Stoneham number explained

In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Stoneham (1920–1996). For coprime numbers b, c > 1, the Stoneham number α_b,c is defined as

\alpha _{b,c} = \sum _{n = c ^{k>1}} (1)/(b ^{nn}) = \sum ^{inf ty} _{k =1} (1)/(b ^{c ^{k}} c ^{k})

It was shown by Stoneham in 1973 that α_b,c is b-normal whenever c is an odd prime and b is a primitive root of c². In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of α_b,c.^[1]

References

• .
• Book: Bugeaud, Yann . Distribution modulo one and Diophantine approximation . Cambridge Tracts in Mathematics . 193 . Cambridge . . 2012 . 978-0-521-11169-0 . 1260.11001.
• 0276.10028 . Stoneham . R.G. . On absolute $(j,ε)$-normality in the rational fractions with applications to normal numbers . . 22 . 3 . 277–286 . 1973 . 10.4064/aa-22-3-277-286 . free .
• 0276.10029 . Stoneham . R.G. . On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers . . 22 . 4 . 371–389 . 1973 . 10.4064/aa-22-4-371-389 . free .

Notes and References

1. Bailey . David H. . Crandall . Richard E. . 2002 . Random Generators and Normal Numbers . Experimental Mathematics . 11. 4 . 527–546 . 10.1080/10586458.2002.10504704 . 8944421 .