Indian Statistical Institute ( ISI ) B.Math & B.Stat : Number Theory

Consider the equation $x^2 + y^2 = 2007$. How many solutions $(x, y)$ exist such that $x$ and $y$ are positive integers? $2007 = 2000 + 7 \equiv 0 + 3 \equiv 3 (mod \quad 4)$. Now we know that square on an integer is either divisible by $4$ or leaves a remainder $1$ when divided by $4$, said otherwise $x \in \mathbb{Z} \implies x^2 \equiv 0 \quad or \quad 1 (mod \quad 4)$. Thus for integers $x$ and $y$, $x^2+y^2 \equiv 0 \quad or \quad 1 \quad or \quad 2 (mod \quad 4)$. Since we have different remainders $mod \quad 4$ on the two sides, it follows there cannot be any solution in $\mathbb{Z}$ hence no solution in $\mathbb{Z^+}$