651 5717 5813 5869 5987 6079

2707 INTE 2851 2939 3023 3119 3209 3301 3361 3461 3533 3607 3677 3767 3851 3923 4013 4093 A177 4259 4349 4447 4519 4621 4691 4789 4889 4967 5023 5113 5227 5309 5413 5479 5063 5653 5737 5821 5879 6007 6089

Page 2789 2857 2953 3037 3121 3217 3307 3371 3463 3939 3613 3691 3769 3853 3929 4019 4099 4201 A261 4357 4451 4523 4637 4703 A793 4903 4969 5039 5119 5231 5323 5417 5483 5569 5657 5741 5827 5881 6011 6091

DE KONINCK

2713 2091 2861 2957 3041 3137 3221 3313 3373 3467 3041 3617 3697 3779 3863 3931 4021 A111 A211 4271 4363 4457 4547 4639 A721 4799 4909 4973 5051 5147 5233 5333 5419 5501 5973 5659 5743 5839 5897 6029 6101

2719 21 G 2879 2963 3049 3163 3229 3319 3389 3469 3547 3623 3701 3793 3877 3943 4027 4127 4217 4273 4373 4463 4549 4643 4723 4801 4919 4987 5059 5153 5237 5347 5431 5503 5581 5669 5749 5843 5903 6037 6113

2129 2801 2887 2969 3061 3167 3251 3323 3391 3491 3557 3631 3709 3797 3881 3947 4049 4129 4219 4283 4391 4481 4561 4649 4729 4813 4931 4993 5077 5167 5261 5351 5437 5507 5091 5683 5779 5849 5923 6043 6121

2731 2803 2897 2971 3067 3169 3253 3329 3407 3499 3559 3637 3719 3803 3889 3967 A051 4133 4229 A289 4397 4483 4567 4651 4733 4817 4933 4999 5081 5171 5273 5381 5441 5519 5623 5689 5783 5851 5927 6047 6131

2741 2819 2903 2999 3079 3181 3257 3331 3413 doll 3071 3643 3120 3821 3907 3989 4057 4139 4231 A297 4409 4493 4583 4657 A751 4831 4937 5003 5087 5179 5279 5387 5443 5521 5639 5693 oot 5857 5939 6053 6133

THOSE

6143 6229 6311 6373 6481 6577 6679 6763 6841 6947 7001 1109 (211 7307 7417 7907 7973 7649 C27 7841 7927 8039 8117 8221 8293 8389 8513 8599 8681 8747 8837 8933 9013 9127 9203 9293 9391 9461 9539 9643 9739 9817 9901

6151 6247 6317 6379 6491 6581 6689 6779 6857 6949 7013 7121 7213 7309 7433 7517 OMG 7669 7741 7853 7933 8053 8123 8231 8297 8419 8521 8609 8689 8753 8839 8941 9029 9133 9209 9311 9397 9463 9547 9649 9743 9829 9907

6163 6257 6323 6389 6521 6599 6691 6781 6863 6959 7019 2% 7219 7321 7451 7523 7583 7673 7753 7867 7937 8059 8147 8233 8311 8423 8527 8623 8693 8761 8849 8951 9041 9137 9221 9319 9403 9467 9551 9661 9749 9833 9923

6173 6263 6329 6397 6529 6607 6701 6791 6869 6961 7027 7129 1229 7331 7457 7529 7589 7681 7757 7873 7949 8069 8161 8237 8317 8429 8537 8627 8699 8779 8861 8963 9043 9151 9227 9323 9413 9473 9587 9677 9767 9839 9929

FASCINATING

6197 6269 6337 6421 6547 6619 6703 6793 6871 6967 7039 7151 7237 7333 7459 7537 7591 7687 7759 (3 7951 8081 8167 8243 8329 8431 8539 8629 8707 8783 8863 8969 9049 9157 9239 9337 9419 9479 9601 9679 9769 9851 9931

