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

Compiled by Wilfrid Keller

News Flash!
On January 14, 2025, Valter Cavecchia and PrimeGrid
discovered that
99 · 25798449 + 1 divides F5798447.
This is the third largest Fermat number known to be composite !

On January 10, 2025, Ryan Propper, Serge Batalov
and FermatSearch showed that
39701 · 2414679 + 1 divides F414677.

The other previously discovered factors

Date Divisibility Discoverer
Dec 07, 2024 1995420873 · 27222 + 1 divides F7220 Lélio de Paula
Jul 04, 2024 56073 · 2223382 + 1 divides F223380 Propper & Batalov
May 24, 2024 322072887044529 · 2253 + 1 divides F251 Gary Gostin
Jul 28, 2023 13243 · 2699764 + 1 divides F699760 Propper & Batalov
Jul 27, 2023 43390164659167 · 2164 + 1 divides F160 Erwin Doescher

* * *

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 L. Euler
52347 7 1732 L. Euler
6 1071 8 1855 T. Clausen; F. Landry 1880
262814145745 8 1855 T. Clausen; F. Landry & H. Le Lasseur 1880
7 116503103764643 9 13 Sep 1970 M. A. Morrison & J. Brillhart
11141971095088142685 9 13 Sep 1970 M. A. Morrison & J. Brillhart
8 604944512477 11 1980 R. P. Brent & J. M. Pollard
[59 digits] 11 1980 R. P. Brent & J. M. Pollard
9 37 16 1903 A. E. Western
[46 digits] 11 15 Jun 1990 A. K. Lenstra, M. S. Manasse & a larger team
[96 digits] 11 15 Jun 1990 A. K. Lenstra, M. S. Manasse & a larger team
10 11131 12 15 Aug 1953 J. L. Selfridge
395937 14 1962 J. Brillhart
[37 digits] 12 20 Oct 1995 R. P. Brent
[248 digits] 13 1995 R. P. Brent
11 39 13 1899 A. Cunningham
119 13 1899 A. Cunningham
10253207784531279 14 17 May 1988 R. P. Brent
434673084282938711 13 13 May 1988 R. P. Brent
[560 digits] 13 20 Jun 1988 R. P. Brent & F. 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 I. M. Pervushin; E. Lucas
397 16 1903 A. E. Western
973 16 1903 A. E. Western
11613415 14 1974 J. C. Hallyburton & J. Brillhart
76668221077 14 25 Jul 1986 R. Baillie
[50 digits] 15 27 Mar 2010 M. Vang, Zimmermann & Kruppa
13 41365885 16 1974 J. C. Hallyburton & J. Brillhart
20323554055421 17 05 Jan 1991 R. E. Crandall
6872386635861 19 09 May 1991 R. E. Crandall
609485665932753836099 19 16 Jun 1995 R. P. Brent
14 [49 digits] 16 03 Feb 2010 T. Rajala & Woltman
15 579 21 1925 M. B. Kraitchik
17753925353 17 04 Aug 1987 G. B. Gostin
1287603889690528658928101555 17 03 Jul 1997 R. E. Crandall & C. van Halewyn
16 1575 19 14 Aug 1953 J. L. Selfridge
180227048850079840107 20 Dec 1996 R. E. Crandall & K. Dilcher
17 59251857 19 30 May 1978 G. B. Gostin
[44 digits] 19 15 Mar 2011 D. Bessell & Woltman
18 13 20 1903 A. E. Western (Prime: P. Seelhoff 1886)
9688698137266697 23 16 Apr 1999 R. E. Crandall, R. J. McIntosh & C. Tardif
19 33629 21 16 Nov 1962 H. Riesel
308385 21 1963 C. P. Wrathall
8962167624028624126082526703 22 18 Jul 2009 D. Bessell & Woltman
21 534689 23 1963 C. P. Wrathall
22 3853959202444067657533632211 24 26 Mar 2010 D. Bessell & Woltman
23 5 25 1878 I. M. Pervushin
25 48413 29 1963 C. P. Wrathall
1522849979 27 07 Aug 1985 G. B. Gostin
16168301139 27 16 Dec 1987 P. B. McLaughlin
26 143165 29 1963 C. P. Wrathall
27 141015 30 1963 C. P. Wrathall
430816215 29 21 Feb 1985 G. B. Gostin
28 25709319373 36 05 Feb 1997 T. Taura
29 1120049 31 17 Oct 1980 G. B. Gostin & P. B. McLaughlin
30 149041 32 1963 C. P. Wrathall
127589 33 1963 C. P. Wrathall

50 digit k = 17353230210429594579133099699123162989482444520899
49 digit k = 1784180997819127957596374417642156545110881094717
44 digit k = 14783975791554494074552473179612897725474511

