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.

Definitions and Notes

Any generalized Fermat number F_b,n = (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 b^n/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.

Record Primes of this Type

rank prime digits who when comment 1 4 · 511786358 + 1 8238312 A2 Oct 2024 Generalized Fermat 2 38432361048576 + 1 6904556 L6094 Dec 2024 Generalized Fermat 3 19637361048576 + 1 6598776 L4245 Sep 2022 Generalized Fermat 4 19517341048576 + 1 6595985 L5583 Aug 2022 Generalized Fermat 5 10590941048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat 6 9194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat 7 81 · 220498148 + 1 6170560 L4965 Jun 2023 Generalized Fermat 8 4 · 58431178 + 1 5893142 A2 Jan 2024 Generalized Fermat 9 4 · 311279466 + 1 5381674 A2 Sep 2024 Generalized Fermat 10 25 · 213719266 + 1 4129912 L4965 Sep 2022 Generalized Fermat 11 81 · 213708272 + 1 4126603 L4965 Oct 2022 Generalized Fermat 12 81 · 213470584 + 1 4055052 L4965 Oct 2022 Generalized Fermat 13 18099898524288 + 1 3805113 x50 Sep 2025 Generalized Fermat 14 17177670524288 + 1 3793205 L5186 Oct 2025 Generalized Fermat 15 16211276524288 + 1 3780021 L6006 Aug 2025 Generalized Fermat 16 15958750524288 + 1 3776446 L5639 Jul 2025 Generalized Fermat 17 15852200524288 + 1 3774921 L5526 Jul 2025 Generalized Fermat 18 4 · 55380542 + 1 3760839 L4965 Feb 2023 Generalized Fermat 19 13520762524288 + 1 3738699 L6221 Feb 2025 Generalized Fermat 20 13427472524288 + 1 3737122 L5775 Mar 2025 Generalized Fermat

Related Pages

• Generalized Fermat Numbers by Yves Gallot
• Smallest base values yielding Generalized Fermat Primes by Yves Gallot

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 G_n = 6²ⁿ + 1 and H_n = 10²ⁿ + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813

Riesel69b

H. Riesel, "Common prime factors of the numbers A_n =a²ⁿ+1," BIT, 9 (1969) 264-269.  MR 41:3381