## Diophantine Equation

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

prove that

$

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.