Representing Integers as the Sum of Two Squares in the Ring \Z_n

2014/04/01 by Joshua Harrington, Lenny Jones, Harrington, Joshua +3
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1404.0187

openalex publication_date 2014/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \pmod4, have 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.

Related

• Representing numbers as the sum of squares and powers in the ring\n \ℤ_n
• The Sum of Squares function in the ring ℤ_n
• Sums of integers and sums of their squares
• Continuants and some decompositions into squares
• Finite sequences of integers expressible as sums of two squares