6199 6271 6343 6427 6551 6637 6709 6803 6883 6971 7043 (key) 7243 7349 TAT7 7541 7603 7691 7789 7879 7963 8087 8171 8263 8353 8443 8543 8641 8713 8803 8867 8971 9059 9161 9241 9341 9421 9491 9613 9689 9781 9857 9941

NUMBERS

6203 6277 6353 6449 6553 6653 6719 6823 6899 6977 7057 (awe 7247 7351 7481 7547 7607 7699 7793 7883 7993 8089 8179 8269 8363 8447 8563 8647 8719 8807 8887 8999 9067 9173 9257 9343 9431 9497 9619 9697 9787 9859 9949

6211 6287 6359 6451 6563 6659 6733 6827 6907 6983 7069 7187 7253 7369 7487 7549 7621 7703 7817 7901 8009 8093 8191 8273 8369 8461 8573 8663 8731 8819 8893 9001 9091 9181 9277 9349 9433 9511 9623 9719 ROM 9871 9967

All

6217 6299 6361 6469 6569 6661 6737 6829 6911 6991 7079 7193 7283 7393 7489 7559 7639 TLE 7823 7907 8011 8101 8209 8287 8377 8467 8581 8669 8737 8821 8923 9007 9103 9187 9281 9371 9437 9521 9629 9721 9803 9883 9973

6221 6301 6367 6473 6571 6673 6761 6833 6917 6997 7103 7207 7297 7411 7499 7561 7643 7723 7829 7919 8017 8111 8219 8291 8387 8501 8597 8677 8741 8831 8929 9011 9109 9199 9283 9377 9439 9533 , 9631 9733 9811 9887

i?

Sy

=

ua

tH air

a

cae

ie

ks ad

ae

Tia? Js

ink

:

Ree

are tno pe

kb

443

5

-

Cres

_

ra

imp ron san 5

a

Gr z

«bet

te

hd

Abe. 4

rs

_

re ae

; Say.

a

ae

ae

rss

&@ re -

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.

Ne oy Hi

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wi4r ite

.

wins d

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2g

THOSE

FASCINATING

NUMBERS

413

Bibliography

[1]

T. Agoh, K. Dilcher & L.Skula, Fermat Number Theory 66 (1997), no. 1, 29-50.

[2|

T. Agoh, K. Dilcher & L. Skula, Wilson quotients for composite moduli, Math.

quotients for composite moduli,

J.

Comp. 67 (1998), no. 222, 843-861.

[3

W.R. Alford, A.Granville & C.Pomerance, There are Carmichael numbers, Ann. of Math. 140 (1994), 703-722.

1.0. Angell & H.J.Godwin,

infinitely

many

On truncatable primes, Math. Comp. 31 (1977),

265-267.

[5]

W.S. Anglin, The square pyramid puzzle, Amer. Math. Monthly 97 (1990), 120-124.

[6]

[7]

W.S. Anglin, Mathematics: A Concise History and Philosophy, Springer Verlag, 1996. J. Arkin, V.E. Hoggatt & E.G.Strauss,

On Euler’s solution of a problem

of

Diophantus, Fibonacci Quart. 17 (1979), 333-339.

(8)

[9]

E.T. Avanesov, Solution of a problem on figurate numbers, (1966), 409-420.

Acta Arith. 12

A. Bager, Problem E2833*, Amer. Math. Monthly 87 (1980), 404, Solution in

the same journal 88 (1981), 622.

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R. Baillie, Table of ¢(n) = ¢(n + 1), Math. Comp. 30 (1976), 189-190.

[11]

A. Baker & H. Davenport, The equations 327 —2 = y” and 8x —7 = z?, Quart.

[12]

R. Balasubramanian, J.M. Deshouillers & F. Dress, Probleme de Waring pour les bicarrés. I. Schéma de la solution, C.R. Acad. Sci. Paris, Série I Math. 303

J. Math. Oxford Ser. (2) 20 (1969), 129-137.

(1986), no. 4, 85-88.

[13]

W.W. Rouse Ball & H.S.M. Coxeter, Mathematical Recreations and Essays, Dover Publications, 1987.

