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Number Theory

Source: Polish Math Olympiad 2021 2nd round p3 day 1

February 13, 2021

number theorySum of Squares

Problem Statement

Positive integers

a,b,z

satisfy the equation

ab=z^2+1

. Prove that there exist positive integers

x,y

such that

\frac{a}{b}=\frac{x^2+1}{y^2+1}