On the Equations xn±yn = zm

Abstract

1. We determine first an infinity of solutions of the equation x n-yn=zm when m, n are relatively prime. Let p be an odd prime; then q= p m has a primitive root g, say, and so, if ϕ is Euler's function, x= g ϕ(q) satisfies x n= gnϕ(q)) = l (mod q).