Hi,
Need help with this proof. (a/p) is legendre symbol not to be confused with 1 divided by p;
Prove
(1.2/p) + (2.3/p) + (3.4/p) + ..... + ((p-2)(p-1)/p) = -1
Thanks
ok! first note that $\displaystyle \sum_{k=1}^{p-2}\left ( \frac{k(k+1)}{p} \right)=\sum_{k=1}^{p-1}\left ( \frac{k(k+1)}{p} \right)=S_1.$ for any $\displaystyle 0 \leq j \leq p-1,$ let $\displaystyle S_j=\sum_{k=1}^{p-1}\left ( \frac{k(k+j)}{p} \right).$ then $\displaystyle S_0=p-1$ and for any $\displaystyle 1 \leq j \leq p-1: \ S_j=S_1.$ why?
also note that for any integer $\displaystyle k: \ \ \sum_{j=0}^{p-1}\left ( \frac{k+j}{p} \right)= \sum_{j=1}^{p-1}\left ( \frac{j}{p} \right)=0.$ therefore: $\displaystyle (p \ - \ 1)(S_1+1)=\sum_{j=0}^{p-1}S_j=\sum_{k=1}^{p-1} \sum_{j=0}^{p-1}\left ( \frac{k(k+j)}{p} \right)=\sum_{k=1}^{p-1}\left ( \frac{k}{p} \right)\sum_{j=0}^{p-1}\left ( \frac{k+j}{p} \right)=0.$ thus $\displaystyle S_1=-1.$