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AN ARITHMETICAL EXCURSION VIA STONEHAM NUMBERS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 96 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 27 March 2014, pp. 303-315
- Print publication:
- June 2014
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- Article
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Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ be a prime and 
$b$ a primitive root of 
$p^2$. In this paper, we give an explicit formula for the number of times a value in 
$\{0,1,\ldots,b-1\}$ occurs in the periodic part of the base-
$b$ expansion of 
$1/p^m$. As a consequence of this result, we prove two recent conjectures of Aragón Artacho et al. [‘Walking on real numbers’, Math. Intelligencer35(1) (2013), 42–60] concerning the base-
$b$ expansion of Stoneham numbers.
