The number 1013 is prime. Write 2026 as a sum of two squares of integers, and explain why the representation is unique once you allow for swapping the two terms and changing signs.

2026 = 45^2 + 1^2 = 2025 + 1, and this is the only representation up to order and sign.

Fermat's theorem: an odd prime is a sum of two squares exactly when it is 1 modulo 4. Which residue is 1013?

中文 / English

Olympiad

Competition-shaped. The first move is rarely the obvious one; the proof carries the weight.

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