Phi2(x) = x+1 Phi2(2^2^m) = 2^2^m+1 = F(m) divisors: k*2^n+1 where n≥m+2 (information extracted from Wilfrid Keller's page updated 2025-12-05) Fermat primes 0 3 1 5 2 17 3 257 4 65537 Completely factored 5 641 * 6700417 6 274177 * 67280421310721 7 59649589127497217 * 5704689200685129054721 8 1238926361552897 * 93461639715357977769163558199606896584051237541638188580280321 9 2424833 * 7455602825647884208337395736200454918783366342657 * P(99) 10 45592577 * 6487031809 * 4659775785220018543264560743076778192897 * P(252) 11 319489 * 974849 * 167988556341760475137 * 3560841906445833920513 * P(564) cofactors 5 52347 7 1732 Euler 6 262814145745 8 1855 Clausen; Landry & Le Lasseur 1880 7 11141971095088142685 9 13 Sep 1970 Morrison & Brillhart 8 [59 digits] 11 1980 Brent & Pollard 9 [96 digits] 11 15 Jun 1990 Lenstra, Manasse & a larger team 10 [248 digits] 13 1995 Brent 11 [560 digits] 13 20 Jun 1988 Brent & Morain 9 [46 digits] 11 15 Jun 1990 Lenstra, Manasse & a larger team 10 [37 digits] 12 20 Oct 1995 Brent 11 434673084282938711 13 13 May 1988 Brent 11 10253207784531279 14 17 May 1988 Brent 7 116503103764643 9 13 Sep 1970 Morrison & Brillhart 8 604944512477 11 1980 Brent & Pollard 10 395937 14 1962 Brillhart 10 11131 12 15 Aug 1953 Selfridge 6 1071 8 1855 Clausen; Landry 1880 11 119 13 1899 Cunningham 11 39 13 1899 Cunningham 9 37 16 1903 Western 5 5 7 1732 EulerECM
12 114689 * 26017793 * 63766529 * 190274191361 * 1256132134125569 * 568630647535356955169033410940867804839360742060818433 * C(1133) 13 2710954639361 * 2663848877152141313 * 3603109844542291969 * 319546020820551643220672513 * C(2391) 14 116928085873074369829035993834596371340386703423373313 * C(4880) 15 1214251009 * 2327042503868417 * 168768817029516972383024127016961 * C(9808) 16 825753601 * 188981757975021318420037633 * C(19694) 17 31065037602817 * 7751061099802522589358967058392886922693580423169 * C(39395) 18 13631489 * 81274690703860512587777 * C(78884) 19 70525124609 * 646730219521 * 37590055514133754286524446080499713 * C(157770) 20 C(315653) 21 4485296422913 * C(631294) 22 64658705994591851009055774868504577 * C(1262577) 23 167772161 * C(2525215) 24 C(5050446) 25 25991531462657 * 204393464266227713 * 2170072644496392193 * C(10100842) 26 76861124116481 * C(20201768) 27 151413703311361 * 231292694251438081 * C(40403531) 28 1766730974551267606529 * C(80807103) 29 2405286912458753 * C(161614233) 30 640126220763137 * 1095981164658689 * C(323228467) 12 [50 digits] 15 27 Mar 2010 Vang, Zimmermann & Kruppa 14 [49 digits] 16 03 Feb 2010 Rajala 17 [44 digits] 19 15 Mar 2011 Bessell 15 1287603889690528658928101555 17 03 Jul 1997 Crandall & van Halewyn 19 8962167624028624126082526703 22 18 Jul 2009 Bessell 22 3853959202444067657533632211 24 26 Mar 2010 Bessell 13 609485665932753836099 19 16 Jun 1995 Brent 16 180227048850079840107 20 Dec 1996 Crandall & Dilcher [composite cofactors]Trial division
