No integer solutions to equation

Question

I'm trying to show that:

$x^2 + 14y^2 = \pm 2$ has no integer solutions.

My initial thought was the usual approach of considering modulo 7, but this seems to fail. My next guess was to consider modulo 5, which yields $\pm 2 = (x+y)(x-y)$ but this seems to fail too. I'm not sure where to go next.

Answer 1

Consider that for $x=0$ or $y=0$ there is no solutions and we have for $x,y\geq 1$ that $x^2+14y^2\geq 1+14\geq 15> 2$