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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see also Sum of two squares $n = a^2 + b^2$ – draks ... Oct 17 '13 at 10:33
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2This isn't actually true...it could be $p \equiv 0 \pmod 4$. Do you mean to assume that $p$ is prime? – DanielV Oct 17 '13 at 10:35
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Related : http://math.stackexchange.com/questions/122048/1-is-a-quadratic-residue-modulo-p-if-and-only-if-p-equiv-1-pmod4 – lab bhattacharjee Oct 17 '13 at 10:37
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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?
DonAntonio
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