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 αb,c is defined as
α b , c = ∑ n = c k > 1 1 b n n = ∑ k = 1 ∞ 1 b c k c k {\displaystyle \alpha _{b,c}=\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}}
It was shown by Stoneham in 1973 that αb,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 αb,c.
References
Bailey, D. H.; Crandall, R. E. (2002), "Random generators and normal numbers" (PDF), Experimental Mathematics, 11 (4): 527–546, doi:10.1080/10586458.2002.10504704, S2CID 8944421. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001. Stoneham, R.G. (1973). "On absolute $(j,ε)$-normality in the rational fractions with applications to normal numbers". Acta Arithmetica. 22 (3): 277–286. doi:10.4064/aa-22-3-277-286. Zbl 0276.10028. Stoneham, R.G. (1973). "On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers". Acta Arithmetica. 22 (4): 371–389. doi:10.4064/aa-22-4-371-389. Zbl 0276.10029.