Generalized Fermat Divisors (bases 3,5,6,10,12)

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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(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 · 2^5082306+1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
2 7 · 2^3511774+1 1057151 p236 Nov 2008 Divides GF(3511773, 6)
3 7 · 2^2915954+1 877791 g279 Jun 2008 Divides GF(2915953, 12) [g322]
4 3 · 2^2478785+1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
5 3 · 2^2291610+1 689844 L753 Aug 2008 Divides GF(2291607, 3), GF(2291609, 5)
6 7 · 2^2167800+1 652574 g279 Apr 2007 Divides Fermat F(2167797), GF(2167799, 5), GF(2167799, 10)
7 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)
8 7 · 2^2139912+1 644179 g279 May 2007 Divides GF(2139911, 12)
9 17 · 2^1990299+1 599141 g267 Mar 2006 Divides GF(1990298, 3)
10 13 · 2^1861732+1 560439 g267 Nov 2005 Divides GF(1861731, 6)
11 3 · 2^1832496+1 551637 p189 Jul 2007 Divides GF(1832490, 3), GF(1832494, 5)
12 5 · 2^1777515+1 535087 p148 Apr 2005 Divides GF(1777511, 5), GF(1777514, 6)
13 13 · 2^1499876+1 451509 g267 Dec 2004 Divides GF(1499875, 3)
14 7 · 2^1491852+1 449094 p166 Mar 2005 Divides GF(1491851, 6)
15 15 · 2^1418605+1 427044 g279 Apr 2006 Divides GF(1418600, 5), GF(1418601, 6)
16 17 · 2^1388355+1 417938 g267 Sep 2005 Divides GF(1388354, 10)
17 11 · 2^1343347+1 404389 p169 May 2005 Divides GF(1343346, 6)
18 5 · 2^1320487+1 397507 g55 Mar 2002 Divides GF(1320486, 12)
19 5 · 2^1282755+1 386149 g55 Jun 2002 Divides GF(1282754, 3), GF(1282748, 5)
20 15 · 2^1276177+1 384169 g279 Feb 2006 Divides GF(1276174, 3), GF(1276174, 10)

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