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

Special Numbers in the Ring Zn

Samuel S. Gross
Noblis, Inc.
Falls Church, VA 22042

Joshua Harrington
Department of Mathematics
Cedar Crest College
Allentown, PA 18104


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 a2 + b2 - dc2 = 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 Zn satisfying ma2 + b2 - dc2 (mod n). By refining a recent result of Harrington, Jones, and Lamarche on representing integers as the sum of two non-zero squares in Zn, we establish a complete characterization of all such pairs.

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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.

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