Introduction to Arithmetic Factorization and Congruences from the Standpoint of Abstract Algebra

Citation preview

THE AMERICAN

MATHEMATICAL MONTHLY THE OFFICIAL

THE MATHEMATICAL

JOURNAL

OF

ASSOCIATION OF AMERICA, INC.

NUMBER 8

VOLUME 65

PART E

Introduction to Arithmetic Factorizationand CongruencesfromtheStandpoint of AbstractAlgebra HI. S. VANDIVERand MILOW. WEAVER

Number7 of the HWRBERTELLSWORTHSLAUGHJT MEMORIALPAPERS

1958

OCTOBER ONE DOLLAR A COPY

The AMERICAN MATHEMATICAL MONTHLY (FOUNDED IN 1894 BY BENJAmINF. FINKBL)

Editor RALPH D. JAmEs, RICHARD V. ANDREE JOHN A. BROWN ROY DUBISCH

HOWARD EVES

ASSOCIATE EDITORS G. E. HAY L. M. KELLY

JOHN R. MAYOR

LLOYD J. MONTZINGO,JR. IVAN NIVEN

C. 0. OAKLEY J. M. H. OLMSTED E. P. STARKE

H. S. ZUCKERMAN

of to theEditor,R. D. JAz, Department EDITORIAL CORRESPONDENCE shouldbe addressed shouldbe typewritten 8, Canada.Articles of BritishColumbia,Vancouver University Mathematics, the possiblecareshouldbe takenin preparing and double-spaced on 8t by 11 paper.The greatest copy. andauthorsshouldkeepa complete manuscript Association toF. R. OISON,Mathematical ADVERTISING CORRESPONDENCE shouldbeaddressed 14,N.Y. Buffalo ofBuffalo, ofAmerica, University of the Association as wellas correspondence NOTICE OF CHANGE OF ADDRESS by members H. M. GEEu., to theMoNTHLT shouldbe sentto theSecretary-Treasurer, subscriptions regarding Buffalo 14,N. Y. ofBuffalo, University AsociationofAmerica, Mathematical THIS IS THE OFFICIAL JOURNALOF THE

MATHEMATICAL ASSOCIATION OF AMERICA, INC. Mathematics) ofCollegiate (Devotedto theInterests OFFICERS OF THE ASSOCLITION

President,G. B. PRICE,Universityof Kansas G. B. THOMAS,JR.,MassachusettsInstituteofTechnology FirstVice-President, B. W. JONES,Universityof Colorado Second Vice-President, H. M. GERMAN,Universityof Buffalo Secretary-Treasurer,

Associate Secretary,LLOYD J. MONTZINGO,JR., University of Buffalo Editor,R. D. JAmEs,Universityof BritishColumbia

Additional Members of the Board of Governors:iH. M. BACON, JEwELL H. BUsHEY, LAMBERTOCESARI, H. S. M. COXETER, P. H. DAUS, W. L. DuREN, JR., H. P. EVANS, HOWARDEVES, WILLiAmFELLER, N. J. Fnm, J. S. FRAmi, A. S. HoUsEHOLDER,G. B. Hun', W. N. Hu"', G. K. KALISCH, E. C. KH:nuR,J. C. KNIpp, W. G. LEAVITT,Z. L. LOFLIN, A. T. LONSETH, SAUNDERSMACLANE, J. R. MAYOR, E. J. MCSHANE, G. M. MERRIMAN,R. S. PARK, GEORGE P6LYA, 0. J. RALER, C. R. SHERER,M. F. SmiLEY,R. G. SMITH,B. M. STEWART,F. M. STEWART, W. R. UTZ, JR., BERNARD VINOGRADE, CHARLEa WExLER, C. R. WYLIE, JR. Mathematical a subscription to theAmerican oftheAsociation(including Annualduesformembers thesubscription priceis $8.00. are$5.00.Fornon-members Monthly) PUBmLISHD

BY THB AsOCIATION

at Menasha, Wisconsin,and Buffalo,N. Y.,

ofJanuary, February, March,April,May,June-July, duringthemonths December. November, October, August-September,

formailing at Menasha,Wis.Acceptance Enteredas secondclassmatterat thepostoffice in forin theActofFebruary 28, 1925,embodied at specialrateofpostageprovided 4, Section538,P. L. and R., authorized April1, 1928. Paragraph PRINTED IN TEl

UNITED STATES OP AMCA

INTRODUCTION TO ARITHMETIC FACTORIZATION AND CONGRUENCES FROM THE STANDPOINT OF ABSTRACT ALGEBRA By H. S. VANDIVER,

Professorof Mathematics and

MILO W. WEAVER, Assistant Professorof Mathematics The Universityof Texas

The Seventh HERBERT

ELLSWORTH MEMORIAL

SLAUGHT

PAPER

Published as a supplementto the AMERICAN Volume 65

MATHEMATICAL

MONTHLY

Number 8 October, 1958

TABLE OF CONTENTS CHAPTER

I.