(17) 40 74327396788321657 42 01 Mar 2021 Morelli (16) 18 9688698137266697 23 16 Apr 1999 Crandall, McIntosh & Tardif 39 2864929972774011 41 07 Dec 2012 Rajala 65 1210895760431083 68 29 Jan 2013 Maznichenko 103 7595155767819149 105 31 Dec 2020 Strasser 118 1527888802614951 120 28 Jan 2019 Strasser 132 1075441212722595 135 19 Dec 2018 Strasser (15) 42 111318179143061 45 20 Mar 2011 Maznichenko 251 322072887044529 253 24 May 2024 Gostin (14) 13 20323554055421 17 05 Jan 1991 Crandall 52 81909357657279 54 15 Mar 2010 Vonck 86 20018578522347 88 07 Jun 2012 Dangler 160 43390164659167 164 27 Jul 2023 Doescher 195 48595346636925 197 26 Jun 2014 Batalov 274 58130105122923 277 05 Apr 2018 Maznichenko 287 21346190509623 290 25 Dec 2015 Maznichenko 425 44893411020831 427 25 Jun 2023 Gostin (13) 31 5463561471303 33 12 Apr 2001 Kruppa & Forbes 48 2139543641769 50 28 May 2001 Bodschwinna 83 1595863660157 87 12 May 2005 Danilov 96 3334131633063 101 25 Nov 2008 Ptáček 166 2674670937447 171 10 Jul 2012 Maznichenko 287 8662575142599 289 28 Nov 2016 Maznichenko 459 5449229488169 465 08 May 2013 Propper (12) 13 6872386635861 19 09 May 1991 Crandall 43 212675402445 45 23 Dec 2000 Samidoost 88 119942751127 90 13 Mar 2001 Nohara 90 198922467387 92 26 Mar 2001 Grobstich 94 482524552001 97 18 Apr 2001 Grobstich 299 272392805475 304 20 Nov 2010 Komkov 417 303472680883 420 30 Jan 2013 Maznichenko 666 217924552867 668 26 May 2014 Maznichenko 1379 205071338665 1382 24 Nov 2021 Gostin 1436 713828860473 1438 21 Apr 2023 Gostin 1493 623619365837 1495 09 Feb 2023 Gostin 1784 321941255325 1788 31 May 2023 Gostin (11) 12 76668221077 14 25 Jul 1986 Baillie 15 17753925353 17 04 Aug 1987 Gostin 25 16168301139 27 16 Dec 1987 McLaughlin 28 25709319373 36 05 Feb 1997 Taura 133 88075576149 135 24 Jul 2001 Samidoost 172 20569603303 174 11 May 2001 Durman 286 78472588395 288 02 Nov 2002 Danilov 297 72677552745 301 13 Feb 2003 Danilov 517 84977118993 520 25 Jan 2010 Maiorov 1132 10111717305 1136 26 Dec 2012 Batalov 1296 58951871175 1299 30 Oct 2016 Gostin 1680 26778342591 1683 05 Dec 2016 Gostin 2144 19789270693 2148 02 Nov 2018 Gostin 2587 15249465809 2591 09 Dec 2019 Gostin 3135 36812439837 3144 27 Nov 2025 Gostin (10) 25 1522849979 27 07 Aug 1985 Gostin 37 1275438465 39 29 Apr 1991 Gostin 77 5940341195 79 08 Apr 1998 Taura 107 1289179925 111 08 Nov 1992 Gostin 116 3433149787 120 08 Aug 1999 Taura 164 1835601567 167 31 May 1993 Gostin 284 1061341513 286 25 Dec 2000 Durman 343 4844391185 345 29 Aug 2001 Danilov 480 5673968845 484 24 Nov 2001 Danilov 569 6616590375 575 25 Feb 2003 Kuzmin 600 6213186413 605 29 Sep 2005 Vuori 943 4785972759 954 28 Mar 2012 Sen 1160 2018719057 1162 15 May 2005 