On sparse perfect powers

TL;DR

This paper investigates the infinite occurrence of perfect powers with exactly k non-zero digits in base x, except for a specific known finite case, extending understanding of digit patterns in perfect powers.

Contribution

It proves the infinitude of perfect powers with a fixed number of non-zero digits in any base x, except for the special case where x=2 and k=4, which is known to be finite.

Findings

Infinitely many perfect powers with k non-zero digits in base x for all cases except x=2, k=4.

Confirmed the known finiteness for the case x=2, k=4.

Extended the understanding of digit patterns in perfect powers.

Abstract

This work is devoted to proving that, given an integer $x \ge 2$, there are infinitely many perfect powers, coprime with $x$, having exactly $k \ge 3$ 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.