Random generators and normal numbers

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Random Generators and Normal Numbers David H. Bailey1 and Ri hard E. Crandall2 30 O tober 2001

Abstra t

Pursuant to the authors' previous haoti -dynami al model for random digits of fundamental onstants [3℄, we investigate a omplementary, statisti al pi ture in whi h pseudorandom number generators (PRNGs) are entral. Some rigorous results su h as the following are a hieved: Whereas the fundamental onstant log 2 = 2 + 1=(n2 ) is not yet known to be 2-normal (i.e. normal to base 2), we are able to establish b-normality (and trans enden y) for onstants of the form 1=(nb ) but with the index n onstrained to run over ertain subsets of Z + . In this way we demonstrate, for example, that the onstant 2 3 = =3 32 33 1=(n2 ) is 2-normal. The onstants share with ; log 2 and others the property that isolated digits an be dire tly al ulated, but for the new lass su h

omputation is extraordinarily rapid. For example, we nd that the googol-th (i.e. 10100 th) binary bit of 2 3 is 0. We also present a olle tion of other results | su h as density results and irrationality proofs based on PRNG ideas | for various spe ial numbers.

P

P

;

P

n

n

n

Z

n

n

;

;

;:::

;

Lawren e Berkeley National Laboratory, Berkeley, CA 94720, USA, dhbaileylbl.gov. Bailey's work is supported by the Dire tor, OÆ e of Computational and Te hnology Resear h, Division of Mathemati al, Information, and Computational S ien es of the U.S. Department of Energy, under ontra t number DE-AC03-76SF00098. 2 Center for Advan ed Computation, Reed College, Portland, OR 97202 USA, randallreed.edu. 1

1

1. Introdu tion

We all a real number b-normal if, qualitatively speaking, its base-b digits are \truly random." For example, in the de imal expansion of a number that is 10-normal, the digit 7 must appear 1=10 of the time, the string 783 must appear 1=1000 of the time, and so on. It is remarkable that in spite of the elegan e of the lassi al notion of normality, and the sobering fa t that almost all real numbers are absolutely normal (meaning bnormal for every bp= 2; 3; : : :), proofs of normality for fundamental onstants su h as log 2; ;  (3) and 2 remain elusive. In [3℄ we proposed a general \Hypothesis A" that

onne ts normality theory with a ertain aspe t of haoti dynami s. In a subsequent work, J. Lagarias [28℄ provided interesting viewpoints and analyses on the dynami al

on epts. In the present paper we adopt a kind of omplementary viewpoint, fo using upon pseudorandom number generators (PRNGs), with relevant analyses of these PRNGs arried out via exponential-sum and other number-theoreti al te hniques. One example of su

ess along this pathway is as follows: Whereas the possible b-normality of the fundamental

onstant 1 log 2 = 2 + n2

X

n

n

Z

remains to this day unresolved (for any b), we prove that for ertain subsets S bases b, the sum 1

X



Z + and

2 nb

n

n

S

is indeed b-normal (and trans endental). An attra tive spe ial ase is a number we denote 2 3, obtained simply by restri ting the indi es in the log 2 series de nition to run over powers of 3: ;

X

;

X

1 1 1 = 3k =1 3 2 =3k 1 n2 = 0:0418836808315029850712528986245716824260967584654857 : : : 10 = 0:0AB8E38F684BDA12F684BF35BA781948B0FCD6E9E06522C3F35B : : :16 ;

2 3 =

n

n

k

k

>

whi h number we now know to be 2-normal (and thus 16-normal as well; see Theorem 2.2(6)). It is of interest that until now, expli it b-normal numbers have generally been what one might all \arti ial," as in the ase of the 2-normal, binary Champernowne

onstant:

C2 = 0:(1)(10)(11)(100)(101)(110)(111)


2 ; with the (
) notation meaning the expansion is onstru ted via on atenation of registers. Now with numbers su h as 2 3 we have b-normal numbers that are \natural" in the sense that they an be des ribed via some kind of analyti formulation. Su h talk is of ourse heuristi ; the rigor omes in the theorems of the following se tions. ;

2

In addition to the normality theorems appli able to the restri ted sums mentioned above, we present a olle tion of additional results on irrationality and b-density (see ensuing de nitions). These side results have arisen during our resear h into the PRNG

onne tion.

