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Representing Integers as the Sum of Two Squares in the Ring ****Z**_{n}

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Joshua Harrington, Lenny Jones, and Alicia Lamarche

Department of Mathematics

Shippensburg University

Shippensburg, PA 17257

USA

**Abstract:**

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 **Z**_{n},
the ring of integers modulo *n*, we
give a characterization of all integers *n* ≥ 2 such that every
*z* ∈ **Z**_{n}
can be written as the sum of two squares in **Z**_{n}.

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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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