Representing numbers as the sum of squares and powers in the ring $\mathbb{Z}_n$

TL;DR

This paper investigates how numbers can be expressed as sums of two squares in modular arithmetic and discusses the density of such numbers in the natural numbers, with implications for sums involving powers of 2.

Contribution

It provides a general analysis of sum-of-two-squares representations in $Z_n$ and explores their density among positive integers, extending classical results.

Findings

Characterization of sum-of-two-squares representations in $Z_n$

Insights into the density of integers representable as sums of two squares and powers of 2

Connections between modular representations and natural number properties

Abstract

We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum of two squares and powers of $2$ in $\mathbb{N}$.