2. Nomen lature and fundamentals We rst give some ne essary nomen lature relevant to base-b expansions. For a real number  2 [0; 1) we shall assume uniqueness of base-b digits, b an integer  2; i.e.  = 0:b1b2


with ea h b 2 [0; b 1℄, with a ertain termination rule to avoid in nite tails of digit values b 1. One way to state the rule is simply to de ne b = bb  ; another way is to onvert a trailing tail of onse utive digits of value b 1, as in 0:4999


! 0:5000


for base b = 10. Next, denote by fg, or  mod 1, the fra tional part of , and denote by jjjj the loser of the absolute distan es of  mod 1 to the interval endpoints 0; 1; i.e. jjjj = min(fg; 1 fg). Denote by ( ) the ordered sequen e of elements 0; 1; : : :. Of interest will be sequen es ( ) su h that (f g) is equidistributed in [0; 1), meaning that any subinterval [u; v )  [0; 1) is visited by f g for a (properly de ned) limiting fra tion (v u) of the n indi es; i.e., the members of the sequen e fall in a \fair" manner. We sometimes onsider a weaker ondition that (fa g) be merely dense in [0; 1), noting that equidistributed implies dense. Armed with the above nomen lature, we paraphrase from [3℄ and referen es [27℄ [21℄ [33℄ [25℄ in the form of a olle tive de nition: De nition 2.1 (Colle tion) The following pertain to real numbers  and sequen es of real numbers ( 2 [0; 1) : n = 0; 1; 2; : : :). For any base b = 2; 3; 4 : : : we assume, as enun iated above, a unique base-b expansion of whatever real number is in question. j

j

j

n

n

n

n

n

n

1.  is said to be b-dense i in the base-b expansion of  every possible nite string of

onse utive digits appears. 2.  is said to be b-normal i in the base-b expansion of  every string of k base-b digits appears with (well-de ned) limiting frequen y 1=b . A number that is b-normal for every b = 2; 3; 4; : : : is said to be absolutely normal. (This de nition of normality diers from, but is provably equivalent to, other histori al de nitions [21℄ [33℄.) k

3. The dis repan y of ( ), essentially a measure of unevenness of the distribution in [0; 1) of the rst N sequen e elements, is de ned (when the sequen e has at least N elements) n

D

N

=

sup 

0

a 1 we have k = 364
1093k 2 . (Thus for example the order of 2 modulo 10932 is also 364.) Remark 2. One ould attempt to expand the (b; p)-PRNG de nition to in lude the even-modulus ase pk = 2k , with odd b  3, in whi h ase the rules run like so: If 2k j(b 1) then the order is k = 1, else if 2k j(b2 1) then k = 2, else de ne a by 2a jj(b2 1), when e the order is k = 21+k a . One expe ts therefore that normality results would a

rue for a (b; 2)-PRNG system, odd b  3. However we do not travel this path in the present treatment; for one thing some of the lemmas from the literature are geared toward odd prime powers, so that more details would be required to in lude p = 2 systems. The perturbation rn for a (b; p)-PRNG system is thus of the \ki king" variety; happening inde nitely, but not on every iteration, and in fa t, happening progressively rarely. Between the \ki ks," iterates of the sequen e are, in ee t, repetitions of a ertain type of

9

normalized linear ongruential PRNG. This is best seen by examining a spe i example. Consider su

essive iterates of the (2; 3)-PRNG system, whose iterates run like so: (x0;

x1 ; : : : ; x26 ; : : :)

= (0;

13

26

11

27 40 81

;

27

;

25 27

;

23 27

;

19 27

;

27

;

1

;

2

;

3 3 22 17

27

;

27

;

4

8

;

;

7

9 9 9 7 14 27

;

27

;

;

5 9 1

;

