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x^n = y^n mod n

Source: 2023 Swedish Mathematical Competition p5

March 24, 2024
number theory

Problem Statement

(a) Let xx and yy be integers. Prove that x=yx = y if xn≡ynx^n \equiv y^n mod nn for all positive integers nn.
(b) For which pairs of integers (x,y)(x, y) are there infinitely many positive integers nn such that xn≡ynx^n \equiv y^n mod nn?
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