On sparse 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.

arXiv comment9 pages