Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.3 |

Noblis, Inc.

Falls Church, VA 22042

USA

Joshua Harrington

Department of Mathematics

Cedar Crest College

Allentown, PA 18104

USA

**Abstract:**

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* ≥ 2,
such that for every integer *m* there exist
units *a*, *b*, and *c* in **Z**_{n}
satisfying
*m* ≡ *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.

Received August 1 2015;
revised versions received September 18 2015; September 30 2015.
Published in *Journal of Integer Sequences*, November 25 2015.
Order of authors switched, January 11 2016.

Return to