The Top Twenty--a Prime Page Collection

Generalized Fermat

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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
1475856524288 + 1 2976633 L3230 Aug 2012 Generalized Fermat
2356926524288 + 1 2911151 L3209 Jul 2012 Generalized Fermat
3341112524288 + 1 2900832 L3184 Jun 2012 Generalized Fermat
475898524288 + 1 2558647 p334 Nov 2011 Generalized Fermat
52042774262144 + 1 1654187 L4499 Nov 2016 Generalized Fermat
61828858262144 + 1 1641593 L4200 Aug 2016 Generalized Fermat
71615588262144 + 1 1627477 L4200 May 2016 Generalized Fermat
81488256262144 + 1 1618131 L4249 Mar 2016 Generalized Fermat
91415198262144 + 1 1612400 L4308 Feb 2016 Generalized Fermat
10773620262144 + 1 1543643 L3118 Apr 2012 Generalized Fermat
11676754262144 + 1 1528413 L2975 Feb 2012 Generalized Fermat
12525094262144 + 1 1499526 p338 Jan 2012 Generalized Fermat
13361658262144 + 1 1457075 p332 Nov 2011 Generalized Fermat
14145310262144 + 1 1353265 p314 Feb 2011 Generalized Fermat
1540734262144 + 1 1208473 p309 Mar 2011 Generalized Fermat
1624518262144 + 1 1150678 g413 Mar 2008 Generalized Fermat
179 · 23497442 + 1 1052836 L1780 Oct 2012 Generalized Fermat, divides GF(3497441, 10)
1881 · 23352924 + 1 1009333 L1728 Jan 2012 Generalized Fermat
1944330870131072 + 1 1002270 L4501 Nov 2016 Generalized Fermat
2044085096131072 + 1 1001953 L4482 Oct 2016 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., 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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