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 7 · 2^3511774 + 1 1057151 p236 Nov 2008 Divides GF(3511773, 6)
2 9 · 2^2543551 + 1 765687 L1204 Jun 2011 Divides Fermat F(2543548), GF(2543549, 3), GF(2543549, 6)
3 3 · 2^2478785 + 1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
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 13 · 2^1861732 + 1 560439 g267 Nov 2005 Divides GF(1861731, 6)
6 39 · 2^1824871 + 1 549343 L2664 Sep 2011 Divides GF(1824867, 6)
7 5 · 2^1777515 + 1 535087 p148 Apr 2005 Divides GF(1777511, 5), GF(1777514, 6)
8 87 · 2^1580858 + 1 475888 L2487 Jul 2011 Divides GF(1580856, 6)
9 7 · 2^1491852 + 1 449094 p166 Mar 2005 Divides GF(1491851, 6)
10 15 · 2^1418605 + 1 427044 g279 Apr 2006 Divides GF(1418600, 5), GF(1418601, 6)
11 11 · 2^1343347 + 1 404389 p169 May 2005 Divides GF(1343346, 6)
12 225 · 2⁹⁸⁸⁶⁹⁵ + 1 297630 L1446 Sep 2010 Divides GF(988693, 6)
13 13 · 2⁶⁸⁴⁵⁶⁰ + 1 206075 g267 Jul 2003 Divides GF(684557, 10), GF(684559, 6)
14 351 · 2⁵⁸⁸³²⁵ + 1 177107 L651 Mar 2009 Divides GF(588323, 6)
15 7 · 2⁵⁶¹⁸¹⁶ + 1 169125 g148 Jun 2003 Divides GF(561815, 5); GF(561815, 6) [p149]
16 345 · 2⁵²⁵⁹⁷⁷ + 1 158338 g258 Sep 2007 Divides GF(525974, 6)
17 11 · 2⁵²⁵⁵⁸⁹ + 1 158220 p116 Nov 2003 Divides GF(525588, 6)
18 39 · 2⁵¹²⁹⁹⁷ + 1 154430 g267 Jan 2005 Divides GF(512994, 5), GF(512995, 6)
19 243 · 2⁴⁹⁵⁷³² + 1 149233 L165 May 2007 Divides Fermat F(495728), GF(495726, 3), GF(495728, 6), GF(495727, 12)
20 9 · 2⁴⁶¹⁰⁸¹ + 1 138801 g122 Aug 2003 Divides Fermat F(461076), GF(461077, 3), GF(461077, 6), GF(461077, 12)

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. 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