$x^2 + 2 = 5y$ ($x$ and $y$ positive integers)

Question

Question:

Determine all positive integers $x$ and $y$ that satisfy the equation $x^2 + 2 = 5y$.

Answer 1

Hint: When is $5y - 2$ a square? Modulo 5 this is $5y - 2 = 3$, but squares mod 5 are always 0, 1 or 4 so the equation has no solutions.

Answer 2

HINT $\rm\ \ x^2 \equiv\ \ldots\ \:(mod\ 5)\ $ versus $\rm\ \mathbb Z^2\ \equiv\ \{0,\:\pm1,\:\pm2\}^2\ \equiv\ \ldots \ \:(mod\ 5)$