Differences of 2 squares.

$\begin{aligned} 2018 &= x^2 - y^2 \\ &= (x-y)(x+y) \end{aligned}$

The prime factorization of 2018 is $2018=2\times 1009$ . Since $x$ and $y$ are both integers, there are two cases:

• $x-y=1$ , $x+y=2018$ : In this case, $x$ and $y$ are consecutive, so one is even and one is odd. But their sum, $2018$ , is even, a contradiction.
• $x-y=2$ , $x+y=1009$ : In this case, we have the opposite problem. Because their difference is $2$ , $x$ and $y$ are either both even or both odd. But their sum, $1009$ , is odd, again a contradiction.

So there are no integers $x$ and $y$ that satisfy $x^2-y^2 = 2018$ .