AI 中文总结
本文针对罗斯2024年提出的关于\(\{ -1, 1 \}\)-完美数和非亏数的两个猜想进行研究，通过特定方法对这两个猜想进行反证，得出与原猜想不同的结论。
AI 中文摘要
一个正整数\(n\)若满足\(\sigma(n) \geq 2n\)，则称其为非亏数。设正整数\(n\)的正因数为\(1 = d_0 < d_1 < \cdots < d_k < d_{k + 1} = n\)，\(\mathcal{S}\)为整数集，若存在\(\lambda_j \in \mathcal{S}\)使得\(1 + \sum_{j = 1}^{k} \lambda_j d_j = n\)，则称\(n\)为\(\mathcal{S}\)-完美数。2024年罗斯引入\(\mathcal{S}\)-完美数的研究并得出两个关于\(\{ -1, 1 \}\)-完美数和非亏数的猜想，本文对这两个猜想进行了反证。
英文摘要
A positive integer $n$ is said to be nondeficient if $σ(n) \geq 2n$. Letting the positive divisors of a positive integer $n$ be written as $1 = d_0 < d_1 < \cdots < d_k < d_{k+1} = n$, and letting $\mathcal{S}$ denote a set of integers, if there exist values $λ_j \in \mathcal{S}$ such that $1 + \sum_{j=1}^{k} λ_j d_j = n$, then $n$ is said to be an $\mathcal{S}$-perfect number. Ross, in 2024, introduced the study of $\mathcal{S}$-perfect numbers, and concluded with two conjectures that each concern both $\{ -1, 1 \}$-perfect numbers and nondeficient numbers. We disprove both of these conjectures.
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