Items of Interest

We will discuss Carmichael numbers in connection with solving
xk = x for all x in Zp.
A number n is a Carmichael number if
xn = x for all x in Zn.
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 xn = x for all x in Zp 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.)


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