27

;

1

;

2

;

9 9 2 4 27

;

27

;

8 27

;

16 27

;

5 27

;

10 27

;

20 27

;

;


)

A pattern is lear from the above: The entire sequen e is merely (after the initial 0) a on atenation of subsequen es of respe tive lengths 2
3k 1 , for k = 1; 2; : : :. Ea h subsequen e is the omplete period of a ertain linear ongruential PRNG normalized by 3k . Note that sin e 2 is a primitive root modulo 3k ; k > 0, ea h subsequen e visits, exa tly on e, every integer in [1; 3k 1℄ that is oprime to 3. In general, we an write the following (for simpli ity we use simply  to mean equality on the mod-1 ir le):

 0;  b
0 + r  1p ; b x  ; p

x0 x1

1

2

::: ; bp 2 xp 1  ; p p+1 bp 1 + rp  ; xp  p p2 ::: ; bp2 p 1 (p + 1) xp2 1  ; p2 p2 + p + 1 bp(p 1)(p + 1) 2 = + r ; xp2  p p2 p3 ::: ; and generally speaking,

xpk

1

k  (p

1)=(p

p

k

1)

:

It is evident that upon the 1=pk perturbation, the xn ommen e a run of length '(pk ) = pk 1 (p 1) before the next perturbation, a

ording to the following rule: The subsequen e (xpk 1 ; : : : ; xpk 1 ) runs from 



ak b0 mod pk =pk 10

through 

pk ak b

1 (p 1) 1



mod pk =pk

in lusive, where ak

=

pk p

1 1

is oprime to p. One is aware that, in general, this subsequen e will repeat | i.e., \walk on itself" | unless b is a primitive root of pk . This phenomenon o

urs, of ourse, be ause the powers of b modulo pk have period k whi h period an be less than '(pk ). Later we shall use the symbol Mk to be the multipli ity of these subsequen e orbits; Mk = '(pk )=k . For the moment we observe that the Mk are bounded for all k , in fa t Mk < pa always. We need to argue that (xn ) is equidistributed. For this we shall require an important lemma on exponential sums, whi h lemma we hereby paraphrase using our (b; p)-PRNG nomen lature:



Lemma 4.2 (Korobov, Niederreiter) Given b 2; g d(b; p) = 1, and p an odd prime (so that the order of b modulo pk is k of the text), assume positive integers k; H; J and d = g d(H; pk ) with J k and d < k =1 . Then



J X j k 2iHb =p e j =1

r


2a,

log2 '

p'

1) terms, has dis repan y satisfying

;

where C is a onstant depending only on (b; p) (and thus k -independent).

p

Su h O(log 2 N= N ) results in luding some results on best-possibility of su h bounds, have been a hieved in brilliant fashion by Niederreiter (see [30℄, p. 1009 and [31℄, pp. 169-170). Su h results have histori ally been applied to generators of long period, e.g. PRNGs that have long period for the given modulus. Evidently the only statisti al drawba k for the PRNGs of present interest is that the dis repan y-bound onstant an be rather large (and, of ourse, we need eventually to hain our PRNG subsequen es together to obtain an overall dis repan y). Remark.

Proof.

By the Erd}os{Turan Theorem 2.2(9), for any integer mk 0

D'

< C1 

1 mk

+

m

k X



k 1 1 Mk X

h=1 h



' j =0

e

2ihak

>

0 we have

1

bj =pk

A

;

where Mk is the multipli ity '=k that des ribes how many omplete y les the given subsequen e performs moduloppk . When h < k =1 = pk a the j
j term is, by Lemma 4.2, less than C2 (1 + log k )=  k . Choosing mk = bpk=2 < pk ap onstrains the index p h properly, and we obtain D' < C3 (1= mk + (log(mk ) log k )=  k ). Now we re all p p that Mk = '=k is bounded and observe that C4 ' < mk < C5 ', and the desired dis repan y bound is met. Finally we an address the issue of equidistribution of the (b; p)-PRNG system, by

ontemplating the hain of subsequen es that onstitute the full generator: Theorem 4.5

DN

For the full (b; p)-PRNG sequen e (xn ), the dis repan y for N

< C

log2 N

p

N

>

1 satis es

;

where C is independent of N , and a

ordingly, (xn ) is equidistributed.

