斯托纳姆数

斯托纳姆数（Stoneham numbers）是一种特别的实数，得名自数学家李查·斯托纳姆（Richard G. Stoneham, 1920–1996）。对于互质且大于1的整数b和c，可以定义斯托纳姆数α_b,c 如下：

$\alpha _{b,c}=\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}$

斯托纳姆在1973年证明只要c为奇质数，而b是c²的原根，则斯托纳姆数是以b为底的正规数。

参考资料

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