II.

III.

IV.

V.

VI.

PAGE

Introduction .1 Associativealgebraicsystems. 1.1. Some referencesto associative algebra. 1.2. Some definitions. 1.3. Three theoremson semigroups. Semigroupswitha unique basis, and primes.. 2.1. General uniquely factorablesemigroups. 2.2. Unique factorizationin the multiplicativesemigroupof natural numbers. Ideals and congruencesin theringof rationalintegers. .12 3.1. Principal ideals involvingthe rational integers. . . 3.2. Elements of arithmeticcongruencetheory . . . 3.3. An algorithmfora solution of a linear congruence The ringof residueclasses modulom. . .17 4.1. Residue classes ..17 4.2. An application of semigroupsto a linear congruence 4.3. Absolutely distinctsolutions of polynomial equations ill commutative rings .17 4.4. A theoremon the product of the distinct elements of a finite Abelian group .18 The additivegroupof theringof residueclasses modulo i . . . 5.1. On the generatorsof cyclicgroupswithapplication to the additive group modulo m .20 5.2. Simple propertiesof the totieit .22 The multiplicative group@(m) ofresidueclasses Ca modulom; (a, m) = 1 6.1. The theoremsof Euler and Wilson . .24 6.2. Criterionforthe solutionof a quadratic congruencemodulo p . 6.3. The Minkowski-Thuetheoremon linear congruences . 6.4. The expressionof p=4n+1 as the sum of two squares 6.5. Repetitive sets with application to Euler's theorem . 6.6. A basis theoremforfiniteAbelian groups . .27 6.7. A criterionforcyclic groups ..29 6.8. The number of solutions of ax=b (mod m), (a, m) d, with application to cyclic groups ..30 6.9. Primitiveroots modulo m ..31 6.10. A theoremon the cyclic subgroup of @(m) of maximal order, with application to Carmichael numbers . .34 6.11. Integralhomogeneoussymmetricfunctionsdefinedover repetitive sets in commutativeringswith units . .36 6.12. Direct product of semigroupswith application to @(m) iii

5 5 5 6 8 8

8

12 12 14 17

20

24 25 25 26 26

37

iv

TABLE OF CONTENTS

VII. Quadraticreciprocity.42 7.1. Simple propertiesof the Legendre symbol .42 7.2. Gauss' lemma .43 7.3. The law of quadratic reciprocity.44 VIII. The semigroupof thenonunitsof themultiplicativesemigroupof residue classes modulom ..48 8.1. On the groups contained in finitecyclic semigroups 48 8.2. The cyclic semigroupsgenerated by nonunit residue classes . 48 . 8.3. Unique factorizationmodulo m .49 52 IX. The semiringformedby certaingeneralizedresidueclasses . . 9.1. Some finitecommutativesemirings . .52 ..53 9.2. The generalizedresidue classes

INTRODUCTION In the presentpaper* we discuss the beginningsof the theoriesof factorization and congruences,involving rational integers,using some of the simplest concepts in the theories of commutative semigroups (in particular, Abelian groups) and commutativerings. (As an example of the simplicityof the tools used, we nowhereemploy the generalbasis theoremforAbelian groups.) In the main, we develop only those special results fromsemigroup or group theory that we findnecessaryin orderto obtain our numbertheoreticresults.We thereby obtain much more coherence in our account than has been apparent to us in expositions we have observed, published before this, in which nothingbut elementaryalgebra was employed by the writers. Up to the point where we introducethe concept of products of semigroups, at least, we hope this account will be understoodby readerswho have had little more than a firstcourse in numbertheoryor modernalgebra. However, the remainder of the article is more sophisticatedand may appeal mostlyto experienced mathematicians. As far as previous effortsto apply abstract algebra to number theory are concerned,we note that H. Webert made some steps in thisdirection.For example, he treated the set of residue classes modulo m, using the termZathlklassen nach einemModul, by applying some elementaryresultsin group theory.G. A. Miller$ also obtained Fermat's and Wilson's theoremsas well as several other elementaryresults by the use of simple propertiesof groups, and in particular of cyclic groups.Vandiver? treated the theoryof finiteringsand semiringsfrom a standpoint of developing results with direct application to number theory. In his courses in algebra and number theory at The Universityof Texas, he adopted the viewpointof abstract algebra and group theoryin discussingeven * About one-third of Vandiver'sworkon thispaperwas done undera SeniorPostdoctoral ofthework Fellowshipawardedto himby theNationalScienceFoundation,and abouttwo-thirds oftheworkdonebyWeaveron thisarticlewas doneunder donebyVandiverand abouttwo-thirds Basic ResearchGrant3697,whichwas awardedto themby theNationalScienceFoundation. who examinedthe manuscript The writerswishto expresstheirthanksto R. D. Allentharp withgreatcareand madea numberofvaluablesuggestions. FriedrichViewegund Sohn, t Lehrbuchder Algebra,Bd. 2, ZweiteAuflage,Braunschweig, "Das Wichtigste Beispieleinerendlichen vol. 60-68,1899,pp. 302-314.On page60 Weberremarks, Zahlen nach einembeliebeigenModul, commutativen Gruppebietendie Reste der natuirlichen werden."Webermeantto miteinanderverbunden wennsie durchdie gewohnliche Multiplication the residuesmodulom do not, limithimselfhereto residuesprimeto the modulus,as otherwise in general,forma group. $ AnnalsofMath. II, vol. 4, 1903,pp. 188-190;thisMONTHLY, vol. 12, 1905,pp. 41-43; this MONTHLY, vol. 18, 1911,pp. 204-209. ? Trans.Amer.Math Soc.,vol. 13, 1912,pp. 293-304;AnnalsofMath.,II, vol. 18, 1917,pp. 105-114;Proc.Nat. Acad. Sci., vol. 20, 1934,pp. 579-584;Bull. Amer.Math. Soc., vol. 40, 1934, pp. 914-920;Proc.Nat. Acad. Sci.,vol. 21, 1935,pp. 162-165;Proc.Nat. Acad. Sci.,vol. 23, 1937, der Naturforschenden pp. 552-555; thisMONTHLY, vol. 46, 1939,pp. 22-26; Vierteljahrsschrift in Zurich,vol. 85, 1940,pp. 71-86;AnnalsofMath.,II, vol. 48, 1947,pp. 22-28. Gesellschaft

