Generalized Fermat Divisors (base=12)

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Definitions and Notes

The numbers F_b,n =

(with b an integer greater than one) are called the generalized Fermat numbers. (In the Prime database they are denoted GF(n,b) to avoid the use of subscripts.) It is reasonable to conjecture that for each base b, there are only finitely many such primes.

As in the case of the Fermat numbers, many have interested in the form and distribution of the divisors of these numbers. When b is even, each of their divisors must have the form

k^.2^m+1

with k odd and m>n. For this reason, when we find a large prime of the form k^.2ⁿ+1 (with k small), we usually check to see if it divides a Fermat number. For example, Gallot's Win95 program Proth.exe has this test built in for a few choices of b.

The number k^.2ⁿ+1 (k odd) will divide some generalized Fermat number for roughly 1/k of the bases b.

Record Primes of this Type

rank prime digits who when comment

1 7 · 2^2915954+1 877791 g279 Jun 2008 Divides GF(2915953, 12) [g322]
2 3 · 2^2478785+1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
3 3 · 2^2145353+1 645817 g245 Feb 2003 Divides Fermat F(2145351), GF(2145351, 3), GF(2145352, 5), GF(2145348, 6), GF(2145352, 10), GF(2145351, 12)
4 7 · 2^2139912+1 644179 g279 May 2007 Divides GF(2139911, 12)
5 5 · 2^1320487+1 397507 g55 Mar 2002 Divides GF(1320486, 12)
6 25 · 2^1211488+1 364696 g279 Jan 2005 Generalized Fermat, divides GF(1211487, 12)
7 33 · 2^1130884+1 340432 L165 Jul 2006 Divides GF(1130881, 12)
8 113 · 2⁹¹⁶⁸⁰¹+1 275987 L153 May 2009 Divides GF(916800, 5), GF(916800, 12)
9 11 · 2⁸⁸⁶⁰⁷¹+1 266735 g277 Feb 2005 Divides GF(886070, 12)
10 7 · 2⁸⁰⁴⁵³⁴+1 242190 g196 Dec 2003 Divides GF(804533, 12)
11 59 · 2⁷²⁷⁸¹⁵+1 219096 p227 Oct 2008 Divides GF(727814, 12)
12 75 · 2⁷⁰⁵⁶⁸⁸+1 212436 p227 Nov 2008 Divides GF(705684, 12)
13 195 · 2⁵⁸⁵⁹⁸⁸+1 176403 p189 Oct 2007 Divides GF(585985, 12) [K]
14 243 · 2⁴⁹⁵⁷³²+1 149233 L165 May 2007 Divides Fermat F(495728), GF(495726, 3), GF(495728, 6), GF(495727, 12)
15 9 · 2⁴⁶¹⁰⁸¹+1 138801 g122 Aug 2003 Divides Fermat F(461076), GF(461077, 3), GF(461077, 6), GF(461077, 12)
16 1791 · 2⁴⁶⁰⁶⁹⁶+1 138687 p240 Jun 2009 Divides GF(460693, 12)
17 69 · 2³⁹⁴⁵⁷⁴+1 118781 p219 Jan 2008 Divides GF(394572, 12)
18 3 · 2³⁸²⁴⁴⁹+1 115130 g132 Jul 1999 Divides Fermat F(382447), GF(382447, 3), GF(382447, 12), GF(382443, 6)
19 119 · 2³⁷⁶⁹⁵¹+1 113476 g233 Jan 2006 Divides GF(376950, 12)
20 3 · 2³⁶²⁷⁶⁵+1 109204 g245 Jul 2002 Divides GF(362763, 12), GF(362764, 10)

Wilfrid Keller's Prime factors of generalized Fermat numbers F_m(12) and complete factoring status
Wilfrid Keller's Factors of generalized Fermat numbers found after Björn & Riesel

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
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
Riesel69b
H. Riesel, "Common prime factors of the numbers A_n =a^2ⁿ+1," BIT, 9 (1969) 264-269. MR 41:3381