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Either \frac{n^4+m}{m^2+n^2} or \frac{n^4-m}{m^2-n^2} is integer

Source: 2023 Turkey Junior National Olympiad P3

December 22, 2023
number theoryDivisibilityrelatively prime

Problem Statement

Let m,nm,n be relatively prime positive integers. Prove that the numbers n4+mm2+n2n4−mm2−n2\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2} cannot be integer at the same time.
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