Diophantine equation #1(GTM)

$x = \sqrt{15 y ^ 2 +2}$ For there to be integer solutions for this equation, $15 y ^ 2 +2$ must be a square number.

Also $15 y ^ 2 +2$ can only end in a $2$ or a $7$ .

As there are no square numbers which can end in a $2$ or a $7$ , there cannot be any integer solutions.

let's considerate $x^2 - 15y^2 \equiv x^2 \pmod 5$ , so if (x,y) is a pair such that it satisfaies the original diophantine equation, then $x^2 \equiv 2 \pmod 5$ but if $x \equiv 0 \pmod 5$ then $x^2 \equiv 0 \pmod 5$ , if $x \equiv 1 \pmod 5$ then $x^2 \equiv 1 \pmod 5$ , if $x \equiv2 \pmod 5$ then $x^2 \equiv 4 \pmod 5$ , if $x \equiv3 \pmod 5$ then $x^2 \equiv 4 \pmod 5$ and if $x \equiv 4 \pmod 5$ then x^2= 1 (mod 5), so the conclusion is there doesn't exist any solution for this equation. It's impossible x^2=2 (mod 5)