- N does not have any prime factor up to 2^32.
- N cannot be trivially written into a product.
- Take 3 square-free consecutive bases such that (b1/N)=(b2/N)=(b3/N)= -1. N
should be such that b1^((N-1)/2)=b2^((N-1)/2)=b3^((N-1)/2)= -1 (mod N).
- Take a prime base such that (b/N)= -1 and b<>b1<>b2<>b3.
N should pass the strong test in base b.
- or more (if someone has got an idea) !!!
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| Rank | Probable prime | Digits (> 50000) | Who ? | When ? |
| 251 | (7^720497-2^720497)/5 | 608890 | Jon Grantham | 08/2023 |
| 252 | 2^2020893-25 | 608350 | Boyan Hu | 01/2025 |
| 253 | 2^2019811+1867211 | 608024 | Enzo Doria | 04/2018 |
| 254 | F(2904353) | 606974 | Henri Lifchitz | 07/2014 |
| 255 | 2^2014720+71893 | 606492 | Enzo Doria | 04/2018 |
| 256 | (5^866029-3^866029)/2 | 605328 | Jon Grantham | 08/2023 |
| 257 | L(2894561) | 604928 | Ryan Propper | 06/2022 |
| 258 | (18^481147+1)/19 | 603970 | Paul Bourdelais | 07/2011 |
| 259 | 200^262144+119^262144 | 603202 | Kellen Shenton | 08/2022 |
| 260 | (20^463511+1)/21 | 603041 | Paul Bourdelais | 03/2025 |
| 261 | (70^326437-1)/69 | 602307 | Paul Bourdelais | 02/2025 |
| 262 | (83467^121801-1)/(83467^349-1) | 597728 | Ryan Propper | 11/2023 |
| 263 | (61448^124609-1)/(61448^353-1) | 595001 | Ryan Propper | 11/2023 |
| 264 | (15^505117-1)/14 | 594063 | Paul Bourdelais | 01/2015 |
| 265 | 81^310721-80^310721 | 593007 | Paul Bourdelais | 07/2023 |
| 266 | (152^270217-1)/(152-1) | 589570 | Michael Stocker | 06/2015 |
| 267 | Phi(177147,94259) | 587458 | Anonymous | 09/2022 |
| 268 | 2^1950002+1867369 | 587010 | Enzo Doria | 02/2018 |
| 269 | 116265^111218+111218^116265 | 586694 | Ryan Propper | 10/2023 |
| 270 | 2^1940812+3 | 584243 | Paul Bourdelais | 06/2020 |
| 271 | 1933695^2+2^1933695 | 582101 | Ryan Propper | 05/2023 |
| 272 | (5^832151-2^832151)/3 | 581649 | Jon Grantham | 08/2023 |
| 273 | 2^1930263+229883 | 581068 | Enzo Doria | 01/2025 |
| 274 | (129810^113569-1)/(129810^337-1) | 578991 | Ryan Propper | 11/2023 |
| 275 | 2^1923104+153 | 578912 | Stefano Morozzi | 07/2021 |
| 276 | (48^343901-1)/47 | 578179 | Paul Bourdelais | 03/2025 |
| 277 | 116761^87652+87652^116761 | 577122 | Ryan Propper | 10/2023 |
| 278 | 116395^87994+87994^116395 | 575510 | Ryan Propper | 10/2023 |
| 279 | (3^1205459+1)/4 | 575150 | Paul Bourdelais | 08/2015 |
| 280 | 2^1895562+229729 | 570622 | Enzo Doria | 02/2025 |
| 281 | (160972^109561-1)/(160972^331-1) | 568734 | Ryan Propper | 11/2023 |
| 282 | 2^1887092+38191 | 568072 | Enzo Doria | 01/2025 |
| 283 | (37015^124609-1)/(37015^353-1) | 567649 | Ryan Propper | 11/2023 |
| 284 | 2^1882800+384541 | 566780 | Enzo Doria | 12/2017 |
