Colloquium

Starts: September 25, 2009 at 3:00 PM
Location: Jones Hall 301
Contact: Ryan Vinroot

Summary

Speaker: Nick Rogers (University of Rochester)

Full Description

Title: Aliquot Sequences and the Catalan-Dickson Conjecture.

Abstract:
Interest in the sum of the "aliquot parts" of a number -- the divisors
other than the number itself -- dates to the ancient Greeks, who
defined perfect numbers to be those numbers that are equal to the sum
of their aliquot parts.  Viewing this aliquot sum function as a map
from the natural numbers to itself, it is natural to ask what happens
under iteration of this map; the sequence of iterates is called the
aliquot sequence.  Perfect numbers correspond to fixed points, and
amicable and sociable numbers correspond, respectively, to 2-cycles
and k-cycles for k > 2.  A conjecture formulated by Catalan, and
extended by Dickson, states that every aliquot sequence is bounded;
that is, it terminates at 1, or reaches a perfect number or an
amicable or sociable cycle.  In this talk we'll describe some reasons
to believe that the conjecture is actually false, and a new heuristic
that suggests it may be true after all.