3. Do there exist integers $x$ and $y$ such that $x^{2}+2012=y^{2}$? Justify your claim. If such numbers exist, determine all of them.
3. The equation $x^{2}+2012=y^{2}$ can be written as

$$
2012=y^{2}-x^{2} \text { or }(y-x)(y+x)=2012
$$

Further, $(y-x)(y+x)=2 \cdot 2 \cdot 503$.
a) $y+x=2$
a)

$\frac{y-x=1006}{y+x=-2}$

gives the solution $(-502,504)$,

$\underline{y-x=-1006}$ gives the solution $(502,-504)$,

$y+x=-1006$

$y-x=-2 \quad$ gives the solution $(-502,-504)$,

$y+x=1006$

$y-x=2 \quad$ gives the solution $(502,504)$.

4 POINTS
b) $y+x=4$

$\underline{y-x=503}$ (and similar combinations) has no solution in the set of integers.

2 POINTS

$y+x=1$
c) $\begin{aligned} & y+x=1 \\ & y-x=2012\end{aligned}$ (and similar combinations) has no solution in the set of integers.

2 POINTS

TOTAL 10 POINTS