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AN ARITHMETICAL EXCURSION VIA STONEHAM NUMBERS

  • MICHAEL COONS (a1)

Abstract

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.

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References

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[1]Aragón Artacho, F. J., Bailey, D. H., Borwein, J. M. and Borwein, P. B., ‘Walking on real numbers’, Math. Intelligencer 35(1) (2013), 4260.
[2]Bailey, D. H. and Borwein, J. M., ‘Normal numbers and pseudorandom generators’, in: Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday (Springer, New York, 2013).
[3]Bailey, David H. and Crandall, Richard E., ‘Random generators and normal numbers’, Experiment. Math. 11(4) (2003), 527546.
[4]Bailey, David H. and Misiurewicz, Michał, ‘A strong hot spot theorem’, Proc. Amer. Math. Soc. 134(9) (2006), 24952501.
[5]Champernowne, D. G., ‘The construction of decimals normal in the scale of ten’, J. Lond. Math. Soc. 8(4) (1933), 254.
[6]Nishioka, Keiji, ‘Algebraic function solutions of a certain class of functional equations’, Arch. Math. 44 (1985), 330335.
[7]Nishioka, Kumiko, Mahler Functions and Transcendence, Lecture Notes in Mathematics, 1631 (Springer, Berlin, 1996).
[8]Mahler, Kurt, ‘Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen’, Math. Ann. 101(1) (1929), 342366.
[9]Mahler, Kurt, ‘Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen’, Proc. Kon. Nederlandsche Akad. v. Wetenschappen 40 (1937), 421428.
[10]Rosen, Kenneth H., Elementary Number Theory and Its Applications. 5th edn (Addison-Wesley, Reading, MA, 2005).
[11]Schmidt, Wolfgang M., ‘On normal numbers’, Pacific J. Math. 10 (1960), 661672.
[12]Stoneham, R. G., ‘On absolute (j, ε)-normality in the rational fractions with applications to normal numbers’, Acta Arith. 22 (1972/73), 277286.
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AN ARITHMETICAL EXCURSION VIA STONEHAM NUMBERS

  • MICHAEL COONS (a1)

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