Generalized Fermat

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The Prime Pages

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 F_b,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 b^n/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

rank prime digits who when comment

1 24518²⁶²¹⁴⁴+1 1150678 g413 Mar 2008 Generalized Fermat
2 1372930¹³¹⁰⁷²+1 804474 g236 Sep 2003 Generalized Fermat
3 1361244¹³¹⁰⁷²+1 803988 g236 Jul 2004 Generalized Fermat
4 1176694¹³¹⁰⁷²+1 795695 g236 Aug 2003 Generalized Fermat
5 572186¹³¹⁰⁷²+1 754652 g0 Jan 2004 Generalized Fermat
6 386892¹³¹⁰⁷²+1 732377 p259 Oct 2009 Generalized Fermat
7 130816¹³¹⁰⁷²+1 670651 g308 Jul 2003 Generalized Fermat
8 62722¹³¹⁰⁷²+1 628808 g308 Feb 2003 Generalized Fermat
9 81 · 2^1643428+1 494724 g418 Mar 2009 Generalized Fermat
10 81 · 2^1606848+1 483712 gt Mar 2007 Generalized Fermat
11 19502212⁶⁵⁵³⁶+1 477763 p160 Jan 2005 Generalized Fermat
12 17684828⁶⁵⁵³⁶+1 474979 g410 Aug 2007 Generalized Fermat
13 17655444⁶⁵⁵³⁶+1 474932 g410 Aug 2007 Generalized Fermat
14 17629398⁶⁵⁵³⁶+1 474890 g410 Aug 2007 Generalized Fermat
15 2187182⁶⁵⁵³⁶+1 415491 g260 Apr 2009 Generalized Fermat
16 2177038⁶⁵⁵³⁶+1 415359 g260 Nov 2008 Generalized Fermat
17 2162068⁶⁵⁵³⁶+1 415162 g260 Jul 2008 Generalized Fermat
18 1874512⁶⁵⁵³⁶+1 411101 g413 Jun 2008 Generalized Fermat
19 1828502⁶⁵⁵³⁶+1 410393 GF2 Mar 2005 Generalized Fermat
20 1540550⁶⁵⁵³⁶+1 405516 GF2 May 2003 Generalized Fermat

Generalized Fermat Prime Search by Yves Gallot
Smallest base values yielding Generalized Fermat Primes by Yves Gallot

References

BR98
A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441--446. MR 98e:11008 (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
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 G_n = 6^2ⁿ + 1 and H_n = 10^2ⁿ + 1," Math. Comp., 23:106 (1969) 413--415. MR 39:6813
Riesel69b
H. Riesel, "Common prime factors of the numbers A_n =a^2ⁿ+1," BIT, 9 (1969) 264-269. MR 41:3381