Generalized Fermat Divisors (base=3)

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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 3 · 2^5082306+1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
2 3 · 2^2478785+1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
3 3 · 2^2291610+1 689844 L753 Aug 2008 Divides GF(2291607, 3), GF(2291609, 5)
4 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)
5 17 · 2^1990299+1 599141 g267 Mar 2006 Divides GF(1990298, 3)
6 3 · 2^1832496+1 551637 p189 Jul 2007 Divides GF(1832490, 3), GF(1832494, 5)
7 13 · 2^1499876+1 451509 g267 Dec 2004 Divides GF(1499875, 3)
8 5 · 2^1282755+1 386149 g55 Jun 2002 Divides GF(1282754, 3), GF(1282748, 5)
9 15 · 2^1276177+1 384169 g279 Feb 2006 Divides GF(1276174, 3), GF(1276174, 10)
10 3 · 2⁹¹⁶⁷⁷³+1 275977 g245 Jun 2001 Divides GF(916771, 3), GF(916772, 10)
11 35 · 2⁸³¹⁴¹¹+1 250282 g279 Jun 2006 Divides GF(831410, 3)
12 5 · 2⁸¹⁹⁷³⁹+1 246767 g55 Jul 2001 Divides GF(819738, 3)
13 3 · 2⁸⁰¹⁹⁷⁸+1 241420 g372 Sep 2005 Divides GF(801973, 3), GF(801977, 5)
14 35 · 2⁷⁶⁸⁰⁶³+1 231212 L126 Dec 2005 Divides GF(768062, 3)
15 3 · 2⁷⁰⁹⁹⁶⁸+1 213723 g372 May 2005 Divides GF(709962, 3), GF(709963, 5)
16 225 · 2⁶⁰⁵¹⁷²+1 182178 p43 Sep 2005 Generalized Fermat, divides GF(605169, 3)
17 163 · 2⁵⁹⁷⁴⁷⁴+1 179860 p43 May 2005 Divides GF(597473, 3)
18 45 · 2⁵⁷⁴⁵⁰⁶+1 172946 g409 Apr 2007 Divides GF(574504, 3)
19 1143 · 2⁵⁷³⁸⁹⁴+1 172763 L717 Mar 2009 Divides GF(573892, 3)
20 243 · 2⁵⁵⁵⁹⁸⁴+1 167371 L165 Sep 2007 Divides GF(555978, 3)

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
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