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.are all 2020 Mathematics Subject Classification. 11D41, 11P99.