Integers with digits 0 or 1

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Abstract
Let g ⩾ 2 g \geqslant 2 be a given integer and L \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 k > 1 , we study congruences l ≡ a ( mod k ) l \equiv a\;\pmod k , l ∈ L l \in \mathcal {L} and show that such a congruence either has infinitely many solutions, or no solutions in L \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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