Incomplete factorizations in standard notation

F12 = 114689 · 26017793 · 63766529 · 190274191361 · 1256132134125569 ·
568630647535356955169033410940867804839360742060818433 · C1133
F13 = 2710954639361 · 2663848877152141313 · 3603109844542291969 ·
319546020820551643220672513 · C2391
F14 = 116928085873074369829035993834596371340386703423373313 · C4880
F15 = 1214251009 · 2327042503868417 · 168768817029516972383024127016961 · C9808
F16 = 825753601 · 188981757975021318420037633 · C19694
F17 = 31065037602817 ·
7751061099802522589358967058392886922693580423169 · C39395
F18 = 13631489 · 81274690703860512587777 · C78884
F19 = 70525124609 · 646730219521 · 37590055514133754286524446080499713 · C157770
F20 = C315653
F21 = 4485296422913 · C631294
F22 = 64658705994591851009055774868504577 · C1262577
F23 = 167772161 · C2525215
F24 = C5050446
F25 = 25991531462657 · 204393464266227713 · 2170072644496392193 · C10100842
F26 = 76861124116481 · C20201768
F27 = 151413703311361 · 231292694251438081 · C40403531
F28 = 1766730974551267606529 · C80807103
F29 = 2405286912458753 · C161614233
F30 = 640126220763137 · 1095981164658689 · C323228467

Composite Fermat numbers Fm without known factor

m Digits Year Earliest prover
20 315653 1987 D. A. Buell & J. Young
24 5050446 1999 E. W. Mayer, J. S. Papadopoulos & R. E. Crandall

Largest cofactors known to be composite

m Digits Year Earliest prover
21 631294 1993 R. E. Crandall, J. Doenias, C. Norrie & J. Young
22 1262577 2010 D. Domanov; S. Yamada
23 2525215 2000 E. W. Mayer, J. S. Papadopoulos & R. E. Crandall
25 10100842 2009 S. Yamada; A. Höglund
26 20201768 2009 A. Höglund
27 40403531 2010 A. Höglund
28 80807103 2022 E. W. Mayer
29 161614233 2022 E. W. Mayer
30 323228467 2022 E. W. Mayer

For an evaluation of the “Proof Strengths”, please refer to this report.

A more comprehensive presentation of the smaller Fermat numbers
Fm for m ≤ 32 is found here.

