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
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
525 · 213719266 + 1 4129912 L4965 Sep 2022 Generalized Fermat
681 · 213708272 + 1 4126603 L4965 Oct 2022 Generalized Fermat
781 · 213470584 + 1 4055052 L4965 Oct 2022 Generalized Fermat
84896418524288 + 1 3507424 L4245 May 2022 Generalized Fermat
93638450524288 + 1 3439810 L4591 May 2020 Generalized Fermat
109 · 211366286 + 1 3421594 L4965 Mar 2020 Generalized Fermat
113214654524288 + 1 3411613 L4309 Dec 2019 Generalized Fermat
122985036524288 + 1 3394739 L4752 Sep 2019 Generalized Fermat
132877652524288 + 1 3386397 L4250 Jun 2019 Generalized Fermat
142788032524288 + 1 3379193 L4584 Apr 2019 Generalized Fermat
152733014524288 + 1 3374655 L4929 Mar 2019 Generalized Fermat
162312092524288 + 1 3336572 L4720 Aug 2018 Generalized Fermat
172061748524288 + 1 3310478 L4783 Mar 2018 Generalized Fermat
181880370524288 + 1 3289511 L4201 Jan 2018 Generalized Fermat
19475856524288 + 1 2976633 L3230 Aug 2012 Generalized Fermat
20356926524288 + 1 2911151 L3209 Jul 2012 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., 1994.  Providence, RI, 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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