1

2

INTRODUCTION

TO ARITHMETIC

FACTORIZATION

AND CONGRUENCES

the most elementary parts of number theory. This point of view gradually developed in a period of some 25 years. In addition to this, Weaver|| further developed these ideas in his classes at The Universityof Texas duringthe last fiveyears and published some of the resultsobtained therefrom. R. Fueter? applied some of the ideas of abstract algebra, includingmodules and ideals, as well as groups, in connectionwith a developmentof elementary number theory. He also employed such tools in a treatmentof the theory of cyclotomicfields. E. Hecke* gave an expositionof a considerablepart of the theoryof Abelian groupsand applied the resultsto the elementarytheoryof numbersas well as to parts of the theoryof algebraic numbers. H. Hasset consideredthe ring of residue classes modulo m and applied elementarygroup theoryand ringtheoryto developingits properties.He assumed the propertiesofAbelian groupswithoutproofsforhis applications to the theory of the ring mentioned. Later het again considered the ring of residue classes modulo m and used less of group theorythan formerly, but he explained results he obtained by arithmeticalmethods,in termsof groups. Now, under multiplication,the nonzeroelementsof the ring of residue classes forma semigroupwhichis notalways a group.Aside fromtheauthorsofthepresent none of theinvestigators paper, however, abovementionedused semigroupsin their accountsof elementarynumbertheory.Also none of themconsideredthenon-unit elementsin theringof residueclasses modulom wheremattersappear quitecomplicatedunless some of thetheoryof semigroupsis employed.The use of thatsystem, and the introductionof possiblynew conceptsconcerningit, withapplications to factorizationproblems,in our opinion, constitutes themostnovelpart of our treatment.However, in parts of numbertheorywhereaddition and multiplicationare both involved in certainways, the theoryof semigroupsis of little or no value, as in the theoryof continuedfractions,which we do not discuss here. It maybethata readerofthispaper is mainlyinterested in thenumbertheoretical phase of it and may be of theopinion thatit would have beenmuchsimplerto developwhatwe did herein thetheoryof congruences, directly,withouttheuse of abstractalgebra.However,mostof thetoolswe haveset up fromthelattertheorymay be applied extensively, withlittleor no change,in thetheoriesoffinitefields,finite rings,and algebraicnumbers.In fact,in writingthispaper we havehad,amongother ideas, this end in view. Obviously,a purelyarithmeticapproach could notachieve this.? 11Math. Mag., vol. 25, 1952,pp. 125-136;thisMONTHLY,vol. 63, 1956,pp. 387-391. ? Synthetische DritteAuflage,Walterde Gruyterand Co., Berlin,1950. Zahlentheorie,

* Vorlesungen uberdie TheoriederAlgebraischen Zahlen,Akademische Verlagsgesellschaft, Leipzig,1923. t Zahlentheorie, Berlin,1949. Akademie-Verlag uiberZahlentheorie, Springer-Verlag, Berlin,Gottingen, Heidelberg,1950. t Vorlesungen someofwhich ? We notealso thatthearticlecontainsa considerable numberofdefinitions, are usedverylittle.However,in further ofthepresentideas it wouldbe verycondevelopments venientto use themoften.

