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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Integers with digits 0 or $ 1$

Authors: D. H. Lehmer, K. Mahler and A. J. van der Poorten
Journal: Math. Comp. 46 (1986), 683-689
MSC: Primary 11A63; Secondary 11Y99
MathSciNet review: 829638
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Abstract: Let $ g \geqslant 2$ be a given integer and $ \mathcal{L}$ the set of nonnegative integers which may be expressed in base g employing only the digits 0 or 1. Given an integer $ k > 1$, we study congruences $ l \equiv a\;\pmod k$, $ l \in \mathcal{L}$ and show that such a congruence either has infinitely many solutions, or no solutions in $ \mathcal{L}$. There is a simple criterion to distinguish the two cases. The casual reader will be intrigued by our subsequent discussion of techniques for obtaining the smallest nontrivial solution of the cited congruence.

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Article copyright: © Copyright 1986 American Mathematical Society