| 285 | (2^1882787-2^941394+1)/5 | 566775 | Borys Jaworski | 11/2012 |
| 286 | 2^1882474+71389 | 566682 | Enzo Doria | 04/2018 |
| 287 | (6^727496-11)/5 | 566102 | Paul Bourdelais | 08/2019 |
| 288 | 2^1875552+3 | 564598 | Paul Bourdelais | 04/2020 |
| 289 | (145373^109561-1)/(145373^331-1) | 563899 | Ryan Propper | 11/2023 |
| 290 | (10^563535-7)/3 | 563535 | Rytis Slatkevicius | 02/2023 |
| 291 | 3^1180181-2 | 563090 | Jorge Coveiro | 05/2020 |
| 292 | (15063^134689-1)/(15063^367-1) | 561186 | Ryan Propper | 11/2023 |
| 293 | 2^1852265+1890425 | 557588 | Enzo Doria | 11/2017 |
| 294 | 5^797390-2 | 557352 | Paul Bourdelais | 03/2021 |
| 295 | 2^1851447+38573 | 557342 | Enzo Doria | 02/2025 |
| 296 | (10^556441-7^556441)/3 | 556441 | Jon Grantham | 07/2023 |
| 297 | 8^615997-7^615997 | 556301 | Jean-Louis Charton | 07/2009 |
| 298 | (5^795829-2^795829)/3 | 556261 | Jon Grantham | 08/2023 |
| 299 | Tau(773^34960) | 555339 | Nik Lygeros / Olivier Rozier | 09/2015 |
| 300 | ((((((((((1361^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840)^3+1623568 | 555154 | Serge Batalov | 08/2013 |
| 301 | 2^1835428+227221 | 552519 | Enzo Doria | 01/2025 |
| 302 | 2^1835205+225197 | 552452 | Enzo Doria | 10/2024 |
| 303 | 2^1825513+32507 | 549535 | Enzo Doria | 07/2024 |
| 304 | 2^1823921+34937 | 549055 | Enzo Doria | 11/2024 |
| 305 | 2^1822728+228991 | 548696 | Enzo Doria | 02/2025 |
| 306 | 2^1819810+42727 | 547818 | Enzo Doria | 03/2025 |
| 307 | (42^337081-1)/41 | 547165 | Paul Bourdelais | 09/2021 |
| 308 | 114868^57423+57423^114868 | 546667 | Ryan Propper | 10/2023 |
| 309 | 2^1814913+37607 | 546344 | Enzo Doria | 12/2024 |
| 310 | 109092^100999+100999^109092 | 545931 | Ryan Propper | 10/2023 |
| 311 | 2^1812446+34789 | 545601 | Enzo Doria | 02/2025 |
| 312 | 2^1812234+31879 | 545537 | Enzo Doria | 05/2024 |
| 313 | (64814^113569-1)/(64814^337-1) | 544835 | Ryan Propper | 11/2023 |
| 314 | 2^1809045+225641 | 544577 | Enzo Doria | 12/2024 |
| 315 | 2^1804650+230149 | 543254 | Enzo Doria | 02/2025 |
| 316 | 2^1804109+37007 | 543091 | Enzo Doria | 11/2024 |
| 317 | (92170^109561-1)/(92170^331-1) | 542283 | Ryan Propper | 11/2023 |
| 318 | 2^1799578+36883 | 541727 | Enzo Doria | 03/2025 |
| 319 | 2^1797319+29693 | 541047 | Enzo Doria | 11/2024 |
| 320 | (28262^121801-1)/(28262^349-1) | 540608 | Ryan Propper | 11/2023 |
| 321 | (68^294947+1)/69 | 540492 | Paul Bourdelais | 03/2025 |
| 322 | 2^1792175+230543 | 539499 | Enzo Doria | 01/2025 |
| 323 | 2^1791569+33587 | 539317 | Enzo Doria | 09/2024 |
| 324 | (2^1790743-1)/(146840927*158358984977*3835546416767873*20752172271489035681) | 539014 | Dario Alpern | 06/2015 |
| 325 | 10^538640-3 | 538640 | Paul Bourdelais | 01/2021 |
| 326 | (10^538508+53)/9 | 538508 | Rytis Slatkevicius | 01/2023 |
