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

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The Prime Pages

GIMPS has discovered a new largest known prime number: 2^82589933-1 (24,862,048 digits)

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

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 · 2^10829346 + 1 3259959 L3770 Jan 2014 Divides GF(10829343, 3), GF(10829345, 5)
2 3 · 2^7033641 + 1 2117338 L2233 Feb 2011 Divides GF(7033639, 3)
3 3 · 2^5082306 + 1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
4 15 · 2^4800315 + 1 1445040 L1754 Oct 2019 Divides GF(4800313, 3), GF(4800310, 5)
5 15 · 2^4246384 + 1 1278291 L3432 May 2013 Divides GF(4246381, 6)
6 35 · 2^3587843 + 1 1080050 L1979 Jul 2014 Divides GF(3587841, 5)
7 7 · 2^3511774 + 1 1057151 p236 Nov 2008 Divides GF(3511773, 6)
8 9 · 2^3497442 + 1 1052836 L1780 Oct 2012 Generalized Fermat, divides GF(3497441, 10)
9 1005 · 2^3420846 + 1 1029781 L2714 Aug 2017 Divides GF(3420844, 10)
10 193 · 2^3329782 + 1 1002367 L3460 Jul 2014 Divides Fermat F(3329780)
11 7 · 2^2915954 + 1 877791 g279 Jun 2008 Divides GF(2915953, 12) [g322]
12 11 · 2^2897409 + 1 872209 L2973 Feb 2013 Divides GF(2897408, 3)
13 57 · 2^2747499 + 1 827082 L3514 May 2013 Divides Fermat F(2747497)
14 39 · 2^2705367 + 1 814399 L1576 Apr 2013 Divides GF(2705360, 3)
15 11 · 2^2691961 + 1 810363 p286 Mar 2013 Divides GF(2691960, 12)
16 267 · 2^2662090 + 1 801372 L3234 Feb 2015 Divides Fermat F(2662088)
17 169 · 2^2545526 + 1 766282 L2125 Jan 2015 Divides GF(2545525, 10), generalized Fermat
18 9 · 2^2543551 + 1 765687 L1204 Jun 2011 Divides Fermat F(2543548), GF(2543549, 3), GF(2543549, 6), GF(2543549, 12)
19 3 · 2^2478785 + 1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
20 3 · 2^2291610 + 1 689844 L753 Aug 2008 Divides GF(2291607, 3), GF(2291609, 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
Riesel69b
H. Riesel, "Common prime factors of the numbers A_n =a^2ⁿ+1," BIT, 9 (1969) 264-269. MR 41:3381