Pacher 2123 9659599821 2125 16 Dec 2016 Gostin 2286 4004779405 2288 25 Nov 2015 Gostin 3056 3370842847 3058 28 Jan 2017 Gostin 3345 6604326057 3347 13 Dec 2018 Gostin 3576 1132007355 3578 21 Jul 2015 Nohara 3730 1127899657 3732 17 Aug 2015 Nohara 5199 1488947679 5201 24 Jul 2018 Rodenkirch 7220 1995420873 7222 07 Dec 2024 de Paula 27 430816215 29 21 Feb 1985 Gostin 147 124567335 149 01 Dec 1990 Gostin 178 313047661 180 18 Jun 1991 Gostin 184 117012935 187 17 Dec 1990 Gostin 230 372236097 232 13 Apr 2000 Durman 370 573230511 373 22 Aug 2000 Durman 380 321116871 385 26 Jun 2000 Durman 417 118086729 421 13 Jun 1992 Gostin 667 491628159 669 12 Mar 2001 Taura 971 541664191 976 04 Apr 2002 Komin 1201 837747239 1203 12 Jan 2011 Pacher 1710 351276975 1719 09 Mar 2008 Pacher 1722 364182745 1724 19 Apr 2008 Pacher 2420 103257279 2422 07 Nov 2010 Strasser 2606 238451805 2608 11 Mar 2012 Nohara 3314 406860969 3322 18 Jan 2013 Pacher 3703 262254673 3706 17 Jun 2012 Nohara 3850 787703529 3854 13 Aug 2015 Pacher 4184 465917283 4189 27 Dec 2014 Batalov 4260 209161375 4262 14 Mar 2014 Nohara 4652 143918649 4654 09 Oct 2012 Rajala 7137 118807331 7139 21 Feb 2016 Gostin 9863 332436749 9865 23 Mar 2019 Gostin 11075 171369935 11077 29 Jun 2020 Gostin 12 11613415 14 1974 Hallyburton & Brillhart 13 41365885 16 1974 Hallyburton & Brillhart 17 59251857 19 30 May 1978 Gostin 52 21626655 54 1982 Keller 61 54985063 66 17 May 1986 Gostin 64 17853639 67 13 May 1986 Gostin 72 76432329 74 31 May 1986 Gostin 146 37092477 148 23 Dec 1987 Gostin 232 70899775 236 24 Nov 1991 Gostin 251 85801657 254 30 Nov 1991 Gostin 256 36986355 258 20 Jan 1991 Gostin 259 36654265 262 21 Jan 1991 Gostin 338 27654487 342 22 Dec 1990 Gostin 353 18908555 355 25 Dec 1990 Gostin 468 27114089 471 25 Sep 1992 Gostin 579 63856313 581 04 Oct 1999 Taura 620 10084141 624 19 Jan 1992 Gostin 642 52943971 644 08 Aug 1999 Taura 885 16578999 887 12 Jul 1992 Gostin 1114 11618577 1116 22 Jan 2001 Gostin 1394 62705223 1396 06 Dec 2001 Taura 1598 10923781 1600 21 Dec 2000 Taura 3335 43714055 3337 05 Jul 2011 Kamenyuk 3723 13308899 3725 03 Jul 2002 Fougeron 4265 72179955 4269 03 Aug 2012 Nohara 9552 60567741 9555 04 Apr 2016 Gostin 9747 44670651 9749 20 Dec 2014 Nohara 14587 24142479 14590 25 May 2016 Nordin 18345 15071297 18347 07 Sep 2017 Gostin 18933 15175951 18936 13 Sep 2017 Gostin 25599 14698299 25602 03 Mar 2021 Gostin 31880 10981209 31883 10 Jul 2023 Batalov 29 1120049 31 17 Oct 1980 Gostin & McLaughlin 36 3759613 38 02 Jan 1981 Gostin & McLaughlin 75 3447431 77 Apr 1982 Gostin 122 5234775 124 21 May 1986 Gostin 142 8152599 145 22 May 1986 Gostin 301 7183437 304 04 Nov 1990 Gostin 431 5769285 434 30 Nov 1990 Gostin 635 4258979 645 12 Jul 1991 Gostin 4332 2466157 4334 15 Oct 2002 Fougeron 5531 1503975 5533 11 