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

m k n Year Discoverer
31 5463561471303 33 12 Apr 2001 A. Kruppa & T. Forbes
32 1479 34 1963 C. P. Wrathall
36 5 39 1886 P. Seelhoff
3759613 38 02 Jan 1981 G. B. Gostin & P. B. McLaughlin
37 1275438465 39 29 Apr 1991 G. B. Gostin
38 3 41 1903 J. Cullen & A. Cunningham; A. E. Western
2653 40 1963 C. P. Wrathall
39 21 41 1956 R. M. Robinson
2864929972774011 41 07 Dec 2012 T. Rajala & Woltman
40 74327396788321657 42 01 Mar 2021 L. Morelli & Woltman
42 43485 45 1963 C. P. Wrathall
111318179143061 45 20 Mar 2011 R. Maznichenko & Rodenkirch
43 212675402445 45 23 Dec 2000 P. Samidoost & Durman
48 2139543641769 50 28 May 2001 M. Bodschwinna & Durman
52 4119 54 1963 C. P. Wrathall
21626655 54 1982 W. Keller
81909357657279 54 15 Mar 2010 C. Vonck & Durman
55 29 57 1956 R. M. Robinson
58 95 61 1957 R. M. Robinson
61 54985063 66 17 May 1986 G. B. Gostin
62 697 64 1977 D. E. Shippee
63 9 67 1956 R. M. Robinson
64 17853639 67 13 May 1986 G. B. Gostin
65 1210895760431083 68 29 Jan 2013 R. Maznichenko & Woltman
66 7551 69 1977 D. E. Shippee
71 683 73 1977 D. E. Shippee
72 76432329 74 31 May 1986 G. B. Gostin
73 5 75 1906 J. C. Morehead
75 3447431 77 Apr 1982 G. B. Gostin
77 425 79 1957 R. M. Robinson & J. L. Selfridge
5940341195 79 08 Apr 1998 T. Taura
81 271 84 1957 R. M. Robinson & J. L. Selfridge
83 1595863660157 87 12 May 2005 V. A. Danilov & Durman
86 20018578522347 88 07 Jun 2012 M. Dangler & Rodenkirch
88 119942751127 90 13 Mar 2001 T. Nohara & Durman
90 198922467387 92 26 Mar 2001 P. Grobstich & Durman
91 1421 93 1977 D. E. Shippee
93 92341 96 1979 R. Baillie
94 482524552001 97 18 Apr 2001 P. Grobstich & Durman
96 3334131633063 101 25 Nov 2008 M. Ptáček & Durman
99 16233 104 Nov 1979 G. B. Gostin & P. B. McLaughlin;
H. Suyama
103 7595155767819149 105 31 Dec 2020 P. Strasser & Woltman
107 1289179925 111 08 Nov 1992 G. B. Gostin
116 3433149787 120 08 Aug 1999 T. Taura
117 7 120 1956 R. M. Robinson
118 1527888802614951 120 28 Jan 2019 P. Strasser & Woltman
122 5234775 124 21 May 1986 G. B. Gostin
125 5 127 1956 R. M. Robinson
132 1075441212722595 135 19 Dec 2018 P. Strasser & Woltman
133 88075576149 135 24 Jul 2001 P. Samidoost & Durman
142 8152599 145 22 May 1986 G. B. Gostin
144 17 147 1956 R. M. Robinson
146 37092477 148 23 Dec 1987 G. B. Gostin
147 3125 149 Dec 1979 G. B. Gostin & P. B. McLaughlin
124567335 149 01 Dec 1990 G. B. Gostin
150 1575 157 1956 R. M. Robinson
5439 154 May 1980 G. B. Gostin & P. B. McLaughlin;
H. Suyama
160 43390164659167 164 27 Jul 2023 E. Doescher
164 1835601567 167 31 May 1993 G. B. Gostin
166 2674670937447 171 10 Jul 2012 R. Maznichenko & Rodenkirch
172 20569603303 174 11 May 2001 L. N. Durman
178 313047661 180 18 Jun 1991 G. B. Gostin
184 117012935 187 17 Dec 1990 G. B. Gostin
195 48595346636925 197 26 Jun 2014 S. Batalov & G. F. Woltman
201 4845 204 27 Jul 1980 G. B. Gostin & P. B. McLaughlin
205 232905 207 01 Jan 1984 W. Keller
207 3 209 1956 R. M. Robinson
215 32111 217 02 Dec 1980 H. Suyama
226 15 229 1956 R. M. Robinson
228 29 231 1956 R. M. Robinson
230 372236097 232 13 Apr 2000 L. N. Durman
232 70899775 236 24 Nov 1991 G. B. Gostin
250 403 252 1957 R. M. Robinson & J. L. Selfridge
251 85801657 254 30 Nov 1991 G. B. Gostin
322072887044529 253 24 May 2024 G. B. Gostin
255 629 257 1979 R. Baillie
256 36986355 258 20 Jan 1991 G. B. Gostin
259 36654265 262 21 Jan 1991 G. B. Gostin
267 177 271 1957 R. M. Robinson & J. L. Selfridge
268 21 276 1956 R. M. Robinson
274 58130105122923 277 05 Apr 2018 R. Maznichenko
275 22347 279 16 Jan 1984 W. Keller
284 7 290 1956 R. M. Robinson
1061341513 286 25 Dec 2000 L. N. Durman
286 78472588395 288 02 Nov 2002 V. A. Danilov & Durman
287 5915 289 26 Dec 1980 H. Suyama
8662575142599 289 28 Nov 2016 R. Maznichenko
21346190509623 290 25 Dec 2015 R. Maznichenko
