Number Theory Seminar

Tuesday, Sept.30
Nick Rogers, U of R
Heuristics for Aliquot Sequences

ABSTRACT:

For a natural number n, define the aliquot sum function s(n) = sigma(n) - n to be the sum of the divisors of n, other than n itself. The aliquot sequence of a natural number is the sequence obtained by iterating the aliquot sum function. Some classical questions in elementary number theory can be rephrased in terms of aliquot sequences, since perfect numbers correspond to fixed points of s and so-called "amicable" and "sociable" numbers correspond to cycles of s. For any aliquot sequence, there are three possibilities: the sequence terminates at 1, reaches a fixed point or a cycle, or is unbounded. Catalan's aliquot sequence conjecture (sometimes called the Catalan-Dickson conjecture) states that the third possibility never occurs. While some recent circumstantial evidence suggests that the conjecture is actually false, in this talk I'll present a new heuristic argument in favor of the Catalan-Dickson conjecture, by modeling s(n)/n as a random variable and using some basic ideas about random walks.