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

 <br /> <br />
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)  <br /> <br />

where those are the legendre symbols inside.