[14]

A. Balog & 1.Z. Ruzsa, On an additive property of stable sets, Sieve methods, exponential sums and their applications in number theory, Cardiff, 1995, London Mathematical Society Lecture Notes No. 237 (Cambridge University Press, Cambridge, 1997), 55-63.

[15]

W.D. Banks, A.M. Giiloglu, Anatomy of Integers, CRM (2008), 167-173.

[16]

W.D. Banks & F.Luca,

[17]

P.T. Bateman, Problem 10376, Amer. Math. Monthly 104 (1997), 276.

C.W.Nevans & F.Saidak, Descartes numbers, Proceedings and Lecture Notes, AMS, Vol 46

Concatenations

with binary recurrent sequences,

Integer Sequences 8 (2005), no. 1, Article 05.1.3.

J.

414

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[18] C. Bays & R.H. Hudson, Details of the first region of integers « with 13 2(x) < m™3,1(2), Math. Comp. 32 (1978), 571-576. [19] John V. Baxley, Euler’s Constant,

Taylor’s Formula and Slowly Converging

Series, Math. Mag. 65 (1992), 302-313. [20] N.G.W.H. Berger, On even numbers m dividing 2” — 2, Amer. Math. Monthly

58 (1951), 553-555.

[21] B.C. Berndt, Ramanujan’s Notebooks, 1994.

Part IV, New-York,

Springer Verlag,

22] V.Blomer, Binary quadratic forms with large discriminants and sums of two

squarefull numbers, J. Reine Angew. Math. 569 (2004), 213-234. [23] D. Borwein, J.M. Borwein, P.B. Borwein & R. Girgensohn, Giuga’s Conjecture on Primality, Amer. Math. Monthly 103 (1996), 40-49. [24] A.Bremner

&

P.Morton,

A

new

characterization

of the

integer

5906,

Manuscripta Math. 44 (1983), 187-229. [25] J. Brillhart, D.H.Lehmer, J.L. Selfridge, B.Tuckerman & S.S. Wagstaff Jr., Factorisations of b” +1, b = 2,3,5,6,7,10,11,12 up to high powers, Contemporary Mathematics, AMS, Vol. 22, Second Edition, 1988.

[26] Y. Bugeaud, M. Mignotte & Y. Roy, On the diophantine equation (x”" —1)/(x—

1) = y?, Pacific J. Math. 193 (2000), 257-268.

[27| Y. Bugeaud, M. Mignotte & S.Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no.3, 969-1018.

[28] J.P. Bulher, R.E. Crandall & R.W.Sompolski, Irregular primes to one million,

Math. Comp. 59 (1992), 717-722. [29] C. Caldwell, On the primality of n!+1 (1995), 889-890.

and 2-3-5...p+1,

Math. Comp. 64

[30] C. Carter & R.H. Hudson, A new bound for the smallest x with r(x) > Li(z), Math. Comp. 69 (2000), no. 231, 1285-1296.

[31] E. Catalan, Note extraite d’une lettre adressée a l’éditeur, J. Reine Angew. Math. 27 (1844), 192. [32] S. Cavior, The subgroups of dihedral groups, Math. Mag. 48 (1975), 107. [33] S. Chowla, An extension of Heilbronn’s Class Number Theorem, Quarterly J. Math. (Oxford) 5 (1934), 304-307. [34] J.H.E.Cohn, On square Fibonacci (1964), 537-540.

numbers,

Proc.

London

Math.

Soc. 39

[35] J.H-E.Cohn, The diophantine equation x? + 19 = y”, Acta Arith. 39 (1964), 1537-1540.

