Skolem's conjecture confirmed for a family of exponential equations, III

Abstract

We prove Skolem's conjecture for the exponential Diophantine equation an+tbn=±cn under some assumptions on the integers a,b,c,t. In particular, our results together with Wiles' theorem imply that for fixed coprime integers a,b,c Fermat's equation an+bn=cn has no integer solution n≥3 modulo m for some modulus m depending only on a,b,c. We also provide a generalization where in the equation bn is replaced by a product b1k1⋯bℓkℓ.