How can I prove that $x^2=3y+2$ cannot have integer positive solutions?

Question

I need to prove that $x^2=3y+2$ cannot have integer positive solutions?I tried some numbers, could not find solution. But do not know how to prove.

Answer 1

The equation simplifies to $y = \frac{x^2-2}{3}$. Notice that for any integer $x$, $x^2 \equiv 0 \pmod 3$ or $x^2 \equiv 1 \pmod 3$. You can prove this yourself. Thus the numerator is never a multiple of $3$, thus $y$ is never an integer

Answer 2

we have $$3y+2\equiv 2 \mod 3$$ but for $$x \in \mathbb{Z}$$ we get $$0,1,2 \mod 3$$ but we have $$x^2\equiv 0,1 \mod 3$$ contradiction