Representing Integers as the Sum of Two Squares in the Ring $\Z_n$
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 $q\equiv 3 \pmod{4}$, 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 $\Z_n$.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 DOI:
 10.48550/arXiv.1404.0187
 arXiv:
 arXiv:1404.0187
 Bibcode:
 2014arXiv1404.0187H
 Keywords:

 Mathematics  Number Theory