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 75898⁵²⁴²⁸⁸ + 1 2558647 p334 Nov 2011 Generalized Fermat
2 773620²⁶²¹⁴⁴ + 1 1543643 L3118 Apr 2012 Generalized Fermat
3 676754²⁶²¹⁴⁴ + 1 1528413 L2975 Feb 2012 Generalized Fermat
4 525094²⁶²¹⁴⁴ + 1 1499526 p338 Jan 2012 Generalized Fermat
5 361658²⁶²¹⁴⁴ + 1 1457075 p332 Nov 2011 Generalized Fermat
6 145310²⁶²¹⁴⁴ + 1 1353265 p314 Feb 2011 Generalized Fermat
7 40734²⁶²¹⁴⁴ + 1 1208473 p309 Mar 2011 Generalized Fermat
8 24518²⁶²¹⁴⁴ + 1 1150678 g413 Mar 2008 Generalized Fermat
9 81 · 2^3352924 + 1 1009333 L1728 Jan 2012 Generalized Fermat
10 1372930¹³¹⁰⁷² + 1 804474 g236 Sep 2003 Generalized Fermat
11 1361244¹³¹⁰⁷² + 1 803988 g236 Jul 2004 Generalized Fermat
12 1176694¹³¹⁰⁷² + 1 795695 g236 Aug 2003 Generalized Fermat
13 572186¹³¹⁰⁷² + 1 754652 g0 Jan 2004 Generalized Fermat
14 386892¹³¹⁰⁷² + 1 732377 p259 Oct 2009 Generalized Fermat
15 228188¹³¹⁰⁷² + 1 702323 g124 Aug 2010 Generalized Fermat
16 130816¹³¹⁰⁷² + 1 670651 g308 Jul 2003 Generalized Fermat
17 25 · 2^2141884 + 1 644773 L1741 Sep 2011 Divides Fermat F(2141872), GF(2141871, 5), GF(2141872, 10); generalized Fermat
18 62722¹³¹⁰⁷² + 1 628808 g308 Feb 2003 Generalized Fermat
19 9 · 2^1807574 + 1 544135 L2419 Jun 2011 Generalized Fermat
20 49 · 2^1646042 + 1 495510 L2516 Jul 2011 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)
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 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