# Fermat divisor

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 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 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)

- Status of the factorization of Fermat numbers
- The largest known Fermat divisors

**References:**

- DK95
H. DubnerandW. Keller, "Factors of generalized Fermat numbers,"Math. Comp.,64(1995) 397--405.MR 95c:11010