Apr 2002 Gostin 5792 8872947 5794 03 Mar 2003 Fougeron 8298 1054057 8300 01 Nov 2003 Kitchen 9447 5505161 9449 09 Nov 2012 Batalov 12825 1814649 12827 09 Oct 2009 Nohara 17748 3860269 17750 21 Nov 2012 Batalov 37236 1844383 37238 19 Dec 2017 Gostin 38795 1514535 38797 27 Dec 2017 Gostin 19 308385 21 1963 Wrathall 21 534689 23 1963 Wrathall 26 143165 29 1963 Wrathall 27 141015 30 1963 Wrathall 30 149041 32 1963 Wrathall 30 127589 33 1963 Wrathall 205 232905 207 01 Jan 1984 Keller 375 733251 377 09 Jun 1986 Gostin 376 810373 378 09 Jun 1986 Gostin 398 120845 401 18 Apr 1984 Keller 723 554815 730 04 Jan 1991 Gostin 851 497531 859 18 Jan 1988 Gostin 1069 137883 1073 01 Sep 1992 Gostin 1990 150863 1993 30 Oct 1995 Taura 2059 591909 2063 09 May 2000 Ballinger 4250 173373 4252 22 Dec 1999 Kerchner 5957 421435 5960 16 Jan 2001 Gostin 6355 115185 6358 14 Nov 2000 Kerchner 7181 168329 7187 26 Dec 2000 Gostin 8269 592131 8271 21 May 2003 Gostin 9691 260435 9693 12 Dec 2001 Gostin 11695 203355 11703 31 Dec 2001 Gostin 30256 121531 30260 16 May 2006 Vuori 38967 177795 38969 10 Jan 2009 Nohara 45278 547653 45280 18 Jan 2018 Gostin 63480 242445 63484 10 Mar 2018 Gostin 66643 749893 66648 11 Aug 2021 Doescher 19 33629 21 16 Nov 1962 Riesel 25 48413 29 1963 Wrathall 42 43485 45 1963 Wrathall 93 92341 96 1979 Baillie 99 16233 104 Nov 1979 Gostin & McLaughlin; Suyama 215 32111 217 02 Dec 1980 Suyama 275 22347 279 16 Jan 1984 Keller 334 27609 341 25 Jan 1984 Keller 416 38039 419 16 Feb 1984 Keller 547 77377 550 05 Jun 1986 Gostin 637 11969 643 08 Mar 1984 Keller 906 57063 908 19 May 1986 Gostin 1082 82165 1084 08 Apr 1991 Gostin 1123 25835 1125 15 Jan 1987 Gostin 1225 79707 1231 08 Jun 1991 Gostin 1229 29139 1233 07 Dec 1987 Gostin 1451 13143 1454 30 Oct 1986 Gostin 1849 98855 1851 01 Sep 1992 Gostin 5320 21341 5323 22 Apr 1998 Taura 13623 48265 13626 14 Dec 2000 Gostin 14528 17217 14530 24 Dec 2000 Gostin 19211 13323 19220 18 May 2001 Gostin 48624 28949 48627 03 May 2008 Nohara 49488 71007 49490 18 Apr 2009 Nohara 100518 79425 100520 22 Oct 2014 Samidoost 134995 47893 134998 22 May 2020 Gostin 223380 56073 223382 04 Jul 2024 Propper & Batalov 414677 39701 414679 10 Jan 2025 Propper & Batalov 699760 13243 699764 28 Jul 2023 Propper & Batalov 106432 30967 106436 13 Nov 2012 Batalov 16 1575 19 14 Aug 1953 Selfridge 32 1479 34 1963 Wrathall 38 2653 40 1963 Wrathall 52 4119 54 1963 Wrathall 66 7551 69 1977 Shippee 91 1421 93 1977 Shippee 147 3125 149 Dec 1979 Gostin & McLaughlin 150 1575 157 1956 Robinson 150 5439 154 May 1980 Gostin & McLaughlin; Suyama 201 4845 204 27 Jul 1980 Gostin & McLaughlin 287 5915 289 26 Dec 1980 Suyama 329 1211 333 12 Jan 1981 Suyama 416 8619 418 1981 Suyama 931 1985 933 22 Dec 1980 Keller 4258 1435 4262 05 Aug 1993 Gostin 6909 6021 6912 12 