On differences of perfect powers and prime powers

Abstract

Given a prime number $q$ and a squarefree integer $C_1$, we develop a method to explicitly determine the tuples $(y,n,\alpha )$ for which the difference $y^n-q^{\alpha }$ has squarefree part equal to $C_1$. Our techniques include the combination of the local information provided by Galois representations of Frey-Hellegouarch curves with the effective resolution of Thue-Mahler equations, as well as the use of improved lower bounds for $q-$adic and complex logarithms. As an application of this methodology, we will completely resolve the case when $1\le C_1\le 20$ and $2\le q<25$.

40 pages