Show that the modified Pell equation $x^2 - 7y^2 = -1$ has no solutions in integers $x,y$. 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.

Add $8y^2+4$ to both sides of the equation getting $x^2+y^2+4 = 8y^2+3$ Now read this equation modulo 4, we get $x^2+y^2\equiv3\pmod 4$. As any square leaves a remainder of 0 or 1 mod 4, adding two of them (the LHS) we cannot get 3 (the RHS).