Representing Integers as the Sum of Two Squares in the Ring ${\mathbb{Z}}_n$

Abstract

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3(\mathrm{mod}4)$, have even exponent in the prime factorization of $z$. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of $z$ as the sum of two squares. Viewing each of these questions in ${\mathbb{Z}}_n$, the ring of integers modulo $n$, we give a characterization of all integers $n\ge 2$ such that every $z\in {\mathbb{Z}}_n$ can be written as the sum of two squares in ${\mathbb{Z}}_n$.