Let $\mathbb{F}_p$ be a field of prime characteristic $p>0$.The question is what we can say about the solvability of the equation $x^2+y^2=-1$ in $\mathbb{F}_p$ for every prime $p$. I have found out that there are $\frac{p-1}{2}$ different squares in the field $\mathbb{F}_p$. I was thinking about the circle of radius $kp-1$ has integer point for some $k$.
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1A duplicate. Did you search the site? – Jyrki Lahtonen Jun 15 '18 at 06:57
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Oops. Not a good target. Looking for an older and a simpler version. – Jyrki Lahtonen Jun 15 '18 at 07:01
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This may be better. – Jyrki Lahtonen Jun 15 '18 at 07:03
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Thank you, I am new here and I don't know how to use the site in the best way of it. – Tomato Jun 15 '18 at 07:22