  Let $F$ be a quadratic form in four variables, let $m\in\mathbb{N}$ and let
$\mathbf{k}\in \mathbb{Z}^4$. We count integer solutions to $F(\mathbf{x})=0$
with $\mathbf{x}\equiv \mathbf{k}\:\mathrm{mod}(m)$. One can compare this to
the similar problem of counting solutions to $F(\mathbf{x})=0$ without the
congruence condition. It turns out that adding the congruence condition
sometimes gives a very different main term than the homogeneous case. In
particular, there are examples where the number of primitive solutions to the
problem is $0$, while the number of unrestricted solutions is nonzero.

Number Theory