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
32985036524288 + 1 3394739 L4752 Sep 2019 Generalized Fermat
42877652524288 + 1 3386397 L4250 Jun 2019 Generalized Fermat
52788032524288 + 1 3379193 L4584 Apr 2019 Generalized Fermat
62733014524288 + 1 3374655 L4929 Mar 2019 Generalized Fermat
72312092524288 + 1 3336572 L4720 Aug 2018 Generalized Fermat
82061748524288 + 1 3310478 L4783 Mar 2018 Generalized Fermat
91880370524288 + 1 3289511 L4201 Jan 2018 Generalized Fermat
10475856524288 + 1 2976633 L3230 Aug 2012 Generalized Fermat
11356926524288 + 1 2911151 L3209 Jul 2012 Generalized Fermat
12341112524288 + 1 2900832 L3184 Jun 2012 Generalized Fermat
1375898524288 + 1 2558647 p334 Nov 2011 Generalized Fermat
148883864262144 + 1 1821535 L4715 Nov 2019 Generalized Fermat
158521794262144 + 1 1816798 L4289 Sep 2019 Generalized Fermat
166291332262144 + 1 1782250 L4864 Dec 2018 Generalized Fermat
176287774262144 + 1 1782186 L4726 Dec 2018 Generalized Fermat
185828034262144 + 1 1773542 L4720 Sep 2018 Generalized Fermat
195205422262144 + 1 1760679 L4201 Jun 2018 Generalized Fermat
205152128262144 + 1 1759508 L4720 Jun 2018 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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