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f is N to N, x^2-y^2+2y(f(x)+f(y)) is a perfect square

Source: 10th European Mathematical Cup - Problem S3

December 22, 2021
functional equationnumber theoryEuropean Mathematical Cupemc

Problem Statement

Let N\mathbb{N} denote the set of all positive integers. Find all functions f:N→Nf:\mathbb{N}\to\mathbb{N} such that x2−y2+2y(f(x)+f(y))x^2-y^2+2y(f(x)+f(y)) is a square of an integer for all positive integers xx and yy.
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