Generalized Fermat Divisors (base=10)

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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 · 2^2167800 + 1 652574 g279 Apr 2007 Divides Fermat F(2167797), GF(2167799, 5), GF(2167799, 10)
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 25 · 2^2141884 + 1 644773 L1741 Sep 2011 Divides Fermat F(2141872), GF(2141871, 5), GF(2141872, 10); generalized Fermat
4 45 · 2^2014557 + 1 606444 L1349 Feb 2012 Divides GF(2014552, 10)
5 135 · 2^1515894 + 1 456332 L1129 Jan 2013 Divides GF(1515890, 10)
6 17 · 2^1388355 + 1 417938 g267 Sep 2005 Divides GF(1388354, 10)
7 107 · 2^1337019 + 1 402485 L2659 Jan 2012 Divides GF(1337018, 10)
8 15 · 2^1276177 + 1 384169 g279 Feb 2006 Divides GF(1276174, 3), GF(1276174, 10)
9 29 · 2^1152765 + 1 347019 g300 Sep 2005 Divides GF(1152760, 10)
10 163 · 2^1129934 + 1 340147 L1751 Dec 2010 Divides GF(1129933, 10)
11 1491 · 2^1050764 + 1 316315 L2826 Feb 2013 Divides GF(1050763, 10)
12 9539 · 2^1034437 + 1 311401 L1502 Jan 2013 Divides GF(1034434, 10)
13 3 · 2⁹¹⁶⁷⁷³ + 1 275977 g245 Jun 2001 Divides GF(916771, 3), GF(916772, 10)
14 13 · 2⁶⁸⁴⁵⁶⁰ + 1 206075 g267 Jul 2003 Divides GF(684557, 10), GF(684559, 6)
15 9265 · 2⁴⁸²⁰⁷² + 1 145123 L635 Oct 2009 Divides GF(482070, 10)
16 6841 · 2⁴⁷⁴³⁴⁸ + 1 142797 L1065 Oct 2009 Divides GF(474347, 10)
17 3911 · 2⁴⁶²⁵⁷⁹ + 1 139254 L679 Jul 2009 Divides GF(462577, 10)
18 4377 · 2⁴⁵⁶⁷⁰⁸ + 1 137487 L872 Jun 2009 Divides GF(456707, 10)
19 9 · 2⁴³⁵⁷⁴³ + 1 131173 g122 Jul 2003 Divides GF(435742, 10)
20 15 · 2⁴⁰³⁹²⁹ + 1 121596 p114 Oct 2002 Divides GF(403927, 10)

Wilfrid Keller's Prime factors of generalized Fermat numbers F_m(10) 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