Quadratic Legendre-symbol correlation
= Quadratic Legendre-symbol correlation
For every odd prime $p$,
$$
\sum_{x\in\mathbb F_p}\left(\frac{x(x+1)}p\right)=-1.
$$
Equivalently, count the $p-1$ solutions of $t^2=y^2-1$ by factoring $(y-t)(y+t)=1$.
= Quadratic Legendre-symbol correlation
For every odd prime $p$,
$$
\sum_{x\in\mathbb F_p}\left(\frac{x(x+1)}p\right)=-1.
$$
Equivalently, count the $p-1$ solutions of $t^2=y^2-1$ by factoring $(y-t)(y+t)=1$.