## Items of Interest

We will discuss Carmichael numbers in connection with solving
x^{k} = x for all x in **Z**_{p}.
A number n is a Carmichael number if
x^{n} = x for all x in **Z**_{n}.
We will show that n has to be square-free, i.e., a product of distinct
primes, and that a square-free integer n is Carmichael if
n is congruent to 1 mod p-1 for all primes p dividing n
(i.e, if x^{n} = x for all x in **Z**_{p} when p
divides n).
Here
is a posting about Carmichael numbers that was
sent to the number theory mailing list.

Andrew Granville
has written extensively on Carmichael numbers, including a joint paper
showing that there are infinitely many Carmichel numbers.
You can find links to his work
here.
You will need a postscript reader to read them. (If you access them
from an X-windows client on our unix cluster,
a postscript reader will launch automatically
if you click on the articles.)

Home page for the course

Mark Steinberger's home page