INTRODUCTION

3

The literatureon elementarynumbertheoryis, of course,immense,and the literatureon abstract algebra, particularlygroupsand theirgeneralizations,likewise. Consequently,when we make referencesin this paper to the work of some otherauthor on some particularidea, thisdoes notmean thatwe necessarilyregard is given him as thefirstmathematicianwho publishedsuch an idea. The reference so thatthereadermightbe able, becauseof it, to augmenthis knowledgeof thetopic beingdiscussed.If some reader thinksit would have been particularlyilluminating to have referredto the work of some author we did not mention,we might be able to list such referencesin a supplement to the presentarticle, if we are advised of these importantomissions. Of course, as time goes on, and particularlyrecently,more and more applications of modernalgebra are being made to various parts of mathematics, pure and applied. The presentcontribution,of course,comes under the heading of an application to numbertheory. Starting with a set of axioms for ordinaryalgebra and some consequences fromthemin the resultsgiven in section 1.1 below, the presentarticle,we think, is self-containedin referenceboth to abstract algebra and numbertheory,aside fromsolution of the problems. The results we needed fromabstract algebra in order to develop theoremsin number theoryare introducedin the text as we needed them,includingthe necessarydefinitions.Even aside fromthe algebraic approach, the proofsof the theoremswe give oftenseem to contain elementsof novelty. The proofsof Theorems 6.9.12 and 6.12.14 are not simple. This has been also true of all other treatmentsof these topics we have noted elsewhere,and seem to be inherentin the nature of the subjects. the difficulties

Chapter I

ASSOCIATIVE ALGEBRAIC SYSTEMS 1.1. Some referencesto associativealgebra.We have alreadytreatedthe

foundationsof associative algebra, which include the foundationsof the theory of integers,in fourpapers published in MathematicsMagazine, each under the title A developmentof associative algebra and an algebraic theoryof numbers, appearing as follows: (I)-Vandiver, (II)-Vandiver, (III)-Vandiver (IV)-Vandiver p. 219.

vol. 25, 1952, pp. 233-250. vol. 27, 1953, pp. 1-18. and Weaver, vol. 29, 1956, pp. 135-149. and Weaver, vol. 30, 1956, pp. 1-8; Errata, vol. 30, 1957,

In view of this, our presentaccount will be naive, but we shall give definitions and proofsin a formwe thinkis usually acceptable in present-daynumbertheory and algebra. 1.2. Some definitions.A semigroupS is an algebraic systemwith operation

(.) and equivalence(=) such that foreach A, B, CE 5,A *B-=X has a unique solution XEE , and (A B) C = A (B. C). The operation symbol is usually .

.

omitted between elements of 5. If e has an element E such that for each AE=EA =A, then E is called the identityof S. It is unique. If for Az(, each Al, A2EG, AlA2=A2A,, e is called commutative or Abelian. If a semigroup (M has an identity E, and for each B G, the equations BX = E and YB =E have solutions, X, YE( , then it is easy to show that X = Y, and (Mis called a group. It follows easily that X is unique, and it is called the inverseof B and is denoted by B-1. If an element U in a semigroupe has an inverse, U is called a unit in 5. If AjA2 ... An= C, then the A's are divisorsorfactorsof C, where C and the A's are in 5. If n > 1, then C is called a product of the A's. The orderof a semigroup e is the number of distinct elements of (. If A E5, the set of nonequivalent powers of A formsa cyclicsemigroup (, and A is said to be a generatorof G. If ( is finite,it contains a maximal subgroup (M which is also cyclic.* The orders of ( and ( are called, respectively,the order and period of A. Let S be a set with equivalence (=) and operations addition (+) and multiplication(X) (the latter sign is usually omitted) such that *

Cf.relation(8.1.2) thatfollows.

5

6

INTRODUCTION TO ARITHMETIC FACTORIZATION AND CONGRUENCES

(i) e is a semigrouprelative to + and (ii) e is a semigrouprelative to X and (iii) Whenever S1, S2, S3G , then S1(S2+S3)

= S1S2+S1S3,

and (S2+S3)S1

= S2S1 +S3S1.