| 327 | 2^1787653+224687 | 538138 | Enzo Doria | 10/2024 |
| 328 | 2^1787167+71963 | 537991 | Enzo Doria | 12/2018 |
| 329 | (7^636043-5^636043)/2 | 537519 | Jon Grantham | 08/2023 |
| 330 | 108309^91610+91610^108309 | 537424 | Ryan Propper | 10/2023 |
| 331 | (12^495953+1)/13 | 535223 | Paul Bourdelais | 11/2018 |
| 332 | 2^1774781+30197 | 534263 | Enzo Doria | 08/2024 |
| 333 | 2^1773644+72511 | 533921 | Enzo Doria | 10/2018 |
| 334 | (92^271639-1)/91 | 533440 | Paul Bourdelais | 02/2022 |
| 335 | 2^1771408+223201 | 533247 | Enzo Doria | 07/2024 |
| 336 | 2^1770613+212597 | 533008 | Enzo Doria | 10/2023 |
| 337 | (11559^131072+1)/2 | 532535 | Serge Batalov | 04/2018 |
| 338 | (85^275881-1)/84 | 532289 | Paul Bourdelais | 02/2022 |
| 339 | 2^1767579+213683 | 532095 | Enzo Doria | 10/2023 |
| 340 | 2^1767218+25249 | 531986 | Enzo Doria | 02/2024 |
| 341 | 2^1762004-3 | 530417 | Paul Bourdelais | 04/2020 |
| 342 | 2^1754760+217465 | 528236 | Enzo Doria | 01/2024 |
| 343 | 2^1753261+25367 | 527785 | Enzo Doria | 11/2023 |
| 344 | 105006^105006-133^133 | 527258 | Sergey Batalov | 05/2024 |
| 345 | 2^1750727+213683 | 527022 | Enzo Doria | 10/2023 |
| 346 | 2^1750208+218371 | 526866 | Enzo Doria | 01/2024 |
| 347 | 2^1743981+28997 | 524991 | Enzo Doria | 03/2024 |
| 348 | 2^1740587+40493 | 523969 | Enzo Doria | 02/2025 |
| 349 | 2^1737680+23161 | 523094 | Enzo Doria | 01/2024 |
| 350 | (73^280697+1)/74 | 523028 | Paul Bourdelais | 02/2022 |
| 351 | 2^1737207+218153 | 522952 | Enzo Doria | 05/2024 |
| 352 | 2^1735354+227509 | 522394 | Enzo Doria | 11/2024 |
| 353 | 2^1734898+225529 | 522257 | Enzo Doria | 09/2024 |
| 354 | 2^1733867+231383 | 521946 | Enzo Doria | 03/2025 |
| 355 | (21^394579+1)/22 | 521719 | Paul Bourdelais | 10/2018 |
| 356 | 2^1732565+226757 | 521555 | Enzo Doria | 12/2024 |
| 357 | 2^1730246+384439 | 520856 | Enzo Doria | 09/2017 |
| 358 | 2^1729303+34313 | 520573 | Enzo Doria | 11/2024 |
| 359 | (55^299087+1)/56 | 520519 | Paul Bourdelais | 02/2022 |
| 360 | 2^1728446+41629 | 520315 | Enzo Doria | 02/2025 |
| 361 | 5^744171-3^744171-1 | 520154 | Jon Grantham | 08/2023 |
| 362 | 2^1727856+31951 | 520137 | Enzo Doria | 04/2024 |
| 363 | (18^414269-1)/17 | 520020 | Paul Bourdelais | 12/2014 |
| 364 | 2^1727029+228257 | 519888 | Enzo Doria | 12/2024 |
| 365 | 2^1726522+30859 | 519735 | Enzo Doria | 06/2024 |
| 366 | 2^1724695+228353 | 519185 | Enzo Doria | 01/2025 |
| 367 | 2^1717949+30077 | 517155 | Enzo Doria | 08/2024 |
| 368 | 2^1717612+31891 | 517053 | Enzo Doria | 04/2024 |
| 369 | (144171^100489-1)/(144171^317-1) | 516776 | Ryan Propper | 11/2023 |
| 370 | 2^1714462+217465 | 516105 | Enzo Doria | 01/2024 |
| 371 | 2^1714021+29777 | 515972 | Enzo Doria | 03/2024 |
