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Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$. Prove that:

$\exists x, y \in \mathbb{Z};\,\, \gcd(x, y) = 1$ such as $x^{2} + ny^{2} \equiv 0\, (\mod p) \iff \Bigg(\displaystyle \frac{-n}{p}\Bigg) = 1$, where $\Bigg(\displaystyle \frac{-n}{p}\Bigg)$ represent the Legendre Symbol.

I've been trying but I have not idea where to start. Thanks in advance.

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Here is an outline, see if you can fill in more details.

First part: if $x^2+ny^2\equiv0\pmod p$ then $y$ is not a multiple of $p$. (If so then $x$ is a multiple of $p$, but this is impossible since $\gcd(x,y)=1$.) So $y$ has a multiplicative inverse $z$ modulo $p$, so $-n\equiv(xz)^2\pmod p$.

Converse: if $\bigl(\frac{-n}p\bigr)=1$ then for some $x$ we have $-n\equiv x^2\pmod p$, that is, $x^2+n\equiv0\pmod p$, and now you should have no trouble finding a suitable value of $y$.

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  • $\begingroup$ ive been trying but i still can get a value of y. $\endgroup$ – Jearson Narvaez Rojas Oct 31 '14 at 7:11

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