On 2-powerfully Perfect Numbers in Three Quadratic Rings

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.

15 pages, 0 figures, Supported by National Science Foundation grant no. 1262930