Generalizing Abundancy Index to Gaussian Integers

Vrishab Krishna

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arXiv · 2004.13487

Generalizing Abundancy Index to Gaussian Integers

Vrishab Krishna

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Abstract

Abundancy index refers to the ratio of the sum of the divisors of a number to the number itself. It is a concept of great importance in defining friendly and perfect numbers. Here, we describe a suitable generalization of abundancy index to the ring of Gaussian integers ($\mathbb{Z}[i]$). We first show that this generalization possesses many of the useful properties of the traditional abundancy index in $\mathbb{Z}$. We then investigate $k$-powerful $\tau$-perfect numbers and prove results regarding their existence in $\mathbb{Z}[i]$.

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Vrishab Krishna. 2020-04-24. Generalizing Abundancy Index to Gaussian Integers. https://arxiv.org/abs/2004.13487

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