Consider the equation $x^2 + y^2 + k = pz$ where $x, y, z, k, z \in Z$ , $p$ is a prime number. For any given prime number $p$ and a given integer $k$, does there always exist integer solutions for $x, y, z$ which satisfy the given equation.
Integer solutions for above would exist if we somehow prove that for all prime $p$ integer solutions for $x,y$ exist such that $x^2 + y^2 \equiv a\pmod p$, for all $a = {0,1,2,.., p-1}$.
If $p$ is odd prime, there are precisely $\frac{p+1}{2}$ quadratic residues modulo $p$ (including $0$).
Let $a \in \{0,1, \ldots, p-1 \}$.
Consider $$B = \{ 0\le b \le p-1: (\exists) x \in \mathbb{Z} \text{ s.t. } x^2 \equiv b\pmod p \}$$ and $$C = \{0 \le c \le p-1 : (\exists) y \in \mathbb{Z} \text{ s.t. } y^2 \equiv a-c\pmod p\ \}$$ Note that $$\big| B \big| = \big| C \big| = \frac{p+1}{2}$$ Thus, $B \cap C$ is nonempty. Hence, there exists some $x,y \in \mathbb{Z}$ such that $x^2+y^2 \equiv a \pmod p$.
