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Generalized Fermat Divisors (bases 3,5,6,10,12) |
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.2m+1with 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.
rank prime digits who when comment 1 3 · 216408818 + 1 4939547 L5171 Oct 2020 Divides GF(16408814, 3), GF(16408817, 5) 2 9 · 213334487 + 1 4014082 L4965 Mar 2020 Divides GF(13334485, 3) 3 9 · 212406887 + 1 3734847 L4965 Mar 2020 Divides GF(12406885, 3) 4 9 · 211500843 + 1 3462100 L4965 Mar 2020 Divides GF(11500840, 12) 5 9 · 211158963 + 1 3359184 L4965 Mar 2020 Divides GF(11158962, 5) 6 3 · 210829346 + 1 3259959 L3770 Jan 2014 Divides GF(10829343, 3), GF(10829345, 5) 7 11 · 29381365 + 1 2824074 L4965 Mar 2020 Divides GF(9381364, 6) 8 17 · 28636199 + 1 2599757 L5161 Feb 2021 Divides GF(8636198, 10) 9 25 · 28456828 + 1 2545761 L5237 Jan 2021 Divides GF(8456827, 12), generalized Fermat 10 11 · 28103463 + 1 2439387 L4965 Mar 2020 Divides GF(8103462, 12) 11 39 · 27946769 + 1 2392218 L5226 Jan 2021 Divides GF(7946767, 12) 12 29 · 27899985 + 1 2378134 L5161 Jan 2021 Divides GF(7899984, 6) 13 29 · 27374577 + 1 2219971 L5169 Oct 2020 Divides GF(7374576, 3) 14 3 · 27033641 + 1 2117338 L2233 Feb 2011 Divides GF(7033639, 3) 15 51 · 26945567 + 1 2090826 L4965 May 2020 Divides GF(6945564, 12) [p286] 16 39 · 26164630 + 1 1855741 L4087 Aug 2020 Divides GF(6164629, 5) 17 21 · 26048861 + 1 1820890 L5106 Jun 2020 Divides GF(6048860, 5) 18 31 · 25560820 + 1 1673976 L1204 Jan 2020 Divides GF(5560819, 6) 19 3 · 25082306 + 1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5) 20 15 · 24800315 + 1 1445040 L1754 Oct 2019 Divides GF(4800313, 3), GF(4800310, 5)
- 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 Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415. MR 39:6813
- Riesel69b
- H. Riesel, "Common prime factors of the numbers An =a2n+1," BIT, 9 (1969) 264-269. MR 41:3381