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 · 2^7033641 + 1 2117338 L2233 Feb 2011 Divides GF(7033639, 3)
2 3 · 2^5082306 + 1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
3 15 · 2^4246384 + 1 1278291 L3432 May 2013 Divides GF(4246381, 6)
4 7 · 2^3511774 + 1 1057151 p236 Nov 2008 Divides GF(3511773, 6)
5 7 · 2^2915954 + 1 877791 g279 Jun 2008 Divides GF(2915953, 12) [g322]
6 11 · 2^2897409 + 1 872209 L2973 Feb 2013 Divides GF(2897408, 3)
7 57 · 2^2747499 + 1 827082 L3514 May 2013 Divides Fermat F(2747497)
8 39 · 2^2705367 + 1 814399 L1576 Apr 2013 Divides GF(2705360, 3)
9 11 · 2^2691961 + 1 810363 p286 Mar 2013 Divides GF(2691960, 12)
10 9 · 2^2543551 + 1 765687 L1204 Jun 2011 Divides Fermat F(2543548), GF(2543549, 3), GF(2543549, 6), GF(2543549, 12)
11 3 · 2^2478785 + 1 746190 g245 Oct 2003 Divides Fermat F(2478782), GF(2478782, 3), GF(2478776, 6), GF(2478782, 12)
12 3 · 2^2291610 + 1 689844 L753 Aug 2008 Divides GF(2291607, 3), GF(2291609, 5)
13 11 · 2^2230369 + 1 671410 L2561 Sep 2011 Divides GF(2230368, 3)
14 7 · 2^2167800 + 1 652574 g279 Apr 2007 Divides Fermat F(2167797), GF(2167799, 5), GF(2167799, 10)
15 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)
16 25 · 2^2141884 + 1 644773 L1741 Sep 2011 Divides Fermat F(2141872), GF(2141871, 5), GF(2141872, 10); generalized Fermat
17 7 · 2^2139912 + 1 644179 g279 May 2007 Divides GF(2139911, 12)
18 45 · 2^2014557 + 1 606444 L1349 Feb 2012 Divides GF(2014552, 10)
19 17 · 2^1990299 + 1 599141 g267 Mar 2006 Divides GF(1990298, 3)
20 101 · 2^1988279 + 1 598534 L3141 May 2013 Divides GF(1988278, 12)

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