What is a PRP ?
A PRP is a probable prime number, a number that nobody knows how to prove
or disprove its primality.
Here are some criterions that can be used to recognize these numbers
:
- 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 ? |
8501 | 2^341246+40755 | 102726 | Lelio R Paula | 11/2016 |
8502 | (2^341256-5)/991699 | 102723 | Lelio R Paula | 12/2013 |
8503 | 7^121547+78 | 102720 | Lelio R Paula | 10/2016 |
8504 | 2^341223-461197 | 102719 | Enzo Doria | 06/2015 |
8505 | 27838^4893+4893^27838 | 102711 | Hans Havermann | 03/2019 |
8507 | 2^341190-20577 | 102709 | Enzo Doria | 06/2013 |
8506 | (27*2^341212+1)/84674449 | 102709 | Lelio R Paula | 12/2013 |
8508 | (25*10^102698+77)/3 | 102699 | Robert Price | 09/2022 |
8509 | 6^131976-65 | 102698 | Lelio R Paula | 10/2016 |
8510 | 2^341134+2533 | 102692 | Henri Lifchitz | 09/2005 |
8511 | 13^92184-54 | 102688 | Lelio R Paula | 10/2016 |
8512 | 2^341116-20873 | 102687 | Enzo Doria | 06/2013 |
8513 | 2^341094-19953 | 102680 | Enzo Doria | 06/2013 |
8514 | 23761^20950+20950^23761 | 102676 | Hans Havermann | 03/2019 |
8516 | 2^341053+129 | 102668 | Lelio R Paula | 08/2011 |
8515 | (4724^27943-1)/4723 | 102668 | Serge Batalov | 06/2022 |
8518 | (31*2^341006-1)/3 | 102655 | Lelio R Paula | 03/2014 |
8517 | 7^121471+94 | 102655 | Lelio R Paula | 10/2016 |
8520 | 2^341004-173 | 102653 | Lelio R Paula | 08/2011 |
8519 | 20^78901+3 | 102653 | Lelio R Paula | 10/2016 |
8521 | (2^341001+1674)/2 | 102652 | Stefano Morozzi | 09/2020 |
8522 | 5^146853-52 | 102646 | Lelio R Paula | 10/2016 |
8523 | 2^340969-216931 | 102642 | Enzo Doria | 05/2014 |
8524 | 122^49196+121 | 102641 | Norbert Schneider | 10/2018 |
8525 | 2^340952+1870021 | 102637 | Enzo Doria | 05/2016 |
8526 | 5*2^340940-3 | 102634 | Lelio R Paula | 08/2014 |
8527 | (6^131891+1)/7 | 102631 | Paul Bourdelais | 01/2010 |
8528 | (11*10^102629-23)/3 | 102630 | Robert Price | 09/2021 |
8529 | 3^215094+5*2^215094 | 102626 | Jon Grantham | 08/2023 |
8530 | 25876^9239+9239^25876 | 102615 | Hans Havermann | 02/2019 |
8531 | (4*10^102612+83)/3 | 102613 | Robert Price | 02/2018 |
8532 | 75^54717+74 | 102598 | Norbert Schneider | 01/2015 |
8533 | (17*10^102597+13)/3 | 102598 | Robert Price | 01/2019 |
8534 | 2^340806-20393 | 102593 | Enzo Doria | 06/2013 |
8535 | (2^340790+5)/138267 | 102583 | Lelio R Paula | 12/2013 |
8536 | 6^131821+77 | 102577 | Lelio R Paula | 10/2016 |
8537 | (203*10^102561-11)/3 | 102563 | Robert Price | 12/2020 |
8538 | 2^340688-29 | 102558 | Henri Lifchitz | 05/2005 |
8539 | 2^340685-109101 | 102557 | Enzo Doria | 04/2014 |
8540 | 65^56567-64 | 102552 | Norbert Schneider | 11/2014 |
8541 | 26230^8121+8121^26230 | 102550 | Hans Havermann | 03/2019 |
8542 | (64*10^102547+161)/9 | 102548 | Robert Price | 08/2022 |
8543 | (8*10^102541+49)/3 | 102542 | Robert Price | 05/2020 |
8544 | (536*10^102533-23)/9 | 102535 | Robert Price | 02/2020 |
8545 | 2^340547-108897 | 102515 | Enzo Doria | 03/2014 |
8546 | (2^340536+778)/2 | 102512 | Stefano Morozzi | 09/2020 |
8547 | 25592^10119+10119^25592 | 102500 | Hans Havermann | 02/2019 |
8548 | (5*10^102492-23)/9 | 102492 | Serge Batalov | 11/2022 |
8549 | 7^121268+82 | 102484 | Lelio R Paula | 10/2016 |
8550 | (6883*2^340443-1)/162401 | 102483 | Lelio R Paula | 05/2015 |
8551 | 2^340434-5955 | 102481 | Lelio R Paula | 11/2016 |
8552 | (7*10^102480-19)/3 | 102481 | Robert Price | 10/2020 |