| 372 | 2^1713855+220703 | 515922 | Enzo Doria | 03/2024 |
| 373 | 2^1712979+222143 | 515659 | Enzo Doria | 05/2024 |
| 374 | 20^396120-3 | 515365 | Paul Bourdelais | 01/2021 |
| 375 | 2^1709785+36557 | 514697 | Enzo Doria | 02/2025 |
| 376 | (136102^100489-1)/(136102^317-1) | 514270 | Ryan Propper | 11/2023 |
| 377 | 2^1708341+221897 | 514262 | Enzo Doria | 09/2024 |
| 378 | (7^608347-5^608347)/2 | 514113 | Jon Grantham | 08/2023 |
| 379 | (62^286483+1)/63 | 513488 | Paul Bourdelais | 02/2022 |
| 380 | 155630^1971+1971^155630 | 512753 | Henri Lifchitz | 11/2024 |
| 381 | 2^1702747+230873 | 512578 | Enzo Doria | 01/2025 |
| 382 | (20^393257+1)/21 | 511638 | Paul Bourdelais | 12/2014 |
| 383 | 2^1699164+40141 | 511500 | Enzo Doria | 01/2025 |
| 384 | 2^1698083+24743 | 511174 | Enzo Doria | 11/2023 |
| 385 | 2^1696453+219257 | 510684 | Enzo Doria | 06/2024 |
| 386 | (124869^100489-1)/(124869^317-1) | 510523 | Ryan Propper | 11/2023 |
| 387 | (124292^100489-1)/(124292^317-1) | 510321 | Ryan Propper | 11/2023 |
| 388 | 10^509546+3 | 509547 | Paul Bourdelais | 01/2021 |
| 389 | 101876^99207+99207^101876 | 509028 | Gabor Levai | 08/2023 |
| 390 | 2^1690434+213589 | 508872 | Enzo Doria | 12/2023 |
| 391 | 2^1688450+221989 | 508275 | Enzo Doria | 07/2024 |
| 392 | 2^1688111+217313 | 508173 | Enzo Doria | 04/2024 |
| 393 | (118270^100489-1)/(118270^317-1) | 508160 | Ryan Propper | 11/2023 |
| 394 | 2^1687922+40849 | 508116 | Enzo Doria | 01/2025 |
| 395 | 2^1687521+230267 | 507995 | Enzo Doria | 01/2025 |
| 396 | 105196^67205+67205^105196 | 507824 | Miklos Levai | 09/2023 |
| 397 | 2^1686452+220681 | 507673 | Enzo Doria | 04/2024 |
| 398 | 2^1685708+27781 | 507449 | Enzo Doria | 01/2024 |
| 399 | (72^273149-1)/71 | 507327 | Paul Bourdelais | 02/2022 |
| 400 | 2^1685092+227191 | 507264 | Enzo Doria | 12/2024 |
| 401 | 105154^66373+66373^105154 | 507052 | Miklos Levai | 09/2023 |
| 402 | 2^1683331+225413 | 506734 | Enzo Doria | 09/2024 |
| 403 | 107890^49423+49423^107890 | 506429 | Miklos Levai | 02/2022 |
| 404 | (14^441697-1)/13 | 506241 | Paul Bourdelais | 12/2014 |
| 405 | 101645^94522+94522^101645 | 505739 | Gabor Levai | 02/2023 |
| 406 | 3^1059503-2^1059503 | 505512 | Mike Oakes | 04/2012 |
| 407 | 2^1678670+21529 | 505331 | Enzo Doria | 11/2023 |
| 408 | 101542^92885+92885^101542 | 504456 | Gabor Levai | 04/2022 |
| 409 | 2^1675306+227839 | 504318 | Enzo Doria | 12/2024 |
| 410 | 2^1673623+42983 | 503811 | Enzo Doria | 03/2025 |
| 411 | 105273^60238+60238^105273 | 503192 | Miklos Levai | 09/2023 |
| 412 | 105098^61113+61113^105098 | 503014 | Miklos Levai | 09/2022 |
| 413 | 2^1670703+60293 | 502932 | Enzo Doria | 11/2017 |
| 414 | (2^1669219-2^834610+1)/5 | 502485 | Borys Jaworski | 01/2011 |