(iv) Substitutiontas in ordinaryarithmeticholds in S. Under these conditionse is called a semiring.The set of units,if thereare any, of the multiplicativesemigroupis called the set of unitsof the semiring.A semiring,whose additive semigroup(ASG) is a commutativegroup,is called a ring. The additive identityof a ringis denoted by 0. Suppose thereis a ring9 whose multiplicativesemigroup(MSG), the additive identityexcluded, formsa commutative group. Such a ringis called afield. The additive inverse of A, if it exists, in a semiringis denoted by -A; A -B is used interchangeablywith A + (-B). Let 5 be a non-emptysubset of a ring 9Z such that for any I,, I2C$a and RCz9, I1-I2 CzS, RI1,Cz, and I1RCz;; then I is called an ideal of T. It is are and I1-(-I2) clear that I is a subringof 9, since I,-I =0, -I, =0-I, also in S. Ideals have been extensivelystudied and have proved valuable in obtaining the structureof algebras. Hereafterin this paper we shall use small Greek lettersto denote elementsof a semiringexcept forsemiringsof rational integers. We shall now give some examples of the concepts definedabove. The sets will be taken fromthe set ofrational numbers.As forsemigroups,such examples are common. Let m be a positive integer;then the set of all positive multiples of m formsa semigroup under addition and also under multiplication.Under addition,the set mkwithk rangingover all integersformsa group. However, the set of rational numbers,not includingzero, formsa group under multiplication. Perhaps the simplest example of a semiringis the set of natural numbers. The set of all positive multiplesof a positive integerm also formsa semiring. If a natural numberm is multipliedby each integerwe obtain a ring. The set of rational numbersformsa field. If a and b are given integers,then the set ax+by with both x and y ranging independentlyover the set of integerssatisfiesthe conditionsin our definition of an ideal in the ring of integers. 1.3. Three theorems on semigroups. We shall prove some theoremsfrom group theorywhich we shall need several times. If S is a semigroup,St: R1, R2, . . . is a subset, finiteor infinite,of S, and SEES, we denote by S9, the set SRi, i = 1, 2, THEOREM 1.3.1. If 5 is a groupofordern and IDis a subgroupof (Moforderd,

thend dividesn. If A generatesa cyclicgroup 5 of ordern and Ar=E, whereE is theidentityof 5, thenn dividesr.

t As thereexist semiringsin whichaddition is not commutative,the orderof the termsshould be preserved here.

ASSOCIATIVE ALGEBRAIC SYSTEMS

7

The firstpart of Theorem 1.3.1 is due to Lagrange. To prove it we shall show that the distinctelementsof (Mare the elementsof disjoint sets, each containing d elements: I, G2 , G3', * * ,, Gn1d,. I contains d elements by hypothesis. We select G2E H, if such exists. Then G2' contains d elements, since from G2H= G2HI',we have G-'G2H= G2'G2H' and EH = EH' and thereforeH = H'. G2S1and I are disjoint; otherwisefromG2H= H' we would have G2C,. We select G3Ej3 , G3ELG21'if such exists and find,similarlyto the above discussion, that G31' contains d elements,no one of which is in ,. And no element of G31' is in G2'; otherwise,we would have G3EG2,. Since O5is finite,'5 is exhausted by a numbern/d of such sets, and we have the firstpart of our theorem. To prove the second statementof the theorem,we use Theorem 2.2.8 below, the proof of which is independent of Theorem 1.3.1, and write r=bn+c, O_ c ql; the argument

is similar for pi 1. We firstnote that the set, denoted by @(m), (6.1.1)

Crl, Cr2,

(ri, m) =

Cre;

* *

1

with e=+(m), formsa group, under multiplication.This followsimmediately fromTheorem 1.3.4 and Theorem 3.2.2. We shall now employ the group (6.1.1) and apply Theorem 4.4.9 to it. We findhere that form>2, k(m) is of the type 2n, by (5.2.4), so that fora given a with (a, m) = 1, (4.4.8) becomes forour special group

II Cri =

(6.1.2)

Cm-lCa)

i=1

whered=(m)/2 and wheret is thenumberofsolutionsin C_,oftherelation (6.1.3)

