Perfect Numbers in the Ring of Eisenstein Integers

TL;DR

This paper explores the properties and existence of perfect numbers within the ring of Eisenstein integers, extending classical number theory concepts into complex quadratic integer domains.

Contribution

It introduces the concept of perfect numbers in the Eisenstein integers and provides initial results and insights into their properties.

Findings

Identification of conditions for perfect numbers in Eisenstein integers

Extension of classical perfect number theory to complex quadratic domains

New results on the structure of perfect numbers in Eisenstein integers

Abstract

One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.