Abstract
Using recent results of the author on the number of solutions of S-unit equations, we give upper bounds for the number of solutions of linear equations in integers that have bounded sum of digits in b-adic expansion. In particular we prove: Let b 1, b 2 be multiplicatively independent natural numbers. Let c ≥ 1 be a constant. Then there exists a constant c 1 dependent only upon c and the number of distinct prime factors of b 1 b 2 such that the number of naturals n, whose sum of digits in base b 1 as well as in base b 2 does not exceed c, is bounded by c 1.
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