297 72677552745 301 13 Feb 2003 V. A. Danilov & Durman
298 247 302 1979 R. Baillie
299 272392805475 304 20 Nov 2010 A. V. Komkov
301 7183437 304 04 Nov 1990 G. B. Gostin
316 7 320 1956 R. M. Robinson
329 1211 333 12 Jan 1981 H. Suyama
334 27609 341 25 Jan 1984 W. Keller
338 27654487 342 22 Dec 1990 G. B. Gostin
343 4844391185 345 29 Aug 2001 V. A. Danilov & Durman
353 18908555 355 25 Dec 1990 G. B. Gostin
370 573230511 373 22 Aug 2000 L. N. Durman
375 733251 377 09 Jun 1986 G. B. Gostin
376 810373 378 09 Jun 1986 G. B. Gostin
380 321116871 385 26 Jun 2000 L. N. Durman
398 120845 401 18 Apr 1984 W. Keller
416 8619 418 1981 H. Suyama
38039 419 16 Feb 1984 W. Keller
417 118086729 421 13 Jun 1992 G. B. Gostin
303472680883 420 30 Jan 2013 R. Maznichenko & Rodenkirch
425 44893411020831 427 25 Jun 2023 G. B. Gostin
431 5769285 434 30 Nov 1990 G. B. Gostin
452 27 455 1956 R. M. Robinson
459 5449229488169 465 08 May 2013 R. Propper & Rodenkirch
468 27114089 471 25 Sep 1992 G. B. Gostin
480 5673968845 484 24 Nov 2001 V. A. Danilov & Durman
517 84977118993 520 25 Jan 2010 V. S. Maiorov & Durman
544 225 547 1979 R. Baillie
547 77377 550 05 Jun 1986 G. B. Gostin
556 127 558 1976 G. Matthew & H. C. Williams
569 6616590375 575 25 Feb 2003 S. Kuzmin & Durman
579 63856313 581 04 Oct 1999 T. Taura
600 6213186413 605 29 Sep 2005 A. Vuori & Durman
620 10084141 624 19 Jan 1992 G. B. Gostin
635 4258979 645 12 Jul 1991 G. B. Gostin
637 11969 643 08 Mar 1984 W. Keller
642 52943971 644 08 Aug 1999 T. Taura
666 217924552867 668 26 May 2014 R. Maznichenko
667 491628159 669 12 Mar 2001 T. Taura
692 717 695 1979 A. O. L. Atkin & N. W. Rickert
723 554815 730 04 Jan 1991 G. B. Gostin
744 17 747 1976 G. Matthew & H. C. Williams
851 497531 859 18 Jan 1988 G. B. Gostin
885 16578999 887 12 Jul 1992 G. B. Gostin
906 57063 908 19 May 1986 G. B. Gostin
931 1985 933 22 Dec 1980 W. Keller
943 4785972759 954 28 Mar 2012 A. Sen & Rodenkirch
971 541664191 976 04 Apr 2002 D. Komin & Durman
1069 137883 1073 01 Sep 1992 G. B. Gostin
1082 82165 1084 08 Apr 1991 G. B. Gostin
1114 11618577 1116 22 Jan 2001 G. B. Gostin
1123 25835 1125 15 Jan 1987 G. B. Gostin
1132 10111717305 1136 26 Dec 2012 S. Batalov & Rodenkirch
1160 2018719057 1162 15 May 2005 M. Pacher & Durman
1201 837747239 1203 12 Jan 2011 M. Pacher, Jobling, Woltman & Fougeron
1225 79707 1231 08 Jun 1991 G. B. Gostin
1229 29139 1233 07 Dec 1987 G. B. Gostin
1296 58951871175 1299 30 Oct 2016 G. B. Gostin
1379 205071338665 1382 24 Nov 2021 G. B. Gostin
1394 62705223 1396 06 Dec 2001 T. Taura
1436 713828860473 1438 21 Apr 2023 G. B. Gostin
1451 13143 1454 30 Oct 1986 G. B. Gostin
1493 623619365837 1495 09 Feb 2023 G. B. Gostin
1551 291 1553 1979 A. O. L. Atkin & N. W. Rickert
1598 10923781 1600 21 Dec 2000 T. Taura
1680 26778342591 1683 05 Dec 2016 G. B. Gostin
1710 351276975 1719 09 Mar 2008 M. Pacher, Jobling, Woltman & Fougeron
1722 364182745 1724 19 Apr 2008 M. Pacher, Jobling, Woltman & Fougeron
1784 321941255325 1788 31 May 2023 G. B. Gostin
1849 98855 1851 01 Sep 1992 G. B. Gostin
1945 5 1947 1957 R. M. Robinson
1990 150863 1993 30 Oct 1995 T. Taura
2023 29 2027 1979 A. O. L. Atkin & N. W. Rickert;
G. V. Cormack & H. C. Williams
2059 591909 2063 09 May 2000 R. Ballinger & Gallot
2089 431 2099 17 Aug 1983 H. Suyama
2123 9659599821 2125 16 Dec 2016 G. B. Gostin
2144 19789270693 2148 02 Nov 2018 G. B. Gostin
2286 4004779405 2288 25 Nov 2015 G. B. Gostin
2420 103257279 2422 07 Nov 2010 P. Strasser
2456 85 2458 1979 A. O. L. Atkin & N. W. Rickert
2587 15249465809 2591 09 Dec 2019 G. B. Gostin
2606 238451805 2608 11 Mar 2012 T. Nohara, Jobling, Woltman & Fougeron
3056 3370842847 3058 28 Jan 2017 G. B. Gostin
3310 5 3313 1979 A. O. L. Atkin & N. W. Rickert;
G. V. Cormack & H. C. Williams
3314 406860969 3322 18 Jan 2013 M. Pacher, Jobling, Woltman & Fougeron
