Multiperfect Numbers in Certain Quadratic Rings

TL;DR

This paper extends the concept of multiperfect numbers to imaginary quadratic rings with unique factorization, exploring $n$-powerfully $t$-perfect numbers using an extended abundancy index.

Contribution

It introduces a new framework for studying multiperfect numbers in quadratic rings via an extended abundancy index, broadening the scope beyond integers.

Findings

Defined $n$-powerfully $t$-perfect numbers in quadratic rings

Extended the abundancy index to imaginary quadratic rings

Identified potential directions for future research

Abstract

Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$-powerfully $t$-perfect numbers in these rings. This definition serves to extend the concept of multiperfect numbers that have been defined and studied in the integers. At the end of the paper, as well as at various points throughout the paper, we point to some potential areas for further research.