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On differences of perfect powers and prime powers
Pedro-José Cazorla García
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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-q^α 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.
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