3335 43714055 3337 05 Jul 2011 M. Kamenyuk, Fougeron & Penné
3345 6604326057 3347 13 Dec 2018 G. B. Gostin
3506 501 3508 26 Jul 1986 G. B. Gostin
3576 1132007355 3578 21 Jul 2015 T. Nohara & Fougeron
3703 262254673 3706 17 Jun 2012 T. Nohara, Jobling, Woltman & Fougeron
3723 13308899 3725 03 Jul 2002 J. Fougeron
3730 1127899657 3732 17 Aug 2015 T. Nohara & Fougeron
3850 787703529 3854 13 Aug 2015 M. Pacher, Jobling, Woltman & Fougeron
4184 465917283 4189 27 Dec 2014 S. Batalov & Fougeron
4250 173373 4252 22 Dec 1999 C. F. Kerchner
4258 1435 4262 05 Aug 1993 G. B. Gostin
4260 209161375 4262 14 Mar 2014 T. Nohara, Jobling & Fougeron
4265 72179955 4269 03 Aug 2012 T. Nohara, Jobling, Woltman & Fougeron
4332 2466157 4334 15 Oct 2002 J. Fougeron
4652 143918649 4654 09 Oct 2012 T. Rajala, Jobling & Fougeron
4724 29 4727 1979 G. V. Cormack & H. C. Williams
5199 1488947679 5201 24 Jul 2018 M. Rodenkirch & Fougeron
5320 21341 5323 22 Apr 1998 T. Taura
5531 1503975 5533 11 Apr 2002 G. B. Gostin
5792 8872947 5794 03 Mar 2003 J. Fougeron
5957 421435 5960 16 Jan 2001 G. B. Gostin
6208 763 6210 05 Aug 1993 G. B. Gostin
6355 115185 6358 14 Nov 2000 C. F. Kerchner
6390 303 6393 06 Aug 1993 G. B. Gostin
6537 17 6539 1979 G. V. Cormack & H. C. Williams
6835 19 6838 30 Sep 1978 W. Keller
6909 6021 6912 12 Nov 1993 G. B. Gostin
7137 118807331 7139 21 Feb 2016 G. B. Gostin
7181 168329 7187 26 Dec 2000 G. B. Gostin
7220 1995420873 7222 07 Dec 2024 L. R. de Paula, Rodenkirch & Fougeron
7309 145 7312 25 Oct 1992 H. Dubner
8239 7473 8242 25 Jun 1998 R. Prethaler & Gallot
8269 592131 8271 21 May 2003 G. B. Gostin
8298 1054057 8300 01 Nov 2003 C. Kitchen & Gallot
8555 645 8557 01 Nov 1993 H. Dubner
9322 8247 9324 16 Nov 1999 C. F. Kerchner
9428 9 9431 01 Apr 1983 W. Keller
9447 5505161 9449 09 Nov 2012 S. Batalov, Jobling & Fougeron
9448 19 9450 30 Aug 1980 W. Keller
9549 1211 9551 15 Feb 1998 T. Taura
9552 60567741 9555 04 Apr 2016 G. B. Gostin
9691 260435 9693 12 Dec 2001 G. B. Gostin
9747 44670651 9749 20 Dec 2014 T. Nohara & Fougeron
9863 332436749 9865 23 Mar 2019 G. B. Gostin
11075 171369935 11077 29 Jun 2020 G. B. Gostin
11695 203355 11703 31 Dec 2001 G. B. Gostin
12185 81 12189 04 Sep 1993 H. Dubner
12825 1814649 12827 09 Oct 2009 T. Nohara, Fougeron & Penné
13250 351 13252 01 Oct 1996 T. Taura
13623 48265 13626 14 Dec 2000 G. B. Gostin
14252 1173 14254 07 Mar 1997 T. Taura
14276 157 14280 31 Aug 1996 T. Taura
14528 17217 14530 24 Dec 2000 G. B. Gostin
14587 24142479 14590 25 May 2016 A. Nordin & Fougeron
15161 55 15164 28 Aug 1993 H. Dubner
17748 3860269 17750 21 Nov 2012 S. Batalov, Jobling & Fougeron
17906 135 17909 31 Aug 1996 T. Taura (Prime: J. Young 1993)
18345 15071297 18347 07 Sep 2017 G. B. Gostin
18749 11 18759 12 Sep 1992 H. Dubner
18757 33 18766 20 Aug 1993 H. Dubner
18933 15175951 18936 13 Sep 2017 G. B. Gostin
19211 13323 19220 18 May 2001 G. B. Gostin
22296 4777 22298 07 Mar 2001 G. B. Gostin
23069 681 23071 Jul 1997 T. Taura; Y. Gallot
(Prime: P. Demichel 1996)
23288 19 23290 14 Dec 1992 H. Dubner
23471 5 23473 05 Sep 1984 W. Keller
24651 99 24653 11 Sep 1996 T. Taura
25006 57 25010 18 Dec 1993 J. Young
25599 14698299 25602 03 Mar 2021 G. B. Gostin
28281 81 28285 18 Aug 1996 T. Taura
30256 121531 30260 16 May 2006 A. Vuori, Fougeron, Woltman & Gallot
31880 10981209 31883 10 Jul 2023 S. Batalov
35563 357 35567 27 Aug 2000 N. S. Melo & Gallot
37236 1844383 37238 19 Dec 2017 G. B. Gostin
38795 1514535 38797 27 Dec 2017 G. B. Gostin
38967 177795 38969 10 Jan 2009 T. Nohara, Fougeron & Penné
41894 4935 41897 15 Aug 2001 J. Fougeron
43665 2495 43667 21 Aug 2001 J. Fougeron
45278 547653 45280 18 Jan 2018 G. B. Gostin
48624 28949 48627 03 May 2008 T. Nohara, Fougeron & Penné
49093 165 49095 19 Aug 1998 Y. Gallot
49488 71007 49490 18 Apr 2009 T. Nohara, Fougeron & Penné
