AoPS - Problem

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Problem

Source: Romania TST for BMO 1994,first problem

Tags: number theory

Medjl

06.09.2017 03:08

Prove that if $n$ is a square-free positive integer, there are no coprime positive integers $x$ and $y$ such that $(x + y)^3$ divides $x^n+y^n$

RagvaloD

06.09.2017 12:33

By contradiction Obvious, that $n$ is odd Let $p|(x+y)$ Then $v_p((x+y)^3)=3v_p(x+y)$ And $v_p(x^n+y^n)=v_p(n)+v_p(x+y)$ As $(x+y)^3|x^n+y^n \to 3v_p(x+y) \leq v_p(n)+v_p(x+y) \to v_p(n) \geq 2v_p(x+y) \geq 2$ so $p^2|n$ - contradiction