Given integersa natural number $n \geq 1$$n$ and an element $0 \leq k \leq n-1$$k \in \mathbb{Z}_n$, how many pairs of integers $(x,y)$solutions are there such that $x^2+y^2 \equiv n$ moduloin $k$$\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? ThatThat is, I'm wondering whether there is a mod-$k$$n$ version of the sum of squares function discussed in this post.
As an example, to write 1 as a sum of squares modulo 7, one has permutations of $0^2+1^2$ but also $2^2+2^2$ and $2^2+5^2$ and $5^2+5^2$.