Cr = Ca,

sincethereis an elementof ordertwo in @(m), and it may be takenas Cm-, whichis not C1form> 2. Let a=1; we obtainfrom(6.1.2)

~~~~e ~ l Cri=

(6.1.4)

(6 . 1.f4)

i=1

t/2

Cm_i.

Whenceformn2, (6. 1. 5)

e

II rj j=1

3(1)

t/2

(modm),

solutionsof x2 1 (modm). wheret is nowthenumberofincongruent Using (6.1.4), (6.1.5) in (6.1.2) gives Cd =

=C1. Whence (6.1.6)

a0(m) 31

C1

or

Cmi-

and by squaring, C+(m)

(modm),

of Fermat's form>2, and it is obviousform= 2. This is Euler'sgeneralization theorem: (6. 1. 7)

aP-1

(mod p),

which followsfrom (6.1.6) when m=p, a prime, since it is obvious that +(p) =p-1. Also, if m=p in (6.1.5), we findWilson's theorem: 24

MULTIPLICATIVE

@(m)

GROUP

(p

(6.1.8)

OF RESIDUE

1)!

-

CLASSES Ca MODULO Mn

25

1 (modp)

-

since there are not more than two solutions of x2=1 (mod p) by the Corollary 4.3.3. Using the known number of solutions in x in x2_1 (mod m) in (6.1.5), then we have a generalizationof (6.1.8) due to Gauss. 6.2. Criterionfor the solution of a quadratic congruence modulo p. Again from(6.1.2) and (6.1.8), with m=p, we have Ca'P-1)2 = C, or Cp-i according as C!= Ca has solutions or has no solutions in C-; whence THEOREM

6.2.1. The congruence a (mod p),

x2 =

withp an odd primeand (a, p) = 1, has solutionsor has no solutionsin x according as (6.2.2)

?+ 1 (modp).

a(P-=1)2

Putting a

-1 in (6.2.2) gives, since p is odd, -1-

(mod p)

(-1)(p1)12

when p_3 (mod 4), and 1_ when p-1

(p-1) 12

(-1)

(mod p)

(mod 4), so that

(6.2.3)

x2-

has solutions if and only if p

1

1 (mod p)

(mod 4).

6.3. The Minkowski-Thue theorem on linear congruences. Suppose m> 1, and (a, m) = 1, and let k denote the least integer > i/rn. Consider the numbers of the form (ay+x) where x and y each range independentlyover the set 0,1, . .*, k - 1. As k2> m, thenit followsthat at least two numbersof this form are congruentmodulo m, and we may set ay, + xi

ay2 + X2 (mod in),

with eitherYl#Y2 or xl 5x2. Whence (6.3.1)

a(yi

-

y2)

3

-

xl

(mod in),

with both x1$x2, Y15Y2, since if eitheryl=y2 or xl=x2, with (a, m) = 1, our assumption, "either YlFHY2 or xl x2," is violated. Hence, setting u = Iyl-Y2 v= I xl-x2 | , we have (6.3.2) with Ot 0. We note that z =1+tpn-k+ci + (tl+pkg) pn-k+c t=t+pkg (mod pn). Hence for all c >0 this cannot have more than the pk -1+t1pn-k+, solutions 0, 1, * *, pk-l, in t1,and therefore(6.9.10) has no more than pk solutions in z. Now the solutions of (6.9.4) forma group whose order divides spk and by Theorem 6.6.1 each such solution is the product of an element which satisfies (6.9.5) by one which satisfies(6.9.6). We have just proved that (6.9.5) has no more than s solutions and (6.9.6) no more than pk solutions; hence (6.9.4) has no more than Spk solutions,and by Theorem 6.7.1, the group (6.9.3) is cyclic. Otherwiseexpressed,forp an odd prime,a primitiveroot modulo pn exists. Obviously primitive roots exist modulo 2 and 4, respectively, but not modulo 2kwith k>2, since a2= 1 (mod 8) forany odd a.

MULTIPLICATIVE

GROUP @(m) OF RESIDUE

CLASSES Ca MODULO m

33

Finally, let m = 2pn, where p is an odd prime. Employ a primitiveroot r modulo pn. Let g be that one of the numbersr and r+pn which is odd. If g belongs to the exponent e modulo m, then re _ ge

1 (mod pn),

whence e is divisible by 4(pn) =+(m). But e?3, * A simplerproof,using group theory,of the fact that primitiveroots modulo pn and 2pn exist when p is odd, may be given by the use of the criterionthat an Abelian group of order h is cyclic if and only if forevery primeq dividing h, the relation

xq = E, where E is the identityof the group, has exactly q solutions. However, as of now, we know of no proofof this which does not involve the use of the basis theoremforAbelian groups,which we are avoiding in the present work.

34

INTRODUCTION TO ARITHMETIC FACTORIZATION AND CONGRUENCES 5, 52, . . ., 5c, _57 - 52 . ,_5c

(6.9.15)

give all the 21-1odd incongruentresidues modulo 21. Since 5 belongs to 2t-2 modulo 2t, the elements of the firstline alone are incongruentmodulo 2t and 5b (mod 2t), a and b each in the similarlyforthe second line. If we have 5a_ , c, then if a>b, range 0, 1, 5a-b

_

-

1 (mod

2t),

g

52(a-)

1 (mod

2t).

2t2 modulo2t, then2(a - b) 0 (mod2t2), Again,since5 belongstotheexponent (mod 2t). However, this a-b _O (mod 2t-3); whencea-b = 2t-3, and 52t 3+1--0

(modulo 2t). This is impossible since 52t - 1=0 (mod 4); hence 52t ++100 proves Theorem 6.9.12, as the generatorsof the cyclic groups of orders 2 and 2t-2 are, respectively,the residue classes determinedby (-1) and 5, modulo 2t.

