Representing Integers as the Sum of Two Squares in the Ring Zn
Joshua Harrington, Lenny Jones, and Alicia Lamarche
Department of Mathematics
Shippensburg, PA 17257
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 ≡ 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 Zn,
the ring of integers modulo n, we
give a characterization of all integers n ≥ 2 such that every
z ∈ Zn
can be written as the sum of two squares in Zn.
Full version: pdf,
(Concerned with sequences
Received April 1 2014;
revised versions received June 10 2014; June 12 2014.
Published in Journal of Integer Sequences, June 21 2014.
Minor revision, July 1 2014. Major revision, March 26 2015.
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