Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Prove it United States

Problem:
Let $n$ be a positive integer. Prove that there exist distinct positive integers $x$ , $y$ , $z$ such that
$x^{n-1} + y^n = z^{n+1}.$

Solution

Solution:
One solution is
$x = 2^{n^2} 3^{n+1}, \quad y = 2^{n^2 - n} 3^n, \quad z = 2^{n^2 - 2n + 2} 3^{n-1}.$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.