Nov 1993 Gostin 8239 7473 8242 25 Jun 1998 Prethaler 9322 8247 9324 16 Nov 1999 Kerchner 9549 1211 9551 15 Feb 1998 Taura 14252 1173 14254 07 Mar 1997 Taura 22296 4777 22298 07 Mar 2001 Gostin 41894 4935 41897 15 Aug 2001 Fougeron 43665 2495 43667 21 Aug 2001 Fougeron 50078 7619 50081 06 Mar 2002 Axelsson 60079 5731 60084 14 Feb 2005 Eaton 79221 6089 79223 01 Aug 2008 Samidoost 138557 7333 138560 12 Mar 2011 D'huyvetters 221670 3771 221676 01 Jul 2011 Doom 226614 4479 226618 08 Jul 2011 Doggart 352279 7905 352281 31 May 2012 Boerner 410105 1207 410108 22 Nov 2005 Tajima 906108 1705 906110 13 Jun 2012 Boniecki 1099061 2145 1099064 18 Jun 2013 Tang 3943261 1085 3943263 03 Dec 2025 Dinkel 12 397 16 1903 Western 12 973 16 1903 Western 15 579 21 1925 Kraitchik 62 697 64 1977 Shippee 71 683 73 1977 Shippee 77 425 79 1957 Robinson & Selfridge 81 271 84 1957 Robinson & Selfridge 250 403 252 1957 Robinson & Selfridge 255 629 257 1979 Baillie 267 177 271 1957 Robinson & Selfridge 298 247 302 1979 Baillie 544 225 547 1979 Baillie 556 127 558 1976 Matthew & Williams 692 717 695 1979 Atkin & Rickert 1551 291 1553 1979 Atkin & Rickert 2089 431 2099 17 Aug 1983 Suyama 3506 501 3508 26 Jul 1986 Gostin 6208 763 6210 05 Aug 1993 Gostin 6390 303 6393 06 Aug 1993 Gostin 7309 145 7312 25 Oct 1992 Dubner 8555 645 8557 01 Nov 1993 Dubner 13250 351 13252 01 Oct 1996 Taura 14276 157 14280 31 Aug 1996 Taura 17906 135 17909 31 Aug 1996 Taura (Prime: J. Young 1993) 23069 681 23071 Jul 1997 Taura (Prime: P. Demichel 1996) 35563 357 35567 27 Aug 2000 Melo 49093 165 49095 19 Aug 1998 Gallot 63679 169 63686 19 May 1998 Dubner 90057 189 90061 20 Sep 1999 Morenus 91213 585 91215 30 Jul 2001 Axelsson 104448 927 104451 30 Nov 2001 Oleynick 142460 159 142462 14 Nov 2001 Melo 287384 211 287388 13 Dec 2004 Fougeron 338295 485 338297 21 Aug 2007 Keller, S. Murata 476624 651 476632 28 Dec 2008 Ueda 495728 243 495732 29 May 2007 Keiser 567233 519 567235 06 Mar 2009 Yamashita 585042 151 585044 17 Mar 2007 Saridis 617813 659 617815 31 Mar 2009 Embling 1246013 329 1246017 06 Jan 2012 Dodson 1494096 131 1494099 07 Feb 2012 Derrera 1747656 183 1747660 10 Mar 2013 van Rooijen 2662088 267 2662090 13 Feb 2015 Parangalan 3329780 193 3329782 25 Jul 2014 Ottusch 4192909 141 4192911 17 Jul 2023 Brown 18 13 20 1903 Western (Prime: Seelhoff 1886) 39 21 41 1956 Robinson 55 29 57 1956 Robinson 58 95 61 1957 Robinson 144 17 147 1956 Robinson 226 15 229 1956 Robinson 228 29 231 1956 Robinson 268 21 276 1956 Robinson 452 27 455 1956 Robinson 744 17 747 1976 Matthew & Williams 2023 29 2027 1979 Atkin & Rickert; Cormack & Williams 2456 85 2458 1979 Atkin & Rickert 4724 29 4727 1979 Cormack & Williams 6537 17 6539 1979 Cormack & Williams 6835 19 6838 30 Sep 1978 Keller 9448 19 9450 30 Aug 1980 Keller 12185 81 