The term "generalized Fermat number" has two different definitions, 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 integer 
and nonnegative integer 
, while Riesel (1994) further generalizes, defining
it to be a number of the form 
for positive
integers 
and 
and nonnegative integer 
. Both definitions generalize the usual Fermat
numbers 
. The following table gives the first few
generalized Fermat numbers 
for 
,
1, ... and 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 | A199591 | 6, 26, 626, 390626, 152587890626, ... |
| 6 | A078303 | 7, 37, 1297, 1679617, 2821109907457, ... |
For the form 
with integers 
and 
,
a generalized Fermat number can be prime only for
even 
.
An odd prime 
has this form iff there exists an
integer 
with 
and 
(Broadhurst 2006). Indeed, these conditions force 
, while 
gives 
.
Many of the largest known prime numbers are generalized Fermat numbers. Dubner found 
(10,861 decimal
digits) and 
(10,071 decimal
digits) in September 1992 (Ribenboim 1996, p. 360). The largest known as
of Oct. 2026 for the form 
is 
,
which has 13,426,224 decimal digits and was found
in Oct. 2025 (PrimePages 2026).
The following table gives the first few generalized Fermat primes 
with 
for various even bases 
.
 | prime  |
| 2 | 5, 17, 257, 65537, ... |
| 4 | 17, 257, 65537, ... |
| 6 | 37, 1297, ... |
."
Accessed Oct. 9, 2026.