Let N = '(p) + '(p2 ) +


+ '(pk 1 ) + J = pk 1 1 + J be the index into the full (xn ) sequen e, where 0 < J < '(pk ), so that we have (k 1) omplete mod-pi

Proof.

12

subsequen es, then J initial terms from a last subsequen e having denominator pk . Also let k > 2a + 1 so that Lemma 4.4 is ee tive. By Lemma 4.3 and the Erd}os{Turan Theorem 2.2(9) we have for any positive integer m 2a

DN
2a + 1) we have:

C3 N < pk < N; C5 N i=k < '(pi ) < C6 N i=k ; C7 N < k < C8N; where the various Cj here are k -and-i-independent, positive onstants. Next, hoose the parameter m = bpk=2 . Now the desired dis repan y bound follows upon repla ement of all pk ; '(pi ); k terms with onstants times powers of N . That (xn ) is equidistributed is a onsequen e of Theorem 2.2(12). Now we an move to the main result For base b > 1 and odd prime p oprime to b, the number

Theorem 4.6

b;p =

X

n1

1

pn bpn

is b-normal.

When b is a primitive root of p and pjj(bp 1 1), it is easy to show that this b;p is b-dense (the (b; p)-PRNG sequen e be omes progressively ner in an obvious way). Proof. For a (b; p)-PRNG system the asso iated onstant as in Theorem 3.1 is = P k pk 1 ) and this is a rational multiple of our  . Theorem 2.2(7) nishes the b;p k1 1=(p b argument. Remark.

Corollary 4.7

2;3 =

The number

1 n2n n=3k >1 X

is 2-normal. 13

It is natural to ask whether b;p is trans endental, whi h question we answer in the positive with: Theorem 4.8 The number b;p for any integers b > 1; p > 2 is trans endental. Remark. Though b;p has been introdu ed for (b; p)-PRNG systems with b > 1; g d(b; p) = 1, p an odd prime, the trans enden y result is valid for any integer pair (b; p) with p > 2 and b > 1. Proof. The elebrated Roth theorem states [36℄ [13℄ that if jP=Q j < 1=Q2+ admits of in nitely many rational solutions P=Q (i.e. if  is approximable to degree 2 +  for some  > 0), then  is trans endental. We show here that b;p is approximable to degree p Æ . Fix a k and write 1 ; b;p = P=Q + n pn

X

n>k

pb

k

where g d(P; Q) = 1 with Q = pk bp . The sum over n gives

jb;p

P=Qj
d 2

fxdg = ff2dE g + f

j

E

1

1

1 A mod 1;

1 + td gg;

where td ! 0. Thus f2d E g is equidistributed i (xn ) is, by Theorem 2.2(10). We believe that at least a weaker, density onje ture should be assailable via the kind of te hnique exhibited in Theorem 5.1, whereby one pro eeds onstru tively, establishing density by for ing the indi ated generator to approximate any given value in [0; 1). P. Borwein has forwarded to us an interesting observation on a possible relation between the number E and the \prime-tuples" postulates, or the more general Hypothesis H of S hinzel and Sierpinski. The idea is | and we shall be highly heuristi here | the 20

fra tional part d(m + 1)=2 + d(m + 2)=22 +


might be quite tra table if, for example, we have + 1 = p1 ; m+2 = 2p2 ;

m

::: m

+n =

; npn ;

at least up to some n = N . where the pi > N are all primes that appear in an appropriate \ onstellation" that we generally expe t to live very far out on the integer line. Note that in the range of these n terms we have d(m + j ) = 2d(j ). Now if the tail sum beyond N is somehow suÆ iently small, we would have a good approximation d(m + N )=2 2m E g

f



(1) + d(2)=2 +


= 2E :

d

But this implies in turn that some iterate f2m E g revisits the neighborhood of an earlier iterate, namely f2E g. It is not lear where su h an argument|espe ially given the heuristi aspe t|should lead, but it may be possible to prove 2-density (i.e. all possible nite bitstrings appear in E ) on the basis of the prime k -tuples postulate. That onne tion would of ourse be highly interesting. Along su h lines, we do note that a result essentially of the form: \The sequen e (f2m E g) ontains a near-miss (in some appropriate sense) with any given element of (fnE g)" would lead to 2-density of E , be ause, of ourse, we know E is irrational and thus (fnE g) is equidistributed.

