On differences of perfect powers and prime powers

Given a prime number q and a squarefree integer C 1 , we develop a method to explicitly determine the tuples ( y , n , α ) for which the difference y n − q α 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 ≤ C 1 ≤ 20 and 2 ≤ q < 25 . • A multi-Frey approach is used combining a Frey–Hellegouarch curve with its quadratic twist. • The reduction of the Frey–Hellegouarch curve is combined with Galois theory. • The local information of the modular method is used in conjunction with Thue–Mahler equations. • An application of linear forms in complex and q -adic logarithms give a bound which is 50% smaller than previous applications.