
Glossary: Prime Pages: Top 5000: 
Only five Fermat primes are known, and the Fermat numbers
grow so quickly that it may be years before the first
undecided case: F_{31} =
is shown prime or compositeunless we luck onto a divisor.
Ever since Euler found the first Fermat divisor (divisor
of a Fermat composite),
factorers have been collecting these rare numbers.
(Luck has prevailed! On 12 April 2001, Alexander Kruppa found that 46931635677864055013377 divides F_{31}, so now F_{33} is the least Fermat with unknown status!) Euler showed that every divisor of F_{n} (n greater than 2) must have the form k^{.}2^{n+2}+1 for some integer k. For this reason, when we find a large prime of the form k^{.}2^{n}+1 (with k small), we usually check to see if it divides a Fermat number. The probability of the number k^{.}2^{n}+1 dividing any Fermat number appears to be 1/k.
See Also: Fermats, CunninghamProject, FermatQuotient Related pages (outside of this work)
References:
Chris K. Caldwell © 19992017 (all rights reserved)
