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

Harrington, Joshua ; Jones, Lenny ; Lamarche, Alicia

Journal of integer sequences, Tome 17 (2014) no. 7

Résumé

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 $q \equiv 3$ (mod 4), occur with 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 $Z_{n}$.

Détail

Classification : 11E25, 11A07
Keywords: sums of squares, ring of integers modulo n, congruence