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
53060772262144 + 1 1700222 L4649 Jun 2017 Generalized Fermat
62676404262144 + 1 1684945 L4591 Mar 2017 Generalized Fermat
72611294262144 + 1 1682141 L4250 Mar 2017 Generalized Fermat
82514168262144 + 1 1677825 L4564 Feb 2017 Generalized Fermat
92042774262144 + 1 1654187 L4499 Nov 2016 Generalized Fermat
101828858262144 + 1 1641593 L4200 Aug 2016 Generalized Fermat
111615588262144 + 1 1627477 L4200 May 2016 Generalized Fermat
121488256262144 + 1 1618131 L4249 Mar 2016 Generalized Fermat
131415198262144 + 1 1612400 L4308 Feb 2016 Generalized Fermat
14773620262144 + 1 1543643 L3118 Apr 2012 Generalized Fermat
15676754262144 + 1 1528413 L2975 Feb 2012 Generalized Fermat
16525094262144 + 1 1499526 p338 Jan 2012 Generalized Fermat
17361658262144 + 1 1457075 p332 Nov 2011 Generalized Fermat
184 · 797468702 + 1 1359920 L4548 Feb 2017 Generalized Fermat
19145310262144 + 1 1353265 p314 Feb 2011 Generalized Fermat
2040734262144 + 1 1208473 p309 Mar 2011 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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