A Fermat number, named after Pierre de Fermat, is a number of the form:
- \(F_{n} = 2^{2^{ \overset{n} {}}} + 1\)
It is a generalization of the Fermat prime, which are exactly primes that are one more than a power of two. If \(2^k + 1\) is prime, then \(k\) must be a power of two, so all Fermat primes are also of the form \(2^{2^n + 1}\). It was originally conjectured that all Fermat numbers were prime, but this has been proven false; in fact, only the first five Fermat numbers (F0 to F4) are known to be prime.
The first Fermat number known to be composite is F5, which is equal to 4,294,967,297. The prime factorizations of this number are 641 and 6,700,417.
F334 is the first Fermat number that is larger than a googolplex. The largest Fermat number which is known to be composite, as of October 2020, is F18,233,954.[1]