Abstract

We solve the equation xa + xb + 1 = yq in positive integers x, y, a, b and q with a > b and q ≥ 2 coprime to φ(x). This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and p-adic logarithms, the hypergeometric method of Thue and Siegel applied p-adically, local methods, and the algorithmic resolution of Thue equations.

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