50078 7619 50081 06 Mar 2002 G. Axelsson, Jobling, Woltman & Gallot
60079 5731 60084 14 Feb 2005 M. R. Eaton, Jobling, Woltman & Gallot
63480 242445 63484 10 Mar 2018 G. B. Gostin
63679 169 63686 19 May 1998 H. Dubner & Gallot
66643 749893 66648 11 Aug 2021 E. Doescher, Penné & Fougeron
79221 6089 79223 01 Aug 2008 P. Samidoost, Fougeron & Penné
83861 99 83863 12 Oct 1998 G. N. Gusev & Gallot
90057 189 90061 20 Sep 1999 D. Morenus & Gallot
91213 585 91215 30 Jul 2001 G. Axelsson, Jobling, Woltman & Gallot
94798 21 94801 Mar 1995 J. Young
95328 7 95330 23 Aug 1994 J. Young
100518 79425 100520 22 Oct 2014 P. Samidoost, Fougeron & Penné
104448 927 104451 30 Nov 2001 A. P. Oleynick & Gallot
106432 30967 106436 13 Nov 2012 S. Batalov, Jobling & Fougeron
113547 39 113549 11 May 1999 J. Renze & Gallot
114293 13 114296 Aug 1995 J. Young
125410 5 125413 Aug 1995 J. Young
134995 47893 134998 22 May 2020 G. B. Gostin
138557 7333 138560 12 Mar 2011 D. D'huyvetters, Reynolds, Penné & Fougeron
142460 159 142462 14 Nov 2001 N. S. Melo & Gallot
146221 57 146223 11 Feb 2000 R. Lewis & Gallot
157167 3 157169 14 Jun 1995 J. Young
213319 3 213321 21 Aug 1996 J. Young
221670 3771 221676 01 Jul 2011 M. Doom, Reynolds, Penné & Fougeron
223380 56073 223382 04 Jul 2024 R. Propper & S. Batalov, Fougeron
226614 4479 226618 08 Jul 2011 P. Doggart, Reynolds, Penné & Fougeron
270091 63 270094 05 Feb 2002 T. Taura & Gallot
282717 51 282719 13 Nov 2002 K. Odermatt & Gallot
287384 211 287388 13 Dec 2004 J. Fougeron
303088 3 303093 13 Jan 1998 J. Young
338295 485 338297 21 Aug 2007 W. Keller, S. Murata, Penné & Fougeron
352279 7905 352281 31 May 2012 J. Boerner, Reynolds, Penné & Fougeron
382447 3 382449 25 Jul 1999 J. B. Cosgrave & Gallot
410105 1207 410108 22 Nov 2005 J. Tajima, Jobling, Woltman & Gallot
414677 39701 414679 10 Jan 2025 R. Propper & S. Batalov, Reynolds, Penné
461076 9 461081 04 Aug 2003 T. Nohara, Jobling, Woltman & Gallot
472097 89 472099 05 Oct 2004 P. Samidoost, Fougeron & Woltman
476624 651 476632 28 Dec 2008 E. Ueda, Reynolds, Penné & Fougeron
495728 243 495732 29 May 2007 R. Keiser, Jobling, Penné & Fougeron
567233 519 567235 06 Mar 2009 S. Yamashita, Reynolds, Penné & Fougeron
585042 151 585044 17 Mar 2007 P. Saridis, Jobling, Penné & Gallot
617813 659 617815 31 Mar 2009 E. Embling, Reynolds, Penné & Fougeron
672005 27 672007 30 Aug 2005 C. Cooper, Jobling, Woltman & Gallot
699760 13243 699764 28 Jul 2023 R. Propper & S. Batalov, Reynolds, Penné
906108 1705 906110 13 Jun 2012 R. Boniecki, Reynolds, Penné & Fougeron
960897 11 960901 23 Feb 2005 M. R. Eaton, Jobling, Woltman & Gallot
1099061 2145 1099064 18 Jun 2013 S. Y. Tang, Reynolds, Penné & Fougeron
1246013 329 1246017 06 Jan 2012 B. Dodson, Reynolds, Penné & Fougeron
1494096 131 1494099 07 Feb 2012 R. Derrera, Reynolds, Penné & Fougeron
1747656 183 1747660 10 Mar 2013 B. van Rooijen, Reynolds, Penné & Fougeron
2141872 25 2141884 09 Sep 2011 G. Granowski, Reynolds, Penné & Fougeron
2145351 3 2145353 21 Feb 2003 J. B. Cosgrave, Jobling, Woltman & Gallot
2167797 7 2167800 17 Apr 2007 C. Cooper, Jobling, Woltman & Gallot
2478782 3 2478785 10 Oct 2003 J. B. Cosgrave, Jobling, Woltman & Gallot
2543548 9 2543551 22 Jun 2011 S. Brown, Reynolds, Penné & Fougeron
2662088 267 2662090 13 Feb 2015 J. Parangalan, Reynolds, Penné & Fougeron
2747497 57 2747499 14 May 2013 M. Bishop, Reynolds, Penné & Fougeron
3329780 193 3329782 25 Jul 2014 R. Ottusch, Reynolds, Penné & Fougeron
4192909 141 4192911 17 Jul 2023 S. Brown, Reynolds, Penné & Fougeron
5523858 13 5523860 22 Jan 2020 S. Brown, Reynolds, Penné & Fougeron
5798447 99 5798449 14 Jan 2025 V. Cavecchia, Reynolds, Penné & Fougeron
7963245 27 7963247 16 Jan 2021 T. Greer, Reynolds, Penné & Fougeron
18233954 7 18233956 05 Oct 2020 R. Propper, Reynolds, Penné & Fougeron

