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I wonder if that is idea is part of what sparked Fermat-Euler’s method of the sum of two squares. They said if p is prime and p = 1 mod 4 (or 2) then p is expressible as p = a^2 + b^2 for some integers a, b. The proof is much more complicated than the proof for this formula is and requires imaginary numbers, but that makes sense because we are dealing with addition inside the squares instead of subtraction.

Nice explanation!

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