6. Spe ial numbers having \nonrandom" digits This se tion is a tour of side results in regard to some spe ial numbers. We shall exhibit numbers that are b-dense but not b-normal, un ountable olle tions of numbers that are neither b-dense nor b-normal, and so on. One reason to provide su h a tour is to dispel any belief that, be ause almost all numbers are absolutely normal, it should be hard to use algebra (as opposed to arti ial onstru tion) to \point to" nonnormal numbers. In fa t it is not hard to do so. First, though, let us revisit some of the arti ially onstru ted normal numbers, with a view to reasons why they are normal. We have mentioned the binary Champernowne, whi h an also be written C2

=

X1

n=1

n

2F (n)

where the indi ated exponent is: ( ) =

F n

X log + n

n

k=1

b

2 k :

Note that the growth of the exponent F (n) is slightly more than linear. It turns out that if su h an exponent grows too fast, then normality is ruined. More generally, there is the 21

lass of Erd}os{Copeland numbers [14℄, formed by (we remind ourselves that the (
) notation means digits are on atenated, and here we on atenate the base-b representations)

 = 0:(a1)b (a2)b




where (an ) is any in reasing integer sequen e with an = O(n1+ ), any  > 0. An example of the lass is 0:(2)(3)(5)(7)(11)(13)(17)


10 ; where primes are simply on atenated. These numbers are known to be b-normal, and they all an be written in the form G(n)=bF (n) for appropriate numerator fun tion G and, again, slowly diverging exponent F . We add in passing that the generalized Mahler numbers (for any g; h > 1)

P

Mb (g ) = 0:(g 0 )b (g 1 )b (g 2 )b




are known at least to be irrational [32℄, [38℄, and it would be of interest to establish perturbation sums in regard to su h numbers. In identally, it is ironi that the some of the methods for establishing irrationality of the Mb (g ) are used by us, below, to establish nonnormality of ertain forms. We have promised to establish that

 =

X n=2

n2



n 1

is 2-dense but not 2-normal. Indeed, in the n2 -th binary position we have the value n, and sin e for suÆ iently large n we have n2 (n 1)2 > 1 + log2 n, the numerator n at bit position n2 will not interfere (in the sense of arry) with any other numerator. One may bury a given nite binary string in some suÆ iently large integer n (we say buried be ause a string 0000101, for example, would appear in su h as n = 10000101), when e said string appears in the expansion. Note that the divergen e of the exponent n2 is a key to this argument that  is 2-dense. As for the la k of 2-normality, it is likewise evident that almost all bits are 0's. Let is hereby onsider faster growing exponents, to establish a more general result, yielding a lass of b-dense numbers none of whi h are b-normal. We start with a simple but quite useful lemma. For polynomials P with nonnegative integer oeÆ ients, deg P > 0, and for any integer b > 1, the sequen e Lemma 6.1

(flogb P (n)g : n = 1; 2; 3; : : :)

is dense in [0; 1). Proof. For d = deg P , let P (x) = ad xd + : : : + a0 . Then logb P (n) = logb ad + d(log n)= log b + O(1=n). Sin e log n = 1 + 1=2 + 1=3 + : : : + 1=n + O(1=n2 ) diverges with n but by vanishing in rements, the sequen e (fd(log n)= log bg) and therefore the desired (flogb P (n)g) are both dense by Theorem 2.2(10). 22

Now we onsider numbers onstru ted via superposition of terms P (n)=bQ(n) , with a growth ondition on P; Q: Theorem 6.2

