Geodesic
Representing integers as the sum of two squares in the ring $\Bbb Z_n$
Journal of integer sequences, Tome 17 (2014) no. 7.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $Z_{n}$.
Classification : 11E25, 11A07
Keywords: sums of squares, ring of integers modulo n, congruence 
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     title = {Representing integers as the sum of two squares in the ring $\Bbb Z_n$},
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Harrington, Joshua; Jones, Lenny; Lamarche, Alicia. Representing integers as the sum of two squares in the ring $\Bbb Z_n$. Journal of integer sequences, Tome 17 (2014) no. 7. http://geodesic.mathdoc.fr/item/JIS_2014__17_7_a2/