Let $\displaystyle p $ be an odd prime and let $\displaystyle N_p $ be the number of solutions $\displaystyle (x,y) \in \mathbb{Z}_{p}^2 $ to the equation $\displaystyle y^2 = x^2 + 1 $

prove that

$\displaystyle

N_p = \sum_{m \in \mathbb{Z}_{p} } \left( 1 + \left( \frac{m}{p} \right) \right) \left( 1 + \left( \frac{m+1}{p} \right) \right)

$

where those are the legendre symbols inside.