# Quadratic Reciprocity problem.. help! [closed]

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

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

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• Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos May 27 '18 at 7:59
• umm.. I tried ((p-2)(p-1)/p)=(1x2/p), and thought there is (p-1)/2 quadratic residue and quadratic nonresidue.. but it didn't help to evaluate that. I think the answer is -1. – 박윤수 May 27 '18 at 8:05
• – Martin Sleziak May 27 '18 at 9:07
• – Martin Sleziak May 27 '18 at 9:07

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 $k$.