Representation of integers by ternary quadratic form $x^2+y^2-z^2$

Question

Let $Q$ be the ternary quadratic form $Q(x,y,z)=x^2+y^2-z^2$. Since $Q(0,p+1,p)=2p+1$ and $Q(1,p+1,p)=2p+3$, we see that for every integer $k$, the equation $E_k:Q(x,y,z)=k$ always has a solution. Is it known for which integers $k$ there are infinitely many solutions to $E_k$ ?

Answer 1

Write $$x^2-k=z^2-y^2$$ A number can be represented as a difference of two squares, so long as it is not twice an odd number. For any $k$, you can find infinitely many $x$ such that $x^2-k$ is not twice an odd number, hence, infinitely many solutions.