Prime factors  k.2n + 1  of Fermat numbers  Fm
and complete factoring status

Compiled by Wilfrid Keller

 News Flash! 
On March 26, 2001, Peter Grobstich found a new factor of a Fermat number
using Leonid Durman's program: 198922467387 . 292 + 1 divides F90.
This is the seventh factor discovered with Durman's program.

  The other most recently discovered factors 
Date  Divisibility Discoverer Program by
Mar 13, 2001     119942751127 . 290 + 1 divides F88     Takahiro Nohara     Leonid Durman
Mar 12, 2001     491628159 . 2669 + 1 divides F667     Tadashi Taura     The discoverer
Mar 07, 2001     4777 . 222298 + 1 divides F22296     Gary Gostin     The discoverer

*   *   *

Prime Fermat numbers

F0 = 3,   F1 = 5,   F2 = 17,   F3 = 257,   F4 = 65537

Completely factored Fermat numbers

m     k     n     Year     Discoverer
5 5 7 1732 Euler
52347 7 1732 Euler
6 1071 8 1880 Landry
262814145745 8 1880 Landry & Le Lasseur
7 116503103764643 9 1970 Morrison & Brillhart
11141971095088142685 9 1970 Morrison & Brillhart
8 604944512477 11 1980 Brent & Pollard
[59 digits] 11 1980 Brent & Pollard
9 37 16 1903 Western
[46 digits] 11 1990 Lenstra, Manasse & a larger team
[96 digits] 11 1990 Lenstra, Manasse & a larger team
10 11131 12 1953 Selfridge
395937 14 1962 Brillhart
[37 digits] 12 1995 Brent
[248 digits] 13 1995 Brent
11 39 13 1899 Cunningham
119 13 1899 Cunningham
10253207784531279 14 1988 Brent
434673084282938711 13 1988 Brent
[560 digits] 13 1988 Brent & Morain

46 digit  k = 3640431067210880961102244011816628378312190597
37 digit  k = 1137640572563481089664199400165229051

Complete factorizations in standard notation

 F5 = 641 . 6700417
 F6 = 274177 . 67280421310721
 F7 = 59649589127497217 . 5704689200685129054721
 F8 = 1238926361552897 . P62
 F9 = 2424833 . 7455602825647884208337395736200454918783366342657 . P99
F10 = 45592577 . 6487031809 . 4659775785220018543264560743076778192897 . P252
F11 = 319489 . 974849 . 167988556341760475137 . 3560841906445833920513 . P564

Factorizations known to be incomplete

m     k     n     Year     Discoverer
12 7 14 1877 Lucas & Pervushin
397 16 1903 Western
973 16 1903 Western
11613415 14 1974 Hallyburton & Brillhart
76668221077 14 1986 Baillie
13 41365885 16 1974 Hallyburton & Brillhart
20323554055421 17 1991 Crandall
6872386635861 19 1991 Crandall
609485665932753836099 19 1995 Brent
15 579 21 1925 Kraitchik
17753925353 17 1987 Gostin
1287603889690528658928101555 17 1997 Crandall & van Halewyn
16 1575 19 1953 Selfridge
180227048850079840107 20 1996 Crandall & Dilcher
17 59251857 19 1978 Gostin
18 13 20 1903 Western
9688698137266697 23 1999 Crandall, McIntosh & Tardif
19 33629 21 1962 Riesel
308385 21 1963 Wrathall
21 534689 23 1963 Wrathall

Incomplete factorizations in standard notation

F12 = 114689 . 26017793 . 63766529 . 190274191361 . 1256132134125569 . C1187
F13 = 2710954639361 . 2663848877152141313 . 3603109844542291969 .
319546020820551643220672513 . C2391
F15 = 1214251009 . 2327042503868417 . 168768817029516972383024127016961 . C9808
F16 = 825753601 . 188981757975021318420037633 . C19694
F17 = 31065037602817 . C39444
F18 = 13631489 . 81274690703860512587777 . C78884
F19 = 70525124609 . 646730219521 . C157804
F21 = 4485296422913 . C631294

Composite cofactors of Fermat numbers  Fm 

m     Digits     Year     Prover
12 1187 1986 Baillie
13 2391 1995 Brent
15 9808 1997 Brent
16 19694 1996 Brent
17 39444 1987 Baillie
18 78884 1999 Crandall
19 157804 1993 Crandall, Doenias, Norrie & Young
21 631294 1993 Crandall, Doenias, Norrie & Young

Composite Fermat numbers  Fm  without known factor

m     Digits     Year     Prover
14 4933 1963 Selfridge & Hurwitz
20 315653 1987 Buell & Young
22 1262612 1993 Crandall, Doenias, Norrie & Young
24 5050446 1999 Mayer, Papadopoulos & Crandall