| 415 | 101311^90816+90816^101311 | 502317 | Gabor Levai | 12/2021 |
| 416 | 101643^87368+87368^101643 | 502254 | Gabor Levai | 11/2022 |
| 417 | (161502^96721-1)/(161502^311-1) | 502121 | Ryan Propper | 11/2023 |
| 418 | 101863^84922+84922^101863 | 502085 | Gabor Levai | 10/2022 |
| 419 | 2^1666440+212671 | 501649 | Enzo Doria | 09/2023 |
| 420 | 100291^100291+172^172 | 501582 | Serge Batalov | 04/2024 |
| 421 | 2^1666177+226697 | 501570 | Enzo Doria | 10/2024 |
| 422 | 100207^99856+99856^100207 | 500973 | Gabor Levai | 11/2021 |
| 423 | 101839^83028+83028^101839 | 500969 | Gabor Levai | 11/2022 |
| 424 | 100263^98600+98600^100263 | 500702 | Gabor Levai | 09/2021 |
| 425 | 2^1662045+37607 | 500326 | Enzo Doria | 11/2024 |
| 426 | 101761^82168+82168^101761 | 500126 | Gabor Levai | 03/2022 |
| 427 | 2^1661176+336631 | 500064 | Tony Prest / Never Odd Or Even | 09/2019 |
| 428 | 101070^88579+88579^101070 | 500027 | Gabor Levai | 01/2022 |
| 429 | (85^259159-1)/84 | 500025 | Paul Bourdelais | 11/2021 |
| 430 | 2^1660973+368139 | 500003 | Tony Prest / Never Odd Or Even | 08/2020 |
| 431 | 100000^100000+85707^85707 | 500001 | Henri Lifchitz | 01/2025 |
| 432 | 2^1660964-382425 | 500000 | Tony Prest / Never Odd Or Even | 05/2020 |
| 433 | (87^257689-1)/86 | 499791 | Paul Bourdelais | 01/2021 |
| 434 | (64*10^499210+53)/9 | 499211 | Serge Batalov | 03/2015 |
| 435 | 2^1656126+223227 | 498544 | Enzo Doria | 03/2025 |
| 436 | Tau(631^32370) | 498503 | Nik Lygeros / Olivier Rozier | 05/2015 |
| 437 | 29^339887-28^339887 | 497051 | Jean-Louis Charton | 05/2019 |
| 438 | 2^1650975+185225 | 496993 | Stefano Morozzi | 09/2024 |
| 439 | 2^1650959+198513 | 496989 | Stefano Morozzi | 09/2024 |
| 440 | 2^1650942+181575 | 496984 | Stefano Morozzi | 09/2024 |
| 441 | 2^1650905+193947 | 496972 | Stefano Morozzi | 09/2024 |
| 442 | 2^1650877+186537 | 496964 | Stefano Morozzi | 01/2025 |
| 443 | 2^1650867+192615 | 496961 | Stefano Morozzi | 09/2024 |
| 444 | 2^1650854+188973 | 496957 | Stefano Morozzi | 01/2025 |
| 445 | 2^1650837+199641 | 496952 | Stefano Morozzi | 01/2025 |
| 446 | 2^1650819+198315 | 496947 | Stefano Morozzi | 07/2024 |
| 447 | 2^1650788+185823 | 496937 | Stefano Morozzi | 09/2024 |
| 448 | 2^1650749+185235 | 496925 | Stefano Morozzi | 06/2024 |
| 449 | 2^1650695+183141 | 496909 | Stefano Morozzi | 04/2024 |
| 450 | 2^1650640+185311 | 496893 | Stefano Morozzi | 03/2024 |
| 451 | 2^1650593+187871 | 496879 | Stefano Morozzi | 03/2024 |
| 452 | 2^1650538+196863 | 496862 | Stefano Morozzi | 03/2024 |
| 453 | 2^1650484+181413 | 496846 | Stefano Morozzi | 03/2024 |
| 454 | 2^1650473+193151 | 496842 | Stefano Morozzi | 03/2024 |
| 455 | 2^1650443+193613 | 496833 | Stefano Morozzi | 01/2024 |