THOSE

FASCINATING

NUMBERS

A415

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Index (Unless indicated otherwise, the number at the end of each line is the one where the definition can be found in the text.)

champion, 77 colossally abundant, 55 440 convenient, 37 Cullen, 141 deficient, 90 dihedral-perfect, 130 dihedral 3-perfect, 5472 Erddés-Nicolas, 2016 Euclid, 211 Euler, 272 Fermat, 17 Fibonacci, 55 Giuga, 30

196-algorithm, 196 abc Conjecture, page xvii Bachet Equation, 225 Beal Conjecture, 122

Catalan Conjecture, 9 Chain Cunningham, 1 122659

Goodstein! 3>.249? 83:21 23 Granville, 126 Hamilton, 923 happy, 1880 harmonic, 140 Heegner, 163 highly composite, 180 horse, 13 ideal, 390 insolite, 111

p? +1, 271 Chinese Remainder Theorem, page xvii

Euler Constant, page xiii Euler pseudoprime, 1 905 Fermat’s Last Theorem, page 55 Fermat’s Little Theorem, page xvii

Keith, 197 k-composite, 1428 k-hyperperfect, 21 k-perfect, 6 k-powerful, 841

Height of a prime number, 283 Hypothesis H, page xvii

Index of composition, 629 693 Index of isolation, 2737

Lucas, 613 Lucas-Carmichael, 399

Kaprechar Constant, 495

Markoff, 433 multi-perfect, 140

Median value, 37 Mirimanoff Congruence, 1 006003

narcissistic, 88 Niven, 110

non deficient primitive, 945 palindrome, 26 pentagonal, 210 perfect, 6 perfect Canada, 125 Perrin, 271 441 persistence k, 679

Number abundant, 348 Amicable, 220 Apéry, 1445 automorphic, 76 Bell, 52 Bernoulli, 30 bizarre, 70 Canada perfect, 125 Carmichael, 561 Catalan, 14

Phibonacci, 1037

powerful, 23 prime, 2 pseudo-perfect, 12

425

426

JEAN-MARIE

Ramanujan, | 729 Ruth-Aaron, 714 Sastry, 183 self contained, 293 self described, 6 210 001 000 self replicating, 954 Sierpinski, 78 557 Smith, 22 S-perfect, 126 squarefull, 23 square pyramidal, 208 335 Stanek Stern, 137

sublime, 12 superabundant, 110 880 super-prime, 73 939 133 symmetric, 35853 tetrahedral, 10 triangular, 3 trimorphic, 491 tri-perfect, 120 vampire, 1260 voracious, 1807

unitary hyperperfect, 288 unitary perfect, 6 Wieferich, 16 547 533 489 305 Wilson, 5971 Woodall, 115

Palindrome (see palindrome number) Prime Euler pseudoprime, 1905 Fermat, 17 Fibonacci pseudoprime, 323 irregular, 59 Lucas, 613 Mersenne, 3 regular (see irregular prime) twin, 35 Wieferich, 1093 Wilson, 5

Wolstenholme, 16 843 Prime Number Theorem, page xvii Pseudoprime in base a, 91 Riemann Hypothesis, 5041 Riemann Zeta Function, 177

Schinzel Hypothesis (see Hypothesis H)

DE KONINCK

Strong pseudoprime, 2047 Sub-factorial function, 148 349 Syracuse Conjecture, 41 von Sterneck Conjecture, 7 725 038 629

Waring Problem, 4 Wieferich prime pair, 2903

—

—

—

+

!IN I Franklin Pierce Universit

00187232

_DATE DUE

GAYLORD

PRINTED IN U.S.A.

rs 077 alee SEAR Gea. Who would have thought that listing the positive integers along with their most remarkable properties could end up being such an engaging and stimulating adventure? The author uses this approach to explore elementary and advanced topics in classical number theory. A large variety of numbers are contemplated: Fermat numbers, Mersenne primes, powerful numbers, sublime numbers, Wieferich primes, insolite numbers, Sastry numbers, voracious numbers, to name only a few. The author also presents short proofs of miscellaneous results and constantly challenges the reader with a variety of old and new number theory conjectures. This book becomes a platform for exploring new concepts such as the index of composition and the index of isolation of an integer. In addition, the book displays several tables of particular families of numbers, including the list of all 88 narcissistic numbers and the list of the eight known numbers which are not prime powers but which can be written as the sum of the cubes of their prime factors, and in each case with the algorithm used to

ISBN 978-0-8218-4807-4

9"780821"848074

MBK/64

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Photo Robitaille Marc