On the digits in the base-$b$ expansion of smooth numbers (Analytic Number Theory and Related Areas)

Abstract

Erdös [4] conjectured that, for any integer mgeq 9, the digit 2 appears at least once in the ternary expansion of 2^{m}. More precisely, Dupuy and Weirich [3] conjectured that. for any sufficiently large m, the digits 0, 1, and 2 appear "uniformly" in the ternary expansion of 2^{m}. This is still open. Stewart [10] obtained a lower bound for the number of nonzero digits in the ternary expansion of 2^{m}, thus giving (very) partial results of "uniformity". In this report, we investigate the number of nonzero digits in the base-b expansion of more general smooth numbers and introduce the main results established in [2].