| 456 | 2^1650426+183543 | 496828 | Stefano Morozzi | 01/2024 |
| 457 | 2^1650388+189741 | 496817 | Stefano Morozzi | 10/2023 |
| 458 | 2^1650267+193743 | 496780 | Stefano Morozzi | 09/2023 |
| 459 | 2^1650196+191835 | 496759 | Stefano Morozzi | 07/2023 |
| 460 | 2^1650190+185733 | 496757 | Stefano Morozzi | 07/2023 |
| 461 | 2^1650141+189525 | 496742 | Stefano Morozzi | 07/2023 |
| 462 | 2^1650103+189519 | 496731 | Stefano Morozzi | 05/2023 |
| 463 | 2^1648342+215029 | 496201 | Enzo Doria | 01/2024 |
| 464 | 2^1646078+36937 | 495519 | Enzo Doria | 10/2024 |
| 465 | 2^1645332+34951 | 495295 | Enzo Doria | 12/2024 |
| 466 | (136688^96721-1)/(136688^311-1) | 495136 | Ryan Propper | 11/2023 |
| 467 | 2^1643411+26213 | 494717 | Enzo Doria | 03/2024 |
| 468 | 2^1642104+227491 | 494323 | Enzo Doria | 10/2024 |
| 469 | 2^1640056+216241 | 493707 | Enzo Doria | 02/2024 |
| 470 | 2^1636027+25763 | 492494 | Enzo Doria | 01/2024 |
| 471 | 2^1635235+227693 | 492255 | Enzo Doria | 10/2024 |
| 472 | 2^1632498+30397 | 491431 | Enzo Doria | 08/2024 |
| 473 | 2^1631030+73519 | 490989 | Enzo Doria | 08/2023 |
| 474 | (2^1629469-1)/644908484660139264379223 | 490496 | Gord Palameta | 11/2017 |
| 475 | 2^1627834+225529 | 490027 | Enzo Doria | 08/2024 |
| 476 | 2^1626709+30407 | 489689 | Enzo Doria | 07/2024 |
| 477 | 2^1626438+38557 | 489607 | Enzo Doria | 11/2024 |
| 478 | 2^1621040+39415 | 487982 | Enzo Doria | 11/2024 |
| 479 | 2^1619885+28229 | 487634 | Enzo Doria | 04/2024 |
| 480 | 10^487291+7 | 487292 | Alfred Reich | 06/2021 |
| 481 | 2^1617677+227927 | 486970 | Enzo Doria | 10/2024 |
| 482 | 2^1617579+1889543 | 486940 | Enzo Doria | 09/2017 |
| 483 | 97481^97481-70^70 | 486325 | Henri Lifchitz | 10/2024 |
| 484 | 2^1613632+215011 | 485752 | Enzo Doria | 11/2023 |
| 485 | 2^1612600+27331 | 485441 | Enzo Doria | 03/2024 |
| 486 | 2^1610146+24739 | 484703 | Enzo Doria | 11/2023 |
| 487 | 2^1609360+43711 | 484466 | Enzo Doria | 03/2025 |
| 488 | L(2316773) | 484177 | Serge Batalov | 06/2017 |
| 489 | (57^275669-1)/56 | 484039 | Paul Bourdelais | 04/2021 |
| 490 | 2^1602171+40943 | 482302 | Enzo Doria | 01/2025 |
| 491 | 2^1601964+215491 | 482240 | Enzo Doria | 11/2023 |
| 492 | 96672^96672-1825^1825 | 481939 | Henri Lifchitz | 12/2024 |
| 493 | 2^1600308+34051 | 481741 | Enzo Doria | 09/2024 |
| 494 | 2^1599624+20011 | 481535 | Enzo Doria | 07/2023 |
| 495 | 2^1599205+27047 | 481409 | Enzo Doria | 03/2024 |
| 496 | 2^1599120-5 | 481384 | Stefano Morozzi | 12/2022 |
| 497 | 10^480684-3 | 480684 | Paul Bourdelais | 01/2021 |
| 498 | 2^1595483+184133 | 480289 | Enzo Doria | 07/2017 |
| 499 | 2^1594947+3 | 480127 | Paul Bourdelais | 07/2019 |
| 500 | 2^1594420+36781 | 479969 | Enzo Doria | 10/2024 |