Generalized Fermat

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

(up) Definitions and Notes

Any generalized Fermat number Fb,n = b^2^n+1 (with b an integer greater than one and n greater than zero) which is prime is called a generalized Fermat prime (because they are Fermat primes in the special case b=2).

Why is the exponent a power of two? Because if m is an odd divisor of n, then bn/m+1 divides bn+1, so for the latter to be prime, m must be one. Because the exponent is a power of two, it seems reasonable to conjecture that the number of Generalized Fermat primes is finite for every fixed b.

(up) Record Primes of this Type

rankprime digitswhowhencomment
125241902097152 + 1 13426224 L4245 Oct 2025 Generalized Fermat
24 · 511786358 + 1 8238312 A2 Oct 2024 Generalized Fermat
338432361048576 + 1 6904556 L6094 Dec 2024 Generalized Fermat
419637361048576 + 1 6598776 L4245 Sep 2022 Generalized Fermat
519517341048576 + 1 6595985 L5583 Aug 2022 Generalized Fermat
610590941048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat
79194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
881 · 220498148 + 1 6170560 L4965 Jun 2023 Generalized Fermat
94 · 58431178 + 1 5893142 A2 Jan 2024 Generalized Fermat
104 · 311279466 + 1 5381674 A2 Sep 2024 Generalized Fermat
1125 · 213719266 + 1 4129912 L4965 Sep 2022 Generalized Fermat
1281 · 213708272 + 1 4126603 L4965 Oct 2022 Generalized Fermat
1381 · 213470584 + 1 4055052 L4965 Oct 2022 Generalized Fermat
1418099898524288 + 1 3805113 x50 Sep 2025 Generalized Fermat
1517177670524288 + 1 3793205 L5186 Oct 2025 Generalized Fermat
1616211276524288 + 1 3780021 L6006 Aug 2025 Generalized Fermat
1715958750524288 + 1 3776446 L5639 Jul 2025 Generalized Fermat
1815852200524288 + 1 3774921 L5526 Jul 2025 Generalized Fermat
194 · 55380542 + 1 3760839 L4965 Feb 2023 Generalized Fermat
2013520762524288 + 1 3738699 L6221 Feb 2025 Generalized Fermat

(up) References

BR98
A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441--446.  MR 98e:11008 (Abstract available)
DG2000
H. Dubner and Y. Gallot, "Distribution of generalized Fermat prime numbers," Math. Comp., 71 (2002) 825--832.  MR 2002j:11156 (Abstract available)
DK95
H. Dubner and W. Keller, "Factors of generalized Fermat numbers," Math. Comp., 64 (1995) 397--405.  MR 95c:11010
Dubner86
H. Dubner, "Generalized Fermat primes," J. Recreational Math., 18 (1985-86) 279--280.  MR 2002j:11156
Morimoto86
M. Morimoto, "On prime numbers of Fermat types," Sûgaku, 38:4 (1986) 350--354.  Japanese.  MR 88h:11007
Pi1998
Pi, Xin Ming, "Searching for generalized Fermat primes," J. Math. (Wuhan), 18:3 (1998) 276--280.  MR 1656292
Pi2002
Pi, Xin Ming, "Generalized Fermat primes for b < 2000, m< 10," J. Math. (Wuhan), 22:1 (2002) 91--93.  MR 1897106
RB94
H. Riesel and A. Börn, Generalized Fermat numbers.  In "Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics," W. Gautschi editor, Proc. Symp. Appl. Math. Vol, 48, Amer. Math. Soc., Providence, RI, 1994.  pp. 583-587, MR 95j:11006
Riesel69
H. Riesel, "Some factors of the numbers Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813
Riesel69b
H. Riesel, "Common prime factors of the numbers An =a2n+1," BIT, 9 (1969) 264-269.  MR 41:3381
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