On differences of perfect powers and prime powers

Abstract

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Given a prime number q and a squarefree integer C1, we develop a method to explicitly determine the tuples (y, n, α) for which the difference yn − qα has squarefree part equal to C1. 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 ≤ C1 ≤ 20 and 2 ≤ q < 25.