Nonnormality of the Stoneham Constants
David H Bailey
Lawrence Berkeley Laboratory
http://crd.lbl.gov/~dhbailey

1Normal numbers
Given an integer b > 1, a real number x is b-normal (or “normal base b”) if
every m-long string of digits in the base-b expansion of x appears with
precisely the expected limiting frequency b-m.
Using measure theory, it is can be shown that almost all real numbers are bnormal for a given integer base b. In fact, almost all reals are b-normal
for all integer bases b > 1 simultaneously (i.e., are “absolutely normal”).
These are widely believed to be b-normal, for all integer bases b > 1:
π = 3.1415926535…
e = 2.7182818284…
sqrt(2) = 1.4142135623…
log(2) = 0.6931471805…
Every irrational algebraic number (this conjecture is due to Borel).

But there are no normality proofs for any of these constants in any base, nor
are there any nonnormality results for any of these constants.
Until recently, normality proofs were known only for a few relatively contrived
examples such as Champernowne’s constant =0.123456789101112131415… (which is 10-normal).
There are no proofs of absolute normality for any explicit constant.
2

Random walk on the first two billion
bits of pi

DHB, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu and A. Yee, “An Empirical Approach to theNormality of Pi,” manuscript, 26 Nov 2011, available athttp://crd.lbl.gov/~dhbailey/dhbpapers/normality.pdf.
This also includes numerous statistical analyses of the digits of pi.
3A result for algebraic numbersIf x is algebraic of degree d > 1, then its binary expansion through position nmust have at least C n1/d 1-bits, for all sufficiently large n and some C that
depends on x.
Example: The first n binary digits of sqrt(2) must have at least sqrt(n) ones.
In this example, the result follows by noting that the one-bit count of theproduct of two integers is less than or equal to the product of the one-bitcounts of the two integers. The more general result above requires amore sophisticated approach.
However, note that these results are still a far cry from full normality.

DHB, J. M. Borwein, R. E. Crandall and C. Pomerance, “On the Binary Expansions of Algebraic Numbers,”
Journal of Number Theory Bordeaux, vol. 16 (2004), pg. 487-518.
4

Pseudorandom sequences and thenormality of log 2 and pi
Consider the sequence x0 = 0, andxn =✓1
2xn 1 +n◆

mod 1

If it can be demonstrated that this sequence is equidistributed in the unit
interval, then this would imply that log 2 is 2-normal.
Similarly, consider the sequence x0 = 0, andxn =✓16xn 1 +120n
89n + 161024n3 + 712n2 206n + 212512n4

◆

mod 1

If it can be shown that this sequence is equidistributed in the unit interval,then this would imply that pi is 16-normal (and hence 2-normal).
DHB and R. E. Crandall, “On the Random Character of Fundamental Constant Expansions,” Experimental
Mathematics, vol. 10, no. 2 (Jun 2001), pg. 175-190.

5

A class of provably normal constants
DHB and Richard Crandall have shown that an infinite class of constants is bnormal (and thus bm-normal for any positive integer m), for instance:
2,3=

1n 23n3n=1=0.041883680831502985071252898624571682426096 . . .10=

0.0ab8e38f 684bda12f 684bf 35ba781948b0f cd6e9e0 . . .16

This particular constant was proven 2-normal by Stoneham in 1971. We
extended this to the case where (2,3) are any pair (b,c) of relatively prime
integers > 1, and also to an uncountable class (here rn is n-th bit of r in [0,1)):1(r)=2,3n 23n +rn
3n=1
More recently, DHB and Michal Misiurewich established the α2,3 result more
simply by means of a “hot spot” lemma proved using ergodic theory.
DHB and M. Misiurewicz, “A Strong Hot Spot Theorem," Proceedings of the American Mathematical Society,vol. 134 (2006), no. 9, pg. 2495-2501.
DHB and R. E. Crandall, “Random Generators and Normal Numbers,” Experimental Mathematics, vol. 11, no.
4 (2002), pg. 527-546.
6

A nonnormality result
Although α2,3 is provably 2-normal, surprisingly enough it is NOT 6-normal.
Note that we can0write1
blog3 nc1XXmn m n 3m
nn m n 3 A@mod 1 +326 ↵2,3 mod 1 =32m=1m=blog3 nc+1

The first portion of this expression is zero, since all of the terms in thesummation are integers. When n = 3m, the second portion is accurately
approximated by the first term of the series. Thus,m

63m↵2,3 mod 1 ⇡3 3
43m+1

Because this is so small for large m, this means the base-6 expansion ofα2,3 has long stretches of zeroes beginning at positions 3m + 1. Thisobservation can be fashioned into a rigorous proof of nonnormality.
DHB and J. M. Borwein, “Normal Numbers and Pseudorandom Generators,” available at
http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf.
7α2,3 in base-6 digits
0.
013014043000333425113050213000000124355504543223301150024352532055135234354101
043000000000000000051411300540405554553031442504334351012413452351125142125134
505503545015053522052044340452151505102411552500425130051124454001044131150032
420303213000000000000000000000000000000000000000000142120343111214520135254453
421134122402205253010542044235524110554150155204350414555400310145303033532002
534340401301240104453254343502142020432415025555101004043300045544114501031331
451151014451412344334234124005513133350454235305531511535015334524354502500555
214530542343421530350125024205404135451231323245353031534552304115020154242121
145201542222534340340450530123325534440443103332445332141415014233454542412432
031253400501341502455144043000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000031335054
244443111105553414105201454021341231300142433313311551144002552443344143003224
335300102353515310402031320020533130021221200323413302222122521000035254543002
545040035245112205252324300200332523104022522341033520112253412034105440205323
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
114422021202242031254124153534242220443313314424101440231132320424442150031332
450441322425500055412431400314402120525313133331045531400355154135352044001245
423040225112500244121430424020245242243022325455215430225105432551034135040520
213000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000005304302452321100553442520302

8A general nonnormality result for theStoneham alpha constants
Given co-prime integers b > 1 and c > 1, and integers p, q, r > 0, with neitherb nor c dividing r, let B = bp cq r, and assume this condition holds:

D = cq/p r1/p /bc 1 < 1
Then the constant
1X1↵b,c =nncc bn=1
is not B-normal. Thus, for example, αb,c is b-normal but not bc-normal.
It is not known whether or not this is a complete catalog of the bases forwhich a Stoneham constant is nonnormal.
Ref: DHB, “Nonnormality of the Stoneham constants, manuscript, 8 Dec 2011, available at
http://crd.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf.
9Questions· There are many cases still not covered -- e.g., is α2,3 3-normal, or not?
· Can we prove normality for sums of alphas, i.e. α2,3 + α2,5?
§This may be possible, although quite difficult – we are working on this now.

· Can we prove nonnormality for sums of alphas, i.e., α2,3 + α3,2?
§

Yes – we are working on this now.

· Are there any “relatives” of the Stoneham constants that permit similar
proofs of normality or nonnormality?
· Can absolute normality (i.e., b-normal for all integer bases b > 1simultaneously) be established for ANY explicit constant? Almost all real
numbers are absolutely normal, but there are no explicit examples, to ourknowledge.
· Can normality (to any base) be established for any of the “natural”
irrational constants of mathematics – e, pi, sqrt(2), log(2), zeta(3), etc.?
· For that matter, can nonnormality be proven for any of the natural
irrational constants?
10