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

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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 · 2^16408818 + 1 4939547 L5171 Oct 2020 Divides GF(16408814, 3), GF(16408817, 5)
2 9 · 2^13334487 + 1 4014082 L4965 Mar 2020 Divides GF(13334485, 3)
3 9 · 2^12406887 + 1 3734847 L4965 Mar 2020 Divides GF(12406885, 3)
4 9 · 2^11500843 + 1 3462100 L4965 Mar 2020 Divides GF(11500840, 12)
5 9 · 2^11158963 + 1 3359184 L4965 Mar 2020 Divides GF(11158962, 5)
6 3 · 2^10829346 + 1 3259959 L3770 Jan 2014 Divides GF(10829343, 3), GF(10829345, 5)
7 11 · 2^9381365 + 1 2824074 L4965 Mar 2020 Divides GF(9381364, 6)
8 17 · 2^8636199 + 1 2599757 L5161 Feb 2021 Divides GF(8636198, 10)
9 25 · 2^8456828 + 1 2545761 L5237 Jan 2021 Divides GF(8456827, 12), generalized Fermat
10 11 · 2^8103463 + 1 2439387 L4965 Mar 2020 Divides GF(8103462, 12)
11 39 · 2^7946769 + 1 2392218 L5226 Jan 2021 Divides GF(7946767, 12)
12 29 · 2^7899985 + 1 2378134 L5161 Jan 2021 Divides GF(7899984, 6)
13 29 · 2^7374577 + 1 2219971 L5169 Oct 2020 Divides GF(7374576, 3)
14 3 · 2^7033641 + 1 2117338 L2233 Feb 2011 Divides GF(7033639, 3)
15 51 · 2^6945567 + 1 2090826 L4965 May 2020 Divides GF(6945564, 12) [p286]
16 39 · 2^6164630 + 1 1855741 L4087 Aug 2020 Divides GF(6164629, 5)
17 21 · 2^6048861 + 1 1820890 L5106 Jun 2020 Divides GF(6048860, 5)
18 31 · 2^5560820 + 1 1673976 L1204 Jan 2020 Divides GF(5560819, 6)
19 3 · 2^5082306 + 1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
20 15 · 2^4800315 + 1 1445040 L1754 Oct 2019 Divides GF(4800313, 3), GF(4800310, 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