Special numbers in the ring $\Bbb Z_n$

Harrington, Joshua ; Gross, Samuel S.

Journal of integer sequences, Tome 18 (2015) no. 11.

Résumé

Summary: In a recent article, Nowicki introduced the concept of a special number. Specifically, an integer $d$ is called $special$ if for every integer $m$ there exist solutions in non-zero integers $a, b, c$ to the equation $a^{2} + b^{2} - dc^{2} = m$. In this article we investigate pairs of integers $(n, d)$, with $n \ge 2$, such that for every integer $m$ there exist units $a, b$, and $c$ in $Z_{n}$ satisfying $m \equiv a^{2} + b^{2} - dc^{2} (mod n)$. By refining a recent result of Harrington, Jones, and Lamarche on representing integers as the sum of two non-zero squares in $Z_{n}$, we establish a complete characterization of all such pairs.

Détail

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