Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.4

Representing Integers as the Sum of Two Squares in the Ring Zn

Joshua Harrington, Lenny Jones, and Alicia Lamarche
Department of Mathematics
Shippensburg University
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 zZn can be written as the sum of two squares in Zn.

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(Concerned with sequences A240109 A240370 A243609.)

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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