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This question already has an answer here:

If $p$ is a the sum of two squares are integers $a$ and $b$ s.t. $p=a^2 + b^2$ then $p=1$ mod $4$. I need help proving that.

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marked as duplicate by draks ..., Nick Peterson, Dan Rust, Norbert, Lord_Farin Oct 17 '13 at 11:30

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Hints:

Take $\;\Bbb Z/4\Bbb Z=\{0,1,2,3\}\pmod 4\;$ and observe what's the general form of its squares.

Now take the expression $\;a^2+b^2\pmod 4\;$ . Taking into account the first point and the fact that $\;p\;$ is a prime, what can you deduce?

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