Perfect Eisenstein integers

TL;DR

This paper extends classical number theory concepts like the sum-of-divisors function and perfect numbers to Eisenstein integers and other algebraic number fields, providing new characterizations and conditions.

Contribution

It introduces necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect, generalizing previous work on Gaussian integers and exploring odd cases and other cyclotomic fields.

Findings

Characterization of even Eisenstein integers as (norm-)perfect

Conditions for odd norm-perfect Eisenstein integers

Results on rings of integers of cyclotomic fields

Abstract

We generalise the sum-of-divisors-function $\sigma$ and evenness to the rings of integers of certain algebraic number fields. In particular, we present necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect based on the work of McDaniel [McD] on Gaussian integers. Furthermore, some results concerning odd norm-perfect Eisenstein integers and the rings of integers of other cyclotomic fields are proven.