TITLE: Show that the modified Pell equation $x^2 - 7y^2 = -1$ has no solutions in integers $x,y$.
QUESTION [5 upvotes]: Show that the modified Pell equation $x^2 - 7y^2 = -1$ has no solutions in integers $x,y$. (Hint: reduce the equation modulo a suitably chosen prime.)
I think that we can use the Diophantine equation for this, but I don't know where to start. I am new to this material in Number Theory.

REPLY [3 votes]: Take both sides modulo 7. We have $x^2=-1$ modulo 7 which has no solution as 7 is a prime of type 3 mod 4.