There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a
generalized Fermat number as a number of the form 
with 
,
while Riesel (1994) further generalizes, defining it to be a number of the form 
.
Both definitions generalize the usual Fermat numbers 
.
The following table gives the first few generalized Fermat numbers for various bases
.
 | OEIS | generalized
Fermat numbers in base  |
| 2 | A000215 | 3, 5, 17, 257, 65537, 4294967297, ... |
| 3 | A059919 | 4, 10, 82, 6562, 43046722, ... |
| 4 | A000215 | 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... |
| 5 | A078303 | 6, 26, 626, 390626, 152587890626, ... |
| 6 | A078304 | 7, 37, 1297, 1679617, 2821109907457, ... |
Generalized Fermat numbers can be prime only for even 
. More specifically, an odd prime 
is a generalized Fermat prime iff there exists an integer
with 
and 
(Broadhurst 2006).
Many of the largest known prime numbers are generalized Fermat numbers. Dubner found 
(
digits) and 
(
digits) in September 1992 (Ribenboim 1996, p. 360). The largest known as of
January 2009 is 
(https://t5k.org/primes/page.php?id=84401),
which has 
decimal digits.
The following table gives the first few generalized Fermat primes for various even bases 
.
 | prime  |
| 2 | 5, 17, 257, 65537, ... |
| 4 | 17, 257, 65537, ... |
| 6 | 37, 1297, ... |