Prime factors  k.2n + 1  of larger Fermat numbers  Fm

m     k     n     Year     Discoverer
23 5 25 1878 Pervushin
25 48413 29 1963 Wrathall
1522849979 27 1985 Gostin
16168301139 27 1987 McLaughlin
26 143165 29 1963 Wrathall
27 141015 30 1963 Wrathall
430816215 29 1985 Gostin
28 25709319373 36 1997 Taura
29 1120049 31 1980 Gostin & McLaughlin
30 149041 32 1963 Wrathall
127589 33 1963 Wrathall
32 1479 34 1963 Wrathall
36 5 39 1886 Seelhoff
3759613 38 1981 Gostin & McLaughlin
37 1275438465 39 1991 Gostin
38 3 41 1903 Cullen, Cunningham & Western
2653 40 1963 Wrathall
39 21 41 1956 Robinson
42 43485 45 1963 Wrathall
43 212675402445 45 2000 Samidoost & Durman
52 4119 54 1963 Wrathall
21626655 54 1982 Keller
55 29 57 1956 Robinson
58 95 61 1957 Robinson
61 54985063 66 1986 Gostin
62 697 64 1977 Shippee
63 9 67 1956 Robinson
64 17853639 67 1986 Gostin
66 7551 69 1977 Shippee
71 683 73 1977 Shippee
72 76432329 74 1986 Gostin
73 5 75 1906 Morehead
75 3447431 77 1982 Gostin
77 425 79 1957 Robinson & Selfridge
5940341195 79 1998 Taura
81 271 84 1957 Robinson & Selfridge
88 119942751127 90 2001 Nohara & Durman
90 198922467387 92 2001 Grobstich & Durman
91 1421 93 1977 Shippee
93 92341 96 1979 Baillie
99 16233 104 1979 Gostin & McLaughlin; Suyama
107 1289179925 111 1992 Gostin
116 3433149787 120 1999 Taura
117 7 120 1956 Robinson
122 5234775 124 1986 Gostin
125 5 127 1956 Robinson
142 8152599 145 1986 Gostin
144 17 147 1956 Robinson
146 37092477 148 1987 Gostin
147 3125 149 1979 Gostin & McLaughlin
124567335 149 1990 Gostin
150 1575 157 1956 Robinson
5439 154 1980 Gostin & McLaughlin; Suyama
164 1835601567 167 1993 Gostin
178 313047661 180 1991 Gostin
184 117012935 187 1990 Gostin
201 4845 204 1980 Gostin & McLaughlin
205 232905 207 1984 Keller
207 3 209 1956 Robinson
215 32111 217 1980 Suyama
226 15 229 1956 Robinson
228 29 231 1956 Robinson
230 372236097 232 2000 Durman
232 70899775 236 1991 Gostin
250 403 252 1957 Robinson & Selfridge
251 85801657 254 1991 Gostin
255 629 257 1979 Baillie
256 36986355 258 1991 Gostin
259 36654265 262 1991 Gostin
267 177 271 1957 Robinson & Selfridge
268 21 276 1956 Robinson
275 22347 279 1984 Keller
284 7 290 1956 Robinson
1061341513 286 2000 Durman
287 5915 289 1980 Suyama
298 247 302 1979 Baillie
301 7183437 304 1990 Gostin
316 7 320 1956 Robinson
329 1211 333 1981 Suyama
334 27609 341 1984 Keller
338 27654487 342 1990 Gostin
353 18908555 355 1990 Gostin
370 573230511 373 2000 Durman
375 733251 377 1986 Gostin
376 810373 378 1986 Gostin
380 321116871 385 2000 Durman
398 120845 401 1984 Keller
416 8619 418 1981 Suyama
38039 419 1984 Keller
417 118086729 421 1992 Gostin
431 5769285 434 1990 Gostin
452 27 455 1956 Robinson
468 27114089 471 1992 Gostin
544 225 547 1979 Baillie
547 77377 550 1986 Gostin
556 127 558 1976 Matthew & Williams
579 63856313 581 1999 Taura
620 10084141 624 1992 Gostin
635 4258979 645 1991 Gostin
637 11969 643 1984 Keller
642 52943971 644 1999 Taura
667 491628159 669 2001 Taura
692 717 695 1979 Atkin & Rickert
723 554815 730 1991 Gostin
744 17 747 1976 Matthew & Williams
851 497531 859 1988 Gostin
885 16578999 887 1992 Gostin
906 57063 908 1986 Gostin
931 1985 933 1980 Keller
1069 137883 1073 1992 Gostin
1082 82165 1084 1991 Gostin
1114 11618577 1116 2001 Gostin
1123 25835 1125 1987 Gostin
1225 79707 1231 1991 Gostin
1229 29139 1233 1987 Gostin
1451 13143 1454 1986 Gostin
1551 291 1553 1979 Atkin & Rickert
1598 10923781 1600 2000 Taura
1849 98855 1851 1992 Gostin
1945 5 1947 1957 Robinson
1990 150863 1993 1995 Taura
2023 29 2027 1979 Atkin & Rickert; Cormack & Williams
2059 591909 2063 2000 Ballinger & Gallot
2089 431 2099 1983 Suyama
2456 85 2458 1979 Atkin & Rickert
3310 5 3313 1979 Atkin & Rickert; Cormack & Williams
3506 501 3508 1986 Gostin
4250 173373 4252 1999 Kerchner
4258 1435 4262 1993 Gostin
4724 29 4727 1979 Cormack & Williams
5320 21341 5323 1998 Taura
5957 421435 5960 2001 Gostin
6208 763 6210 1993 Gostin
6355 115185 6358 2000 Kerchner
6390 303 6393 1993 Gostin
6537 17 6539 1979 Cormack & Williams
6835 19 6838 1978 Keller
6909 6021 6912 1993 Gostin
7181 168329 7187 2000 Gostin
7309 145 7312 1992 Dubner
8239 7473 8242 1998 Prethaler & Gallot
8555 645 8557 1993 Dubner
9322 8247 9324 1999 Kerchner
9428 9 9431 1983 Keller
9448 19 9450 1980 Keller
9549 1211 9551 1998 Taura
12185 81 12189 1993 Dubner
13250 351 13252 1996 Taura
13623 48265 13626 2000 Gostin
14252 1173 14254 1997 Taura
14276 157 14280 1996 Taura
14528 17217 14530 2000 Gostin
15161 55 15164 1993 Dubner
17906 135 17909 1996 Taura
18749 11 18759 1992 Dubner
18757 33 18766 1993 Dubner
22296 4777 22298 2001 Gostin
23069 681 23071 1997 Demichel & Gallot; Taura
23288 19 23290 1992 Dubner
23471 5 23473 1984 Keller
24651 99 24653 1996 Taura
25006 57 25010 1993 Young
28281 81 28285 1996 Taura
35563 357 35567 2000 Melo & Gallot
49093 165 49095 1998 Gallot
63679 169 63686 1998 Dubner & Gallot
83861 99 83863 1998 Gusev & Gallot
90057 189 90061 1999 Morenus & Gallot
94798 21 94801 1995 Young
95328 7 95330 1994 Young
113547 39 113549 1999 Renze & Gallot
114293 13 114296 1995 Young
125410 5 125413 1995 Young
146221 57 146223 2000 Lewis & Gallot
157167 3 157169 1995 Young
213319 3 213321 1996 Young
303088 3 303093 1998 Young
382447 3 382449 1999 Cosgrave & Gallot

