Generalized Fermat Divisors (base=5)

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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 3 · 2^2291610+1 689844 L753 Aug 2008 Divides GF(2291607, 3), GF(2291609, 5)
3 7 · 2^2167800+1 652574 g279 Apr 2007 Divides Fermat F(2167797), GF(2167799, 5), GF(2167799, 10)
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 3 · 2^1832496+1 551637 p189 Jul 2007 Divides GF(1832490, 3), GF(1832494, 5)
6 5 · 2^1777515+1 535087 p148 Apr 2005 Divides GF(1777511, 5), GF(1777514, 6)
7 15 · 2^1418605+1 427044 g279 Apr 2006 Divides GF(1418600, 5), GF(1418601, 6)
8 5 · 2^1282755+1 386149 g55 Jun 2002 Divides GF(1282754, 3), GF(1282748, 5)
9 113 · 2⁹¹⁶⁸⁰¹+1 275987 L153 May 2009 Divides GF(916800, 5), GF(916800, 12)
10 43 · 2⁸⁹⁴⁷⁶⁶+1 269354 g279 Aug 2006 Divides GF(894765, 5)
11 7 · 2⁸¹¹²³⁰+1 244206 g148 Dec 2002 Divides GF(811228, 5)
12 3 · 2⁸⁰¹⁹⁷⁸+1 241420 g372 Sep 2005 Divides GF(801973, 3), GF(801977, 5)
13 67 · 2⁷⁷³⁵⁶⁶+1 232869 p227 Nov 2008 Divides GF(773564, 5)
14 3 · 2⁷⁰⁹⁹⁶⁸+1 213723 g372 May 2005 Divides GF(709962, 3), GF(709963, 5)
15 33 · 2⁶⁰⁰²⁷⁰+1 180701 L126 Jul 2005 Divides GF(600269, 5)
16 13 · 2⁵⁶²⁴⁵⁶+1 169318 g267 Jun 2003 Divides GF(562454, 5)
17 7 · 2⁵⁶¹⁸¹⁶+1 169125 g148 Jun 2003 Divides GF(561815, 5); GF(561815, 6) [p149]
18 347 · 2⁵⁴⁴⁸⁸⁷+1 164030 L679 Feb 2009 Divides GF(544886, 5)
19 39 · 2⁵¹²⁹⁹⁷+1 154430 g267 Jan 2005 Divides GF(512994, 5), GF(512995, 6)
20 1995 · 2⁴⁷⁹⁸⁴²+1 144451 p240 Aug 2009 Divides GF(479838, 5)

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