Integer solutions of $x^2-5y^2=1342$ with $0\leq x,y<400$

Question

$x^2-5y^2=1342$, where $x,y \in \mathbb N \ and \ x,y<400$., how many pairs of $(x,y)$ possible here.

what would be my approach here?

Answer 1

There are no solutions. Hint: Work modulo $4$.

Answer 2

Suppose $x,y$ is a solution. Calculating modulo $4$, we get $x^2 - y^2 \equiv 2 \pmod{4}$. However, $x^2$ and $y^2$ have a residue of $0,1$ modulo $4$, so there are no solutions.

Another way to see this is to look modulo $5$: a solution $x,y$ satisfies $x^2 \equiv 2 \pmod{5}$. However $x^2$ has a residue of $0,1,4$ modulo $5$, so there are no solutions.