# 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.

### Definitions and Notes

Any generalized Fermat number Fb,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 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.

### Record Primes of this Type

rankprime digitswhowhencomment
119637361048576 + 1 6598776 L4245 Sep 2022 Generalized Fermat
219517341048576 + 1 6595985 L5583 Aug 2022 Generalized Fermat
310590941048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat
49194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
581 · 220498148 + 1 6170560 L4965 Jun 2023 Generalized Fermat
64 · 58431178 + 1 5893142 A2 Jan 2024 Generalized Fermat
725 · 213719266 + 1 4129912 L4965 Sep 2022 Generalized Fermat
881 · 213708272 + 1 4126603 L4965 Oct 2022 Generalized Fermat
981 · 213470584 + 1 4055052 L4965 Oct 2022 Generalized Fermat
104 · 55380542 + 1 3760839 L4965 Feb 2023 Generalized Fermat
1110913140524288 + 1 3689913 L6043 Jun 2024 Generalized Fermat
129332124524288 + 1 3654278 L5025 Jun 2024 Generalized Fermat
138630170524288 + 1 3636472 L5543 Apr 2024 Generalized Fermat
144 · 37578378 + 1 3615806 A2 Sep 2024 Generalized Fermat
156339004524288 + 1 3566218 L1372 Jun 2023 Generalized Fermat
165897794524288 + 1 3549792 x50 Dec 2022 Generalized Fermat
174896418524288 + 1 3507424 L4245 May 2022 Generalized Fermat
183638450524288 + 1 3439810 L4591 May 2020 Generalized Fermat
199 · 211366286 + 1 3421594 L4965 Mar 2020 Generalized Fermat
203214654524288 + 1 3411613 L4309 Dec 2019 Generalized Fermat

### References

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H. Dubner and Y. Gallot, "Distribution of generalized Fermat prime numbers," Math. Comp., 71 (2002) 825--832.  MR 2002j:11156 (Abstract available)
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H. Dubner and W. Keller, "Factors of generalized Fermat numbers," Math. Comp., 64 (1995) 397--405.  MR 95c:11010
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H. Dubner, "Generalized Fermat primes," J. Recreational Math., 18 (1985-86) 279--280.  MR 2002j:11156
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M. Morimoto, "On prime numbers of Fermat types," Sûgaku, 38:4 (1986) 350--354.  Japanese.  MR 88h:11007
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Pi, Xin Ming, "Searching for generalized Fermat primes," J. Math. (Wuhan), 18:3 (1998) 276--280.  MR 1656292
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Pi, Xin Ming, "Generalized Fermat primes for b < 2000, m< 10," J. Math. (Wuhan), 22:1 (2002) 91--93.  MR 1897106
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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
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H. Riesel, "Some factors of the numbers Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813
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H. Riesel, "Common prime factors of the numbers An =a2n+1," BIT, 9 (1969) 264-269.  MR 41:3381