Integers with digits $0$ or $1$

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.