# 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.

### Definitions and Notes

Any generalized Fermat number Fb,n = (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.

### Record Primes of this Type

rankprime digitswhowhencomment
19194441048576 + 1 6253210 L4286 Sep 2017 Generalized Fermat
2475856524288 + 1 2976633 L3230 Aug 2012 Generalized Fermat
3356926524288 + 1 2911151 L3209 Jul 2012 Generalized Fermat
4341112524288 + 1 2900832 L3184 Jun 2012 Generalized Fermat
575898524288 + 1 2558647 p334 Nov 2011 Generalized Fermat
63060772262144 + 1 1700222 L4649 Jun 2017 Generalized Fermat
72676404262144 + 1 1684945 L4591 Mar 2017 Generalized Fermat
82611294262144 + 1 1682141 L4250 Mar 2017 Generalized Fermat
92514168262144 + 1 1677825 L4564 Feb 2017 Generalized Fermat
102042774262144 + 1 1654187 L4499 Nov 2016 Generalized Fermat
111828858262144 + 1 1641593 L4200 Aug 2016 Generalized Fermat
121615588262144 + 1 1627477 L4200 May 2016 Generalized Fermat
131488256262144 + 1 1618131 L4249 Mar 2016 Generalized Fermat
141415198262144 + 1 1612400 L4308 Feb 2016 Generalized Fermat
15773620262144 + 1 1543643 L3118 Apr 2012 Generalized Fermat
16676754262144 + 1 1528413 L2975 Feb 2012 Generalized Fermat
17525094262144 + 1 1499526 p338 Jan 2012 Generalized Fermat
18361658262144 + 1 1457075 p332 Nov 2011 Generalized Fermat
194 · 797468702 + 1 1359920 L4548 Feb 2017 Generalized Fermat
20145310262144 + 1 1353265 p314 Feb 2011 Generalized Fermat

### 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