On Fermat's last theorem.
This paper focuses on the Diophantine equation , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if , where is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].