If $p$ is an odd prime, evaluate $\left(\frac{1\times2}{p}\right)+\left(\frac{2\times3}{p}\right)+\cdots+\left(\frac{(p-2)\times(p-1)}{p}\right)$

I don't know how I use properties of Legendre symbol. Please help!


closed as off-topic by José Carlos Santos, Saad, Martin Sleziak, Jose Arnaldo Bebita-Dris, Namaste May 27 '18 at 13:27

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This is $$S=\sum_{k=1}^{p-1}\left(\frac{k(k+1)}p\right) =\sum_{k=1}^{p-1}\left(\frac{k^*(k+1)}p\right) =\sum_{k=1}^{p-1}\left(\frac{k^*+1}p\right)$$ where $k^*$ denotes the modulo $p$ inverse of $k$.


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