  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 $q\equiv 3 \pmod{4}$, 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 $\Z_n$, the ring of integers
modulo $n$, we give a characterization of all integers $n\ge 2$ such that every
$z\in \Z_n$ can be written as the sum of two squares in $\Z_n$.
