Only five Fermat primes are known, and the Fermat numbers
grow so quickly that it may be years before the first
undecided case: F31 =
is shown prime or composite--unless 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 F31, so now F33 is the least Fermat with unknown status!)
Euler showed that every divisor of Fn (n greater than 2) must have the form k.2n+2+1 for some integer k. For this reason, when we find a large prime of the form k.2n+1 (with k small), we usually check to see if it divides a Fermat number. The probability of the number k.2n+1 dividing any Fermat number appears to be 1/k.
Related pages (outside of this work)