Finiteness Theorems for Perfect Numbers and Their Kin

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.