Finiteness Theorems for Perfect Numbers and Their Kin
Finiteness Theorems for Perfect Numbers and Their Kin
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完美数及其亲属的有限定理
DOI：
10.4169/amer.math.monthly.119.08.670
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
P. Pollack
中科院分区：
文献类型：
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作者：
P. Pollack
摘要 自古以来，一个自然数如果等于它的真因数之和就被称为完美数。例如，6 = 1 + 2 + 3 是一个完全数。 1913 年，迪克森证明，对于每个固定的 k，只有有限多个具有最多 k 个不同质因数的奇完全数。我们展示了这个结果以及许多类似的结果是如何通过将自然数嵌入到超自然数中并在后者上施加适当的拓扑而得出的；顺序紧凑性的概念起着重要作用。
Abstract Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e.g., 6 = 1 + 2 + 3 is a perfect number. In 1913, Dickson showed that for each fixed k, there are only finitely many odd perfect numbers with at most k distinct prime factors. We show how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter; the notion of sequential compactness plays a starring role.