On 2-powerfully Perfect Numbers in Three Quadratic Rings

TL;DR

This paper extends the concept of perfect numbers to imaginary quadratic rings with unique factorization by defining and studying 2-powerfully perfect numbers in three specific rings where 2 is not prime.

Contribution

It introduces the notion of n-powerfully perfect numbers in quadratic rings and analyzes their properties in three specific imaginary quadratic rings with unique factorization.

Findings

Defined n-powerfully perfect numbers in quadratic rings.

Investigated properties of 2-powerfully perfect numbers in three quadratic rings.

Extended the concept of perfect numbers beyond integers.

Abstract

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, the three imaginary quadratic rings with unique factorization in which $2$ is not a prime.