Stoneham number
From HandWiki
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
- [math]\displaystyle{ \alpha_{b,c} = \sum_{n=c^k\gt 1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{c^k}c^k} }[/math]
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.[1]
References
- ↑ Bailey, David H.; Crandall, Richard E. (2002). "Random Generators and Normal Numbers". Experimental Mathematics 11 (4): 527–546. doi:10.1080/10586458.2002.10504704. https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704.
- "Random generators and normal numbers", Experimental Mathematics 11 (4): 527–546, 2002, doi:10.1080/10586458.2002.10504704, http://www.emis.de/journals/EM/expmath/volumes/11/11.4/pp527_546.pdf.
- Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0.
- 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.
- 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.
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