论文标题
将丰富指数概括为高斯整数
Generalizing Abundancy Index to Gaussian Integers
论文作者
论文摘要
丰富索引是指数字本身的分隔线之和之比。在定义友好和完美的数字方面，这是一个非常重要的概念。在这里，我们描述了对高斯整数环的适当概括（$ \ mathbb {z} [i] $）。我们首先表明，这种概括具有$ \ mathbb {z} $中传统丰富索引的许多有用属性。然后，我们调查$ k $ - 供应$τ$ - 完美的数字，并证明了它们在$ \ mathbb {z} [i] $中存在的结果。
Abundancy index refers to the ratio of the sum of the divisors of a number to the number itself. It is a concept of great importance in defining friendly and perfect numbers. Here, we describe a suitable generalization of abundancy index to the ring of Gaussian integers ($\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 $τ$-perfect numbers and prove results regarding their existence in $\mathbb{Z}[i]$.
