On sparse perfect powers

Abstract

This work is devoted to proving that, given an integer x ≥ 2, there are infinitely many perfect powers, coprime with x, having exactly k ≥ 3 non-zero digits in their base x representation, except for the case x = 2, k = 4, for which a known finiteness result by Corvaja and Zannier holds. Introduction Let k and x be positive integers, with x ≥ 2. In this work, we will study perfect powers having exactly k non-zero digits in their representation in a given basis x. These perfect powers are exactly (up to dividing by a suitable factor) the set solutions of the Diophantine equation (1) y = c0 + k−1