Colorado State University

Higher-order Carmichael numbers

By Everett Howe
From Center for Communications Research,
When December 6, 2006
4:00 pm
Where Room WB202, Weber Building
Abstract Fermat's little theorem states that if n is a prime number, then an-a is a multiple of n for every integer a. The converse of Fermat's little theorem is false: there exist composite numbers n such that an-a is a multiple of n for every integer a, the smallest example being n=561=3*11*17.
Such numbers are called Carmichael numbers, after the mathematician Robert D. Carmichael (1879-1967). Erdös gave a heuristic argument that indicated that there should be infinitely many Carmichael numbers, and in a technically difficult 1994 paper Alford, Granville, and Pomerance used an argument based on Erdös's heuristic to prove that for large values of x, there are more than x2/7 Carmichael numbers less than x.
In this talk I will introduce the "higher-order" Carmichael numbers, which behave even more like primes than do the original Carmichael numbers. They arise from considering a natural ring-theoretic interpretation of the definition of the usual Carmichael numbers. A variant of Erdös's argument indicates that there should be infinitely many Carmichael numbers of order m for every m>0, but we do not even know whether there exist any Carmichael numbers of order 3. I will show how Erdös's argument and some non-trivial computation can produce examples of Carmichael numbers of order 2.
Further Information Rachel Pries
There will be Refreshments in WB 117 at 3.30pm
The Colloquium counts as Seminar Credit for Mathematics Students.