Full names of researchers

Pierre de Fermat (1601-1665), Leonhard Euler (Леонард Эйлер, 1707-1783), Thomas Clausen (1801-1885), James Cullen (1867-1933), Allan Joseph Champneys Cunningham (1842-1928), Maurice Borisovich Kraitchik (Морис Борисович Крайчик, 1882-1957), Fortuné Landry (1799-1895), Henri Le Lasseur de Ranzey (1821-1894), François Édouard Anatole Lucas (1842-1891), James Caddall Morehead (1877-1950), Ivan Mikheevich Pervushin (Иван Михеевич Первушин, 1827-1900), Paul Peter Heinrich Seelhoff (1829-1896), Alfred Edward Western (1873-1961).

A. Oliver L. Atkin (1925-2008), Göran Axelsson, Robert Baillie, J. Ray Ballinger (1953-2014), Serge Batalov (Сергей Баталов), David Bessell, Marshall Bishop, Marko Bodschwinna, James Boerner, Robert Boniecki, Richard P. Brent, John Brillhart (1930-2022), J. Scott Brown, Duncan A. Buell, Valter Cavecchia, Gordon V. Cormack, Curtis Cooper, John B. Cosgrave, Richard E. Crandall (1947-2012), Michael Dangler, Vasily A. Danilov (Василий А. Данилов), Patrick Demichel, Rob Derrera, Dirk D'huyvetters, Karl Dilcher, Bruce Dodson, J. Doenias, Erwin Doescher, Peter Doggart, Dmitry Domanov (Дмитрий Доманов), Mark Doom, Harvey Dubner (1928-2019), Leonid N. Durman (Леонид Н. Дурман), Michael R. Eaton, Eric Embling, Tony Forbes, Jim Fougeron, Yves Gallot, Gary B. Gostin, Grzegorz Granowski, Tom Greer, Peter Grobstich, Gennady N. Gusev (Геннадий Н. Гусев), John C. Hallyburton Jr., Andreas Höglund, Alexander Hurwitz, Paul Jobling, Mykola Kamenyuk (Микола Каменюк), Reto Keiser, Wilfrid Keller, Charles F. Kerchner III, Craig Kitchen, Dmitry Komin (Дмитрий Комин), Alexey V. Komkov (Алексей В. Комков), Alexander Kruppa, Sergey Kuzmin (Сергей Кузьмин), Arjen K. Lenstra, Rachel Lewis, Sergei Maiorov (Сергей Майоров), Mark S. Manasse, G. Matthew, Ernst W. Mayer (1963-2023), Roman Maznichenko (Роман Мазниченко), Richard J. McIntosh, Philip P. McLaughlin Jr., Nestor S. de Araújo Melo, François Morain, Luigi Morelli, Dan Morenus, Michael A. Morrison, Souichi Murata, Takahiro Nohara, Adolf Nordin, C. Norrie, Kevin Odermatt, Anton P. Oleynick (Антон П. Олейник), Raymond Ottusch, Maximilian Pacher, Jason S. Papadopoulos, Lélio Ribeiro de Paula, Jay Parangalan, Jean Penné, John M. Pollard, Robert Prethaler, Ryan Propper, Martin Ptáček, Tapio Rajala, John Renze, Geoffrey Reynolds, Neil W. Rickert, Hans Riesel (1929-2014), Raphael M. Robinson (1911-1995), Mark Rodenkirch, Payam Samidoost, Pavlos Saridis, John L. Selfridge (1927-2010), Andriy Sen, David E. Shippee (1946-2003), Peter Strasser, Hiromi Suyama, Jun Tajima, Sai Yik Tang, Tadashi Taura, Claude Tardif, Eric Ueda, C. van Halewyn, Bart van Rooijen, Michael Vang, Cedric Vonck, Asko Vuori, Hugh C. Williams, George F. Woltman, Claude P. Wrathall, Shoichiro Yamada, Senji Yamashita, Jeffrey Young, Paul Zimmermann.

Names given without initials in the discoverer column refer to an indirect involvement of programmers whose software was essential for the discovery of the particular prime factor: Durman (Fermat), Fougeron (FermFact, PFGW), Gallot (Proth), Jobling (NewPGen), Kruppa (specialized GMP-ECM), Penné (LLR), Reynolds (Srsieve), Rodenkirch (GMP-Fermat, gfndsieve), Woltman (PRP, Prime95, mmff), Zimmermann (GMP-ECM).
A semicolon indicates independent discovery by two individuals or teams.

Summary of factoring status for Fermat numbers Fm
as of January 14, 2025

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)
Six prime factors known m = 12*
Four prime factors known m = 13*
Three prime factors known m = 15*, 19*, 25*, 52, 287
Two prime factors known m = 16*, 17*, 18*, 27*, 30*, 36, 38, 39, 42, 77, 147,
150, 251, 284, 416, 417
Only one prime factor known m = 14*, 21*, 22*, 23*, 26*, 28*, 29*, 31, 32, 37, 40,
and 285 values of m with 40 < m ≤ 18233956
Composite but no factor known m = 20, 24
Character unknown m = 33, 34, 35, 41, 44, 45, 46, 47, 49, 50, 51, . . .
* Cofactor composite

373 prime factors known
328 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, 11, 12
Frequency 191, 89, 50, 14, 10, 5, 6, 3, 3, 1, 1

Selected references

Factoring status at earlier stages
  • 1958 Raphael M. Robinson, A report on primes of the form k · 2n + 1 and on factors of Fermat numbers,
    Proc. Amer. Math. Soc. 9 (1958), 673-681. PDF [38 prime factors known: complete list]
  • 1964 Claude P. Wrathall, New factors of Fermat numbers, Math. Comp. 18 (1964), 324-325. PDF
    [51 prime factors known]
  • 1975 John C. Hallyburton, Jr. & John Brillhart, Two new factors of Fermat numbers,
    Math. Comp. 29 (1975), 109-112. PDF [55 prime factors known]
  • 1983 Wilfrid Keller, Factors of Fermat numbers and large primes of the form k · 2n + 1,
    Math. Comp. 41 (1983), 661-673. PDF [90 prime factors known]
  • 1995 Gary B. Gostin, New factors of Fermat numbers, Math. Comp. 64 (1995), 393-395. PDF
    [46 new prime factors; 161 prime factors known overall]
Completed factorizations
  • 1975 Michael A. Morrison & John Brillhart, A method of factoring and the factorization of F7,
    Math. Comp. 29 (1975), 183-205. PDF [Continued fraction method]
  • 1981 Richard P. Brent & John M. Pollard, Factorization of the eighth Fermat number,
    Math. Comp. 36 (1981), 627-630. PDF [Pollard rho algorithm]
  • 1993 A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse & J. M. Pollard, The factorization of the ninth Fermat number,
    Math. Comp. 61 (1993), 319-349. PDF [Number field sieve]
  • 1999 Richard P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429-451. PDF
    [Elliptic curve method]
Proofs of compositeness
  • 1988 Jeff Young & Duncan A. Buell, The twentieth Fermat number is composite,
    Math. Comp. 50 (1988), 261-263. PDF
  • 2003 Richard E. Crandall, Ernst W. Mayer & Jason S. Papadopoulos, The twenty-fourth Fermat number is composite,
    Math. Comp. 72 (2003), 1555-1572. PDF

