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Exponents modulo 4

Source: 2024 Nepal TST P1

April 12, 2024
number theoryParitymodular arithmetic

Problem Statement

Let a,ba, b be positive integers. Prove that if ab+ba≡3(mod4)a^b + b^a \equiv 3 \pmod{4}, then either a+1a+1 or b+1b+1 can't be written as the sum of two integer squares.
(Proposed by Orestis Lignos, Greece)
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