## 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

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