12189 04 Sep 1993 Dubner 15161 55 15164 28 Aug 1993 Dubner 18749 11 18759 12 Sep 1992 Dubner 18757 33 18766 20 Aug 1993 Dubner 23288 19 23290 14 Dec 1992 Dubner 24651 99 24653 11 Sep 1996 Taura 25006 57 25010 18 Dec 1993 Young 28281 81 28285 18 Aug 1996 Taura 83861 99 83863 12 Oct 1998 Gusev 94798 21 94801 Mar 1995 Young 113547 39 113549 11 May 1999 Renze 114293 13 114296 Aug 1995 Young 146221 57 146223 11 Feb 2000 Lewis 270091 63 270094 05 Feb 2002 Taura 282717 51 282719 13 Nov 2002 Odermatt 472097 89 472099 05 Oct 2004 Samidoost 672005 27 672007 30 Aug 2005 Cooper 960897 11 960901 23 Feb 2005 Eaton 2141872 25 2141884 09 Sep 2011 Granowski 2747497 57 2747499 14 May 2013 Bishop 5523858 13 5523860 22 Jan 2020 Brown 5798447 99 5798449 14 Jan 2025 Cavecchia 7963245 27 7963247 16 Jan 2021 GreerTabulation of primes
12 7 14 1877 Pervushin; Lucas 23 5 25 1878 Pervushin 36 5 39 1886 Seelhoff 38 3 41 1903 Cullen & Cunningham; Western 63 9 67 1956 Robinson 73 5 75 1906 Morehead 117 7 120 1956 Robinson 125 5 127 1956 Robinson 207 3 209 1956 Robinson 284 7 290 1956 Robinson 316 7 320 1956 Robinson 1945 5 1947 1957 Robinson 3310 5 3313 1979 Atkin & Rickert; Cormack & Williams 9428 9 9431 01 Apr 1983 Keller 23471 5 23473 05 Sep 1984 Keller 95328 7 95330 23 Aug 1994 Young 125410 5 125413 Aug 1995 Young 157167 3 157169 14 Jun 1995 Young 213319 3 213321 21 Aug 1996 Young 303088 3 303093 13 Jan 1998 Young 382447 3 382449 25 Jul 1999 Cosgrave 461076 9 461081 04 Aug 2003 Nohara 2145351 3 2145353 21 Feb 2003 Cosgrave 2167797 7 2167800 17 Apr 2007 Cooper 2478782 3 2478785 10 Oct 2003 Cosgrave 2543548 9 2543551 22 Jun 2011 Brown 18233954 7 18233956 05 Oct 2020 Propper
1640 Fermat F(0) F(1) F(2) F(3) F(4) prime couldn't prove that F(5) is prime proposed maybe F(m) be always prime 1732 Goldbach proposed the problem to Euler mentioned that all F(m) are coprime [? probable prime base 2] non of them have a prime divisor p<200 1732 Euler proved that if prime p divides F(m) then it must be of the form k*2^(m+1)+1 then showed that F(5) is divisible by 641=10*2^6+1=5*2^7+1 1855 Clausen factored F(6) reported it to Gause, but Gause died in one month! the note was lost till [later] 1877 Pervushin F(12) is divisible by 7*2^14+1 1877 Pervushin F(23) is divisible by 5*2^25+1 He found the prime 3*2^41+1 1903 Cunningham and Western showed that it divides F(38) He started the tabulation of primes 1879 Lucas claimed that he have proved the compositeness of F(6) 1880 Landry refactored F(6) 11 119 13 1899 Cunningham 11 39 13 1899 Cunningham 9 37 16 1903 Western 12 397 16 1903 Western 12 973 16 1903 Western 18 13 20 1903 Western (Prime: Seelhoff 1886) 12 7 14 1877 Pervushin; Lucas 23 5 25 1878 Pervushin 36 5 39 1886 Seelhoff 38 3 41 1903 Cullen & Cunningham; Western 15 579 21 1925 Kraitchik