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Fixed typos in which n and k were inconsistently labeled, and clarified that solutions are elements of Z_n rather than integers.
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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 $5^2+5^2$.

Given integers $n \geq 1$ and $0 \leq k \leq n-1$, how many pairs of integers $(x,y)$ are there such that $x^2+y^2 \equiv n$ modulo $k$? That is, I'm wondering whether there is a mod-$k$ 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 $5^2+5^2$.

Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ 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 $5^2+5^2$.

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Number of ways to write an integer as a sum of squares modulo $k$

Given integers $n \geq 1$ and $0 \leq k \leq n-1$, how many pairs of integers $(x,y)$ are there such that $x^2+y^2 \equiv n$ modulo $k$? That is, I'm wondering whether there is a mod-$k$ 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 $5^2+5^2$.