The Top Twenty--a Prime Page Collection

Generalized Fermat

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GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

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. Comments and suggestions requested.

(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
110590941048576 + 1 6317602 L4720 Nov 2018 Generalized Fermat
29194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
39 · 211366286 + 1 3421594 L4965 Mar 2020 Generalized Fermat
43214654524288 + 1 3411613 L4309 Dec 2019 Generalized Fermat
52985036524288 + 1 3394739 L4752 Sep 2019 Generalized Fermat
62877652524288 + 1 3386397 L4250 Jun 2019 Generalized Fermat
72788032524288 + 1 3379193 L4584 Apr 2019 Generalized Fermat
82733014524288 + 1 3374655 L4929 Mar 2019 Generalized Fermat
92312092524288 + 1 3336572 L4720 Aug 2018 Generalized Fermat
102061748524288 + 1 3310478 L4783 Mar 2018 Generalized Fermat
111880370524288 + 1 3289511 L4201 Jan 2018 Generalized Fermat
12475856524288 + 1 2976633 L3230 Aug 2012 Generalized Fermat
13356926524288 + 1 2911151 L3209 Jul 2012 Generalized Fermat
14341112524288 + 1 2900832 L3184 Jun 2012 Generalized Fermat
1575898524288 + 1 2558647 p334 Nov 2011 Generalized Fermat
169812766262144 + 1 1832857 L4245 Feb 2020 Generalized Fermat
179750938262144 + 1 1832137 L4309 Feb 2020 Generalized Fermat
189450844262144 + 1 1828578 L5020 Jan 2020 Generalized Fermat
199125820262144 + 1 1824594 L5002 Dec 2019 Generalized Fermat
208883864262144 + 1 1821535 L4715 Nov 2019 Generalized Fermat

(up) Related Pages

(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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