Representing integers as the sum of two squares in the ring \(\mathbb Z_n\) (Q2877901)

A classical theorem by 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 \(q \equiv 3 \pmod 4\) occur with even exponent in the prime factorization of \(z\). A somewhat natural question to ask is what positive integers \(z\) can be written as the sum of two nonzero squares? A partial answer is given by \textit{G. H. Hardy} and \textit{E. M. Wright} [An introduction to the theory of numbers. Oxford: Clarendon Press (1938; Zbl 0020.29201)]: for an integer \(n > 1\) there exist \(u, v \in \mathbb{Z}\), with \(\gcd(u, v) = 1\), such that \(n = u^2 + v^2\) if and only if \(-1\) is a quadratic residue modulo \(n\). The same question can be also addressed in the ring \(\mathbb{Z}_n\). In the paper under review the authors characterize, in a precise manner, the values of \(n\) such that every element in \(\mathbb{Z}_n\) can be written as the sum of two nonzero squares. For the sake of completeness, the values of \(n\) such that every \(z \in \mathbb{Z}_n\) can be written as the sum of two squares where the use of zero is allowed as a summand in such a representation of \(z\) is also provided.