[ Parity and Imparity $]$ [ Factorization $]$

Find all pairs of integers $(x, y)$ that satisfy the equation $x^{2}=y^{2}+2 y+13$.

#
Let's represent the right side of the equation as $(y+1)^{2}+12$ and move $(y+1)^{2}$ to the left side. After this, the equation can be written as

$(x-y-1)(x+y+1)=12$

The numbers $x-y-1$ and $x+y+1$ have the same parity, so both must be even.

Therefore, one of them is $\pm 2$, and the other is $- \pm 6$. From this, we easily obtain the answer.

## Answer

$( \pm 4,1),( \pm 4,-3)$.