Special Numbers in the Ring Zn
Samuel S. Gross
Falls Church, VA 22042
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
m ≡ a2 + b2 - dc2 (mod n).
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.
Full version: pdf,
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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