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

For more detailed information,
see http://www.fermatsearch.org/stat/done.php

n Ln
14 -ひく 19 1100000000000000 =わ 1.1 · 1015
20 4000000000000000 =わ 4 · 1015
21 -ひく 23 2250000000000000 =わ 2.25 · 1015
24 -ひく 25 4800000000000000 =わ 4.8 · 1015
26 3000000000000000 =わ 3 · 1015
27 2600000000000000 =わ 2.6 · 1015
28 2000000000000000000 =わ 2 · 1018
29 1100000000000000000 =わ 1.1 · 1018
30 -ひく 34 100000000000000000 =わ 1017
35 480000000000000000 =わ 4.8 · 1017
36 700000000000000000 =わ 7 · 1017
37 260000000000000000 =わ 2.6 · 1017
38 160000000000000000 =わ 1.6 · 1017
39 100000000000000000 =わ 1017
40 -ひく 46 30000000000000000 =わ 3 · 1016
47 -ひく 49 20000000000000000 =わ 2 · 1016
50 -ひく 59 15000000000000000 =わ 1.5 · 1016
60 -ひく 63 5000000000000000 =わ 5 · 1015
64 -ひく 99 1000000000000000 =わ 1015
100 -ひく 109 200000000000000 =わ 2 · 1014
110 -ひく 149 100000000000000 =わ 1014
150 -ひく 159 150000000000000 =わ 1.5 · 1014
160 -ひく 169 200000000000000 =わ 2 · 1014
170 -ひく 199 2000000000000 =わ 2 · 1012
200 -ひく 249 8000000000000 =わ 8 · 1012
250 -ひく 345 150000000000 =わ 1.5 · 1011
346 -ひく 499 100000000000 =わ 1011
500 -ひく 599 40000000000 =わ 4 · 1010
601 -ひく 620 650000000000 =わ 6.5 · 1011
621 -ひく 640 600000000000 =わ 6 · 1011
641 -ひく 900 300000000000 =わ 3 · 1011
901 -ひく 1000 150000000000 =わ 1.5 · 1011
1001 -ひく 1200 100000000000 =わ 1011
1201 -ひく 1600 60000000000 =わ 6 · 1010
1601 -ひく 2000 30000000000 =わ 3 · 1010
2001 -ひく 4000 400000000 =わ 4 · 108
4001 -ひく 5000 200000000 =わ 2 · 108
5001 -ひく 6000 100000000 =わ 108
6001 -ひく 7200 50000000 =わ 5 · 107
7201 -ひく 8000 20000000 =わ 2 · 107
8001 -ひく 12000 10000000 =わ 107
12001 -ひく 14500 70000000 =わ 7 · 107
14501 -ひく 16000 40000000 =わ 4 · 107
16001 -ひく 19000 20000000 =わ 2 · 107
19001 -ひく 22000 12000000 =わ 1.2 · 107
22001 -ひく 25000 8000000 =わ 8 · 106
25001 -ひく 30000 6000000 =わ 6 · 106
30001 -ひく 35000 3000000 =わ 3 · 106
35001 -ひく 40000 2000000 =わ 2 · 106
40001 -ひく 45000 600000 =わ 6 · 105
45001 -ひく 50000 400000 =わ 4 · 105
50001 -ひく 55000 2400000 =わ 2.4 · 106
55001 -ひく 60000 1600000 =わ 1.6 · 106
60001 -ひく 62000 1200000 =わ 1.2 · 106
62001 -ひく 70000 1000000 =わ 106
70001 -ひく 75000 600000 =わ 6 · 105
75001 -ひく 80000 500000 =わ 5 · 105
80001 -ひく 85000 400000 =わ 4 · 105
85001 -ひく 88000 350000 =わ 3.5 · 105
88001 -ひく 90000 300000 =わ 3 · 105
90001 -ひく 100000 250000 =わ 2.5 · 105
100001 -ひく 120000 130000 =わ 1.3 · 105
120001 - 130000 60000
130001 - 150000 50000
150001 - 190000 30000
190001 - 1000000 20000
1000001 - 1760000 10000
1760001 - 3600000 1200
3600001 - 4000000 740

URL: http://www.prothsearch.com/fermat.html
Last modified: August 22, 2025.

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