Proving a non-linear diophantine equation has no solutions How can I prove that 
$x^2 + y^2 + z^2 + 3(x+ y + z) + 5 = 0$
Has no integer solutions?

Assume that $(x,y,z) \in \mathbb{Z}^3$ satisfies $x^2+y^2+z^2 + 3(x+y+z) + 5 = 0$. For simplicity, let $f(X,Y,Z) = X^2+Y^2+Z^2 + 3(X+Y+Z) + 5$. Then $f(x,y,z) \equiv 0 \pmod{2}$. Notice that $a^2 \equiv a \pmod{2}$ implies $x^2+y^2+z^2\equiv 3(x+y+z)\pmod{2}$, and so $f(x,y,z) \equiv 5 \equiv 1 \not\equiv 0 \pmod{2}$. Contradiction.