0, the number

 =

For polynomials P; Q with nonnegative integer oeÆ ients, deg Q > deg P >

X P (n) Q(n)  b

n 1

is b-dense but not b-normal. Proof. The nal statement about nonnormality is easy: Almost all of the base-b digits are 0's, be ause logb P (n) = o(Q(n) Q(n 1)). For the density agrument, we shall show that for any r 2 (0; 1) there exist integers N0 < N1 < : : : and d1 ; d2 ; : : : with Q(Nj 1 ) < dj < Q(Nj ), su h that lim fbdj g = r: !1

j

This in turn implies that (fbd g) : d = 1; 2; : : :g) is dense, hen e  is b-dense. Now for any as ending hain of Ni with N0 suÆ iently large, we an assign integers dj a

ording to

Q(Nj ) > dj = Q(Nj ) + logb r logb P (Nj ) + j > Q(Nj 1 ) where j

2 [0; 1). Then

P (Nj )=bQ(Nj )

dj

= 2j r:

However, (flogb P (n)g) is dense, so we an nd an as ending Nj - hain su h that lim j = 0. Sin e dj < Q(Nj ) we have

fbd g j

=

0 b r + X P (Nj + k)=bQ N (

j

j +k )

1 d A j

mod 1

k>0

and be ause the sum vanishes as j

! 1, it follows that  is b-dense.

Consider the interesting fun tion [27℄, p. 10: 1 bnx X f (x) = : n n=1 2 The fun tion f is reminis ent of a degenerate ase of a generalized polylogarithm form| that is why we en ountered su h a fun tion during our past [3℄ and present work. Regardless of our urrent onne tions, the fun tion and its variants have ertainly been studied, espe ially in regard to ontinued fra tions [16℄ [17℄ [27℄ [8℄ [29℄ [5℄ [1℄ [17℄ [9℄, [10℄. If one 23

plots the f fun tion over the interval x 2 [0; 1), one sees a brand of \devil's stair ase," a

urve with in nitely many dis ontinuities, with verti al-step sizes o

urring in a fra tal pattern. There are so many other interesting features of f that it is eÆ ient to give another olle tive theorem. Proofs of the harder parts an be found in the aforementioned referen es. Theorem 6.3 (Colle tion)

the argument x 2 (0; 1),

For the \devil's stair ase" fun tion

f

de ned above, with

1.

f

is monotone in reasing.

2.

f

is ontinuous at every irrational x, but dis ontinuous at every rational x.

3. For rational

x

= p=q , lowest terms, we have

( ) =

f x

2q

1

1

1 X +

m=1

1 b m=x

2

but when x is irrational we have the same formula without the 1=(2q term (as if to say q ! 1).

1) leading

4. For irrational x = [a1; a2; a3; : : :℄, a simple ontinued fra tion with onvergents (pn =qn ), we have: ( ) = [A1; A2; A3; : : :℄:

f x

where the elements An are: An

= 2qn

2

2an qn 1 1 : 2qn 1 1

Moreover, if (Pn =Qn ) denote the onvergents to f (x), we have Qn

= 2qn

1:

5. f (x) is irrational i

x

is.

6. If x is irrational then f (x) is trans endental. 7. f (x) is never 2-dense and never 2-normal. 8. The range

R=

f

([0; 1)) is a null set (measure zero).

9. The density of 1's in the binary expansion of f (x) is inverse fun tion on the range R, is just 1's density. 24

x

itself; a

ordingly, f 1 , the

Some ommentary about this fas inating fun tion f is in order. We see now how f an be stri tly in reasing, yet manage to \ ompletely miss" 2-dense (and hen e 2-normal) values: Indeed, the dis ontinuities of f are dense. The notion that the range R be a null set is surprising, yet follows immediately from the fa t that almost all x have 1's density equal to 1=2. The beautiful ontinued fra tion result allows extremely rapid omputation of f values. The fra tion form is exempli ed by the following evaluation, where x is the re ipro al of the golden mean and the Fibona

i numbers are denoted F : i

!

