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x^n + y^n = p^k , if n > 1 is odd, and p is an odd prime, then n is a power of p

Source: New Zealand NZMOC Camp Selection Problems 2018 p9

September 19, 2021
diophantineDiophantine equationnumber theoryPower

Problem Statement

Let x,y,p,n,kx, y, p, n, k be positive integers such that xn+yn=pk.x^n + y^n = p^k. Prove that if n>1n > 1 is odd, and pp is an odd prime, then nn is a power of pp.
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