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

Abstract We prove Skolem's conjecture for the exponential Diophantine equation a n + t b n = ± c n 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 a n + b n = c n 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 b n is replaced by a product b 1 k 1 ⋯ b l k l .