Problem

Source: 2024 Nepal TST P1

Tags: number theory, Parity, modular arithmetic



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