Finite connected components of the aliquot graph

Conditional on a strong form of the Goldbach conjecture, we determine all finite connected components of the aliquot graph containing a number less than $10^9$, as well as those containing an amicable pair below $10^{14}$ or one of the known perfect or sociable cycles below $10^{17}$. Along the way we develop a fast algorithm for computing the inverse image of an even number under the sum-of-proper-divisors function.Comment: 10 pages, to appear in Mathematics of Computatio

Odd Perfect Numbers Have At Least Nine Distinct Prime Factors

An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.Comment: 17 page

Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester\u27s bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980. What motivated Sylvester\u27s sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888

Divisibility of the class numbers of imaginary quadratic fields

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.Comment: 10 pages, accepted for publication in Journal of Number Theory (2017