Mathematical
Colloquium
Colloquium
Higher-order Carmichael numbers
| By |  |
Everett Howe |
| From | |
Center for Communications Research, California |
| 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.
The Colloquium counts as Seminar Credit for Mathematics Students.