Summary of factoring status for Fermat numbers  Fm
as of March 28, 2001

Prime   m = 0, 1, 2, 3, 4
Completely factored m = 5, 6, 7, 8 (two factors each), 9 (3 factors),
       10 (4 factors), 11 (5 factors)
Five prime factors known m = 12*
Four prime factors known m = 13*
Three prime factors known m = 15*, 25
Two prime factors known m = 16*, 18*, 19*, 27, 30, 36, 38, 52, 77, 147, 150,
       284, 416
Only one prime factor known m = 17*, 21*, 23, 26, 28, 29, 32, 37, 39, 42, 43, 55
       and 145 values of  m  with  55 < m < 382447
Composite but no factor known   m = 14, 20, 22, 24
Character unknown   m = 31, 33, 34, 35, 40, 41, 44, 45, 46, 47, 48, . . .
* Cofactor composite

218 prime factors known
185 Fermat numbers known to be composite

Count of factors according to difference  n - m

n - m   = 2, 3, 4, 5, 6, 7, 8, 9,   10
Frequency 107,   57,   30,   5,   7,   5,   3,   1,   3

Search limits for  k.2n + 1  as possible factors of some  Fm,  m < n - 2
All odd  k < Ln  tested

For more detailed and up-to-date information,
see http://www.fermatsearch.org/status.htm
and http://www.prothsearch.net/status.html

n Ln
14 274877906944  = 238
15 -16 137438953472  = 237
17 -18 68719476736  = 236
19 -21 549755813888  = 239
22 -36 2000000000000  =2.1012
37 -44 1100000000000  =11.1011
45 -51 146500000000  =1465.108
52 -99 28000000000 =28.109
100 -220       5600000000 =56.108
221 -500 1000000000 =109
501 -960 200000000 =2.108
961 -999 120000000 =12.107
1000 -1999 5650000 =565.104
2000 -4450 1000000 =106
4451 -8158 262144 =219
8159 -9999 23000 
10000 -15999 16000 
16000 -32734 4096 
32735 -100000 300 
100001 -131000 74 
131001 -180000 22 
180001 -310000 8 

URL: http://www.prothsearch.net/fermat.html
Last modified: March 28, 2001.