6.10. A theoremon the cyclic subgroupof @(m) of maximal order,withapplicationto Carmichael numbers. Since the above discussion shows that we do not have primitiveroots correspondingto all moduli, we may consider the problem of determiningthe order of a maximal cyclicgroup (a cyclic group of maximumorder) contained in @(m). This is an extensionof the problemof finding primitiveroots,as when a primitiveroot g, 0 _g 2; X(pt) = 4(pt) ifp is an odd prime; X(2

*2 *

*

pn) -

h, witheach p an odd prime,

(ptn) whereh is the least common multipleof X(2t),X(ptl), * We shall now apply some relations used in the proof of Theorem 6.6.1 to the obtaining of results,in effect,in the theoryof the additive cyclic group of the residueclasses modulo m, as well as in the theoryof the multiplicativegroup @(m), where,if the p's are distinctprimes, M-=P1P2

tl

t2

..P

tn

r.

If we set at-Pti in the statementof Theorem 6.6.1 with r =n and let 5 be the additive group of the residue classes Ca, a=0, 1, * *, mr-1, then (6.6.7), with the remarkfollowingit, gives, using congruences, * Bull. Amer.Math. Soc., vol. 15, 1908-9,pp. 221-222.

MULTIPLICATIVE

(6.10.1)

GROUP (i(m) OF RESIDUE

a i1

( ( /pgi) (mod

35

CLASSES Ca MODULO m

i),

fora given a. Setting (6.10.2)

a

ri (mod pti),

i

=

1, 2,

,,

we obtain (6.10.3)

a -r(m/p't) +

(mod i).

The converse of this resultwas given in (6.6.8). Let a belong to the exponentk modulo m, and take (6.10.2) to the kthpower , n. Then and assume that ri belongs to the exponentei modulo pti,i =1, 2, we have (6.10.4)

r-

Hence k is a multipleof ei, i = 1, 2,

1 (mod p"); ,

i = 1, 2, ...

,

n.

n. Suppose in particularthat we select

an a in (6.10.1) such that ri belongs to X(pi) for i= 1, 2, * * *, n, which is always

possible by Theorem 6.9.12. Then k is the L. C. M. of these X's, which is X(m). Then a belongs to X(m) modulo m, so thatCa generatesa cyclicsubgroupof order X(m) in thegroup of residue classes @(m) of (6.1.1). Further,in the above, the ei, i=1, 2, * * *, n are divisors of X(pt), respectively,by Theorem 1.3.1. Consequently we have (6.10.5)

n b6X()

1 (mod n),

forany b with (b, m) = 1. We may then state THEOREM 6.10.6. If @3(m)is thegroupof residueclasses givenin (6.1.1), then a cyclicsubgroupof maximal orderhas X(m) as its order.

We shall next apply the above result to the followingwell-knownproblem. For what integersn is an-'_ 1 (mod n) forall a's such that (a, n) = 1? Suppose n is such an integer,and let some a, say a,, belong to the exponentX(n), which is always possible,as we have seen. Then it followsthat n 1 (mod X(n)) by Theorem 1.3.1. On the otherhand, if n z 1 (mod X(n)), then by (6.10.5), if (a, n) = 1, it followsthat al(n) =1 = an- (mod n). So we have the criterion:an-1= 1 (mod n) for all a's such that(a, n) = 1, if and onlyif n 1 (mod X(n)). Below the number 2,000 there are only three values for composite n, namely, 561 =3*11 17; 1105 =5.13.17; 1729 = 7 13 19. Such numbers we shall call Carmichaelnumbers,followingthe example of several writers.* * R. D. Carmichael,Bull. Amer.Math. Soc., vol. 16, 1909-10,pp. 237-238;thisMONTHLY, vol. 19, 1912,pp. 22-27.

INTRODUCTION TO ARITHMETIC FACTORIZATION AND CONGRUENCES

36

6.11. Integral homogeneous symmetricfunctions defined over repetitive sets in commutativerings with units. Here we shall discuss a type of function in a commutativefinitering which has a multiplicativeidentitydenoted by 1. We definethe indicatorof T as the numberof distinctunits in its multiplicative semigroup.If F(yo, 'yi, * * *, 'yj) is a functionwith the -y'sin a ring,then F is functionof degree d, written IHSF called an integralhomogeneoussymmetric provided it satisfies: (i) F is a polynomialin the -y's,which is unchangedaside fromthe orderof the termsby any permutationof the -y's. (ii) F(o--yo, a-yi,. .. , ayj) =gdF(yoy, -, * , yj), oECR. Consider an IHSF of degree d formedby an ordinaryrepetitiveset,t say F(31, /32,

. . ., 1k).

If A is a multiplierof the repetitiveset, then A2,. 2 F(#j31,

F

3fk) =

=

#dF,

and

(6.11.1)

1)F = O.

(fd

It is easy to show that no unit ,u satisfies,uF=O for FET, F# O. Now if it is possible to select A and d so that Ad -1 is a unit in 9, then F = 0. This can be done in a numberof special cases. For example, we may consider the application of the above ideas to the residueclasses modulo m. We can select many subsets of this systemwhichform repetitivesets and apply the relation (6.11.1) to obtain various results. However, we may obtain theoremsgiving us more informationconcerningexceptional cases when we considerthe whole ring modulo m. Under multiplicationthe set of distinct residue classes modulo m formsa repetitiveset with any Ca, (a, m) = 1, as multiplier. Let m

