Generalized Fermat Divisors (base=6)

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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(b,n) 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 · 2^2478785+1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
2 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)
3 13 · 2^1861732+1 560439 g267 Nov 2005 Divides GF(1861731, 6)
4 5 · 2^1777515+1 535087 p148 Apr 2005 Divides GF(1777511, 5), GF(1777514, 6)
5 7 · 2^1491852+1 449094 p166 Mar 2005 Divides GF(1491851, 6)
6 15 · 2^1418605+1 427044 g279 Apr 2006 Divides GF(1418600, 5), GF(1418601, 6)
7 11 · 2^1343347+1 404389 p169 May 2005 Divides GF(1343346, 6)
8 13 · 2⁶⁸⁴⁵⁶⁰+1 206075 g267 Jul 2003 Divides GF(684557, 10), GF(684559, 6)
9 7 · 2⁵⁶¹⁸¹⁶+1 169125 g148 Jun 2003 Divides GF(561815, 5); GF(561815, 6) [p149]
10 345 · 2⁵²⁵⁹⁷⁷+1 158338 g258 Sep 2007 Divides GF(525974, 6)
11 11 · 2⁵²⁵⁵⁸⁹+1 158220 p116 Nov 2003 Divides GF(525588, 6)
12 39 · 2⁵¹²⁹⁹⁷+1 154430 g267 Jan 2005 Divides GF(512994, 5), GF(512995, 6)
13 243 · 2⁴⁹⁵⁷³²+1 149233 L165 May 2007 Divides Fermat F(495728), GF(495726, 3), GF(495728, 6), GF(495727, 12)
14 9 · 2⁴⁶¹⁰⁸¹+1 138801 g122 Aug 2003 Divides Fermat F(461076), GF(461077, 3), GF(461077, 6), GF(461077, 12)
15 45 · 2⁴⁵⁸⁷¹²+1 138088 L170 Jun 2007 Divides GF(458709, 5), GF(458710, 6) [K]
16 3 · 2³⁸²⁴⁴⁹+1 115130 g132 Jul 1999 Divides Fermat F(382447), GF(382447, 3), GF(382447, 12), GF(382443, 6)
17 9 · 2³⁰⁴⁶⁰⁷+1 91697 g23 Jul 1998 Divides GF(304604, 6)
18 3 · 2³⁰³⁰⁹³+1 91241 Y Jan 1998 Divides Fermat F(303088); GF(303088, 3), GF(303086, 6), GF(303092, 10), GF(303088, 12), GF(303092, 5) [g0]
19 15 · 2³⁰⁰⁴⁸⁸+1 90458 p114 Feb 2002 Divides GF(300479, 6), GF(300484, 10)
20 357 · 2²⁸⁶⁶⁷²+1 86300 p161 Dec 2005 Divides GF(286670, 6)

Wilfrid Keller's Prime factors of generalized Fermat numbers F_m(6) 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. volume 48, Amer. Math. Soc., Providence, RI, pp. 583-587, 1994. MR 95j:11006
Riesel69
H. Riesel, "Some factors of the numbers G_n = 62ⁿ + 1 and H_n = 102ⁿ + 1," Math. Comp., 23:106 (1969) 413--415. MR 39:6813
Riesel69b
H. Riesel, "Common prime factors of the numbers A_n =a2ⁿ+1," BIT, 9 (1969) 264-269. MR 41:3381