Looking at the Diophantine equation $x^2 - 31y^2 = -1$, I know that it is not solvable but am not sure how to prove why it isn't solvable.

Question

I looked at the period of the continued fraction expansion of the $\sqrt{31}$. I noticed that it has an even period $(8)$ which indicates that it isn't solvable for $-1$. I'm not sure how to go about proving that an even period means the equation is not solvable.

Answer 1

Modulo $4$ we have $x^2\equiv 0$ or $x^2 \equiv 1.$ Also $-31 y^2\equiv 32y^2-31 y^2=y^2,$ and $y^2\equiv 0$ or $y^2\equiv 1 .$ So $x^2+y^2$ cannot be congruent to $-1$ modulo $4.$