A Fermat prime is a Fermat number 
that is prime.
Fermat primes are therefore near-square primes.
Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem
the proof that there are an infinite
number of Fermat primes (Ribenboim 1996, p. 88). At present, however, the only
Fermat numbers 
for 
for which primality or compositeness has been established
are all composite.
The only known Fermat primes are
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
(OEIS A019434). The known primes of the form 
consist of 2 together with the known Fermat primes, giving
the sequence 2, 3, 5, 17, 257, 65537, ... (OEIS A092506).
For 
, 
is a Fermat prime iff the
decimal period of 
is equal to 
. In other words, Fermat primes greater than 5 are full reptend primes.
