Finiteness Theorems for Perfect Numbers and Their Kin

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly.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.

A conjecture of De Koninck regarding particular square values of the sum of divisors function

“…A nice account detailing properties of the so-called 'supernatural topology' in attacking the odd perfect number problem, is given by Pollack in [7].…”

Densities on Dedekind domains, completions and Haar measure

“…In the case D = Z, this is the approach to supernatural numbers taken in the (very readable) paper [36]. Note also that supernatural numbers are a compactification of N (as emphasized in [36]) and that the latter is in natural bijection with the set of ideals of Z. We will think of S(D) as a compactification of I(D).…”

“…Together with [24, Example 7], Proposition 4.7 suggests that a good system of axioms for densities on D n could be obtained taking (F1)-(F6) and (36), with the obvious modifications needed for the higher dimension.…”

Coset topologies on $\mathbb{Z}$ and arithmetic applications

“…These topologies, especially Golomb's one (and its generalizations to other rings), have received a significant amount of attention in recent years: see for example [1], [2], [3], [8], [18], [19], [23] and [26]. Topologies on the integers were also introduced (and arithmetic applications discussed) in sundry other works, such as [4], [20], [22], [24] and more. In particular, Broughan provided in [4] a construction covering most of the ones cited above.…”

“…The current paper will not go very far in this direction 1 , but we will show how Dirichlet's theorem has a natural interpretation as a statement on the closure in Ẑ of the set of primes (Theorem 3.23) and briefly explain the connection with Euler's proof that this set is infinite (Remark 3.24). Moreover, in section 4 we will discuss the connection between Ẑ and the supernatural numbers, whose topological properties were nicely used by Pollack in [20] to recover a number of results about perfect numbers and the like. This paper started as an undergraduate research project in summer 2016: the original goal was to study the closure in Ẑ of arithmetically interesting subsets of Z, but our perusal of literature showed that the profinite viewpoint had much to say also on the general topic of topologies on the integers and actually a large part of that summer went into examining such topologies.…”