=

al

m -pp2

a2

* P*

pak

p,k

where the p's are distinctprimes,and let S be an IHSF of the distinctresidue classes modulo m. Then by (ii) above, we have if d is the degree of S and (b, m) =1, SC,=S. Hence

(6.11.2)

S(Cb

- C1) = 0.

Now assume that d is not a multipleof 0(pi), where 4(pi) representsthe indicator of pi, i = 1, 2, * * , k, and let Cbi be such that biis a primitiveroot modulo Pi. Then set in any finiteringwas definedin Vandiver,Annalsof repetitive t The idea ofan ordinary mayobviouslybe extendedto finite Math., II, vol. 18, 1916-17,pp. 105-114,and the definition setsin any ring.

MULTIPLICATIVE

GROUP @(m)

OF RESIDUE

37

CLASSES Ca MODULO m

d = r4(pi) + di, where 0 p'. We note that Ct =Cp-lcp-ti. On the other hand, consider the cases when CaC8 = Ct,

with t2< pt. Then it followsthat the (p -tj)'s and the t2's constitutethe integers t' (mod p), then tL'+t' 0 (mod p), 1, 2, * , p' in some order.For ifp-tL' which is impossible, since each of the t's is k>O.

Now it is possible to show that the elements Ak,Ak+1, .

(8.1.2)

. .A,

forma cyclic group* 5. For a proof we note firstthat if s-k, the order of (8.1.2), is 1, s-k has no prime divisors, and ($ is obviously cyclic. Suppose s-k>1. We have fort_0 A8+t = Ak+t.

A8-kAk+t =

Assume that (8.1.3)

Ar(s8)Ak+t

= Ak+t

It then followsthat A (r+l)

(s-k)Ak+t

(8.1.4)

=

A 8-kA

r(9-k)Ak+t

= A-kAk+e =

Ak+t.

Then by induction (8.1.4) holds for all r's. Clearly there exists an n such that so from(8.1.4) An(8-k) is an identityfor(5. We also have

An(8-k) C=;

Anj(s-k)Ai

(An (g-k)+l )=

=

= EAi

An(8-k)+i

And the distinctelementsof 5 are An(8-k)+i i= l, 2, * is An(8k)+8kt the above range, the inverse of A8k)?t group.

,(s-k). And fort in Hence (M is a cyclic

8.2. The cyclic semigroupsgenerated by nonunitresidue classes. The nonunit residue classes modulo m were investigatedby Weaver and E. T. Parker.t Some of theirpropertieswill be given below. We firstnotice that the congruence ax=1 (mod m) has no solutions if and only if a is divisible by a factorof m; hence the nonunitsmodulo m are the * F. C. Biesele,An Introduction to theTheoryofSemi-Groups, M. A. thesis,The University of Texas, June1933,p. 9. t Weaver,Math. Mag., vol. 25, 1952,pp. 125-136;E. T. Parker,Proc. Amer.Math. Soc., vol. 5, 1954,pp. 612-616.

48

49

SEMIGROUP OF THE NONUNITS

elementswhose least positive residues have one or more primedivisorsin common with m. Further,(rlr2,m) = 1 ifand only if (ri,m) = 1 and (r2,m) = 1. Hence the nonunitsmodulo m forma semigroupunder multiplication. We wish to discuss the order and period of a nonunit class Ct modulo m. We firstconsider an example. Let m =24 and Ct= Cl0. Then (8.1.1) becomes C10,(C10)2= C4, (Clo)I= C16,with (Clo)4=(Clo)3. Cio has period 1 and order 3. We observe that if we writethe decompositionof 24 into primes: 24= 23 3, the period n of Cio is the exponent to which 10 belongs modulo (24/23) =3, and if j is the least powerof 10 which23divides,thenthe orderof C10is j+n- 1. These propertiesof C10modulo m are general and are stated below as Theorem 8.2.3. If I is an integerand b is the least positive integersuch that 1=0 (mod d), then b is called the nullifyingexponent of I modulo d. In the above, b =j. We returnto the question of the order and period of the nonunit Ct. Let m = mlk,with (ml, k) = 1 whereeach primedivisorof m whichdivides one of the pair t, k divides the other also. Let b be the nullifyingexponent of t modulo k and t belong to n modulo ml. Then allowing 1 as a possible modulus in the congruencesbelow, we have (8.2.1)

tb-=0

(mod k),

tn

1 (mod ml);

whence (8.2.2)

t(tnI-

1)

0 (mod m1k),

tb+n=

tb (modm).

On the other hand if t has order bi+ni -1 and period ni, we have divisible by km1and by Corollary 2.2.3, =bi

0 (mod k),

tni

tbl(tnl-1)

1 (mod ml).

Hence, by definitionof b and n, b1> b, ni> n, but by definitionof order and period, bi+nii,respectively,(8.3.2) gives (8.3.3)

v

x

i=1

q, wj(mod m), PiVi wf j=1 ab

with v, w contained in units. We see that each natural prime that divides one member of (8.3.3) divides the other also, and we may assume that x=y and that (8.3.3) is writtenin such an orderthat pi= qj when subscriptsare =, and it follows fromthe lemma that if ai?bi?b where pi is the largest power of pi dividingm, thenpaivi and piwi are contained in associated classes. On the other hand, if ai