$\tau$-Norm-Perfect and $\tau$-Perfect Eisenstein Integers for $\tau=\omega+2$ and $2$

TL;DR

This paper characterizes certain perfect and norm-perfect Eisenstein integers related to specific algebraic integers and proves the nonexistence of some classes of these numbers, advancing the understanding of perfect numbers in complex quadratic integer rings.

Contribution

It introduces new characterizations of $( au)$-norm-perfect and $( au)$-perfect Eisenstein integers for specific $ au$, and proves nonexistence results for certain classes.

Findings

Characterization of $( au)$-norm-perfect Eisenstein integers for $ au=\omega+2$.

Proof of nonexistence of $2$-norm-perfect Eisenstein integers divisible by 2.

Extension of complex Mersenne number concepts to Eisenstein integers.

Abstract

Using Robert Spira's \cite{D} definitions of complex Mersenne numbers and the complex sum-of-divisors function, we characterize $(\omega+2)$-norm-perfect and $(\omega+2)$-perfect numbers that are divisble by $\omega+2$ and prove the nonexistence of $2$-norm-perfect numbers that are divisible by $2$ in the Eisenstein integers.