2 p 1+ 5 = [2 0 ; 2 1 ; 2 2 ; : : :℄ 1 ; = 1 + 2+ 1 1

f (1= ) = f

F

F

F

2 4+ 1 1 8+

:::

It is the superexponential growth of the onvergents to a typi al f (x) that has enabled trans enden y proofs as in Theorem 6.2(6). An interesting question is whether (or when) a ompanion fun tion

g (x) =

1 fnxg X n=1

2

n

an attain 2-normal values. Evidently

g (x) = 2x f (x); and, given the established nonrandom behavior of the bits of f (x) for any x, one should be able to establish a orrelation between normality of x and normality of g (x). One reason why this question is interesting is that g is onstru ted from \random" real values fnxg (we know these are equidistributed) pla ed at unique bit positions. Still, we did look numeri ally at a spe i irrational argument, namely

x =

X

1 2

1 n(n+1)=2

n

and noted that g (x) almost ertainly is not 2-normal. For instan e, in the rst 66,420 binary digits of g (x), the string '010010' o

urs 3034 times, while many other length-6 strings do not o

ur at all. 7. Con lusions and open problems

Finally, we give a sampling of open problems pertaining to this interdis iplinary eort:



We have shown that for (b; p)-PRNG systems, the numbers  are ea h b-normal. What about -normality of su h a number for not a rational power of b? b;p

25



WhatP te hniques might allow is to relax the onstraint of rapid growth n = p in our sums 1=(nb ), in order to approa h the spe ta ular goal of resolving the suspe ted b-normality of log(b=(b 1))? One promising approa h is to analyze dis repan y for relatively short parts of PRNG y les. In [25℄, p. 171, [26℄ there appear exponential sum bounds for relatively short indi es J into the last y le. It ould be that su h theorems an be used to slow the growth of the summation index n.



It is lear that the dis repan y bound given in Theorem 4.5, based in turn on the Korobov{Niederreiter Lemma 4.2, is \overkill," in the sense that we only need show D = o(N ) to a hieve a normality proof. Does this mean that the numbers  are somehow \espe ially normal"? For su h a question one would perhaps need extra varian e statisti s of a normal number; i.e., some measures beyond the \fair frequen y" of digit strings.

k

n

N

b;p



We have obtained rigorous results for PRNGs that either have a ertain syn hronization, or have extremely small \tails." What te hniques would strike at the intermediate s enario whi h, for better or worse, is typi al for fundamental onstants; e.g., the onstants falling under the umbrella of Hypothesis A?



With our (b; p)-PRNG systems we have established a ountable in nity of expli it b-normal numbers. What will it take to exhibit an un ountable, expli it olle tion?

 

What are the properties of \ ontra ted" PRNGs, as exempli ed in Se tion 4? Does polynomial-time (in log n) resolution of the n-th digit for our  onstants give rise to some kind of \trap-door" fun tion, as is relevant in ryptographi appli ations? The idea here is that it is so very easy to nd a given digit even though the digits are \random." (As in: Multipli ation of n-digit numbers takes polynomial time, yet fa toring into multiples is evidently very mu h harder.) b;p

8. A knowledgments

The authors are grateful to J. Borwein, P. Borwein, D. Bowman, D. Broadhurst, J. Buhler, D. Copeland, H. Ferguson, M. Ja obsen, J. Lagarias, R. Mayer, H. Niederreiter, S. Ploue, A. Pollington, C. Pomeran e, J. Shallit, S. Wagon, T. Wieting and S. Wolfram for theoreti al and omputational expertise throughout this proje t. We would like to dedi ate this work to the memory of Paul Erd}os, whose ingenuity on a ertain, exoti analysis dilemma|the hara ter of the Erd}os{Borwein onstant E |has been a kind of guiding light for our resear h. When we oauthors began|and this will surprise no one who knows of Erd}os|our goals were presumed unrelated to the Erd}os world. But later, his way of thinking meant a great deal. Su h is the persona of genius, that it an speak to us even a ross robust boundaries.

26

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