8553 | (2^340381+47)/7 | 102465 | Lelio R Paula | 12/2013 |
8554 | 111^50095+110 | 102461 | Norbert Schneider | 04/2017 |
8555 | 180^45427-45427^180 | 102451 | Norman Luhn | 09/2024 |
8556 | 2^340291-20301 | 102438 | Enzo Doria | 06/2013 |
8557 | 12^94915+89 | 102431 | Lelio R Paula | 10/2016 |
8558 | (106*10^102422+17)/3 | 102424 | Robert Price | 04/2021 |
8559 | 24466^15349+15349^24466 | 102417 | Hans Havermann | 03/2019 |
8560 | 2^340212+111546435 | 102415 | Paul Bourdelais | 10/2021 |
8561 | 2^340208-20373 | 102413 | Enzo Doria | 06/2013 |
8562 | 6^131598-17273 | 102404 | Enzo Doria | 02/2013 |
8563 | (2^340132+1182)/2 | 102390 | Stefano Morozzi | 09/2020 |
8564 | (2^340133+1166)/2 | 102390 | Stefano Morozzi | 09/2020 |
8565 | 2^340115-21181 | 102385 | Enzo Doria | 06/2013 |
8566 | (55459*2^340090+1)/1591 | 102379 | Lelio R Paula | 07/2014 |
8567 | (2^340064+3630)/2 | 102370 | Stefano Morozzi | 09/2020 |
8568 | 9348*31^68638+13 | 102369 | Eric Chen | 04/2024 |
8569 | 89^52511-88^52511 | 102365 | Robert Price | 06/2015 |
8570 | 3^214539-112756 | 102362 | Enzo Doria | 07/2014 |
8571 | 23159^23450-23450^23159 | 102353 | Norbert Schneider | 12/2019 |
8572 | 2^340000-19877 | 102351 | Enzo Doria | 06/2013 |
8573 | 2^339990-114845 | 102348 | Enzo Doria | 05/2015 |
8574 | 311^41058-41058^311 | 102348 | Norman Luhn | 09/2024 |
8575 | (3^214516-2)/51383 | 102346 | Lelio R Paula | 12/2013 |
8576 | 20^78661-99 | 102341 | Lelio R Paula | 09/2016 |
8578 | 2^339951-19585 | 102336 | Enzo Doria | 05/2013 |
8577 | (17*10^102335-167)/3 | 102336 | Robert Price | 05/2022 |
8579 | (13*2^339943+1)/45 | 102333 | Lelio R Paula | 12/2013 |
8580 | (236282#+121441)/(121441*25499429) | 102332 | Makoto Morimoto | 08/2020 |
8581 | 5*10^102323-91 | 102324 | Robert Price | 03/2021 |
8583 | 2^339889+29327 | 102317 | Yves Minet | 06/2010 |
8582 | (167*10^102315-17)/3 | 102317 | Robert Price | 03/2020 |
8584 | 26348^7641+7641^26348 | 102314 | Hans Havermann | 03/2019 |
8585 | Phi(239414,10)/6333218543 | 102303 | Makoto Kamada | 08/2019 |
8586 | 2^339834-109493 | 102301 | Enzo Doria | 04/2014 |
8587 | (16*10^102299-19)/3 | 102300 | Robert Price | 02/2019 |
8588 | (11^98221-5^98221)/6 | 102286 | Robert Price | 01/2013 |
8589 | 3^214365+1820 | 102279 | Lelio R Paula | 10/2016 |
8590 | 31978^1579+1579^31978 | 102278 | Hans Havermann | 03/2019 |
8591 | (89*10^102262-467)/9 | 102263 | Robert Price | 04/2022 |
8592 | (11*2^339705-1)/183 | 102261 | Lelio R Paula | 03/2014 |
8593 | 2^339694-16995 | 102259 | Lelio R Paula | 09/2016 |
8594 | (56*10^102255+691)/9 | 102256 | Robert Price | 05/2019 |
8595 | 122^49008+121 | 102249 | Norbert Schneider | 10/2018 |
8596 | (2^339652+3178)/2 | 102246 | Stefano Morozzi | 09/2020 |
8597 | 7^120985-38 | 102245 | Lelio R Paula | 10/2016 |
8598 | 10^102244+122961 | 102245 | Patrick De Geest | 12/2016 |
8599 | 10^102244-27 | 102244 | Lelio R Paula | 09/2016 |
8600 | 7^120982-62 | 102242 | Lelio R Paula | 10/2016 |
8601 | (2^339640+2670)/2 | 102242 | Stefano Morozzi | 09/2020 |
8602 | 4*6^131382-5 | 102236 | Lelio R Paula | 11/2014 |
8603 | (2^339607+3290)/2 | 102232 | Stefano Morozzi | 02/2021 |
8604 | 2^339586-1091 | 102226 | Yves Minet | 11/2012 |
8605 | 11^98161-3072000 | 102225 | Enzo Doria | 12/2012 |
8606 | 9*7^120961-4 | 102225 | Lelio R Paula | 11/2014 |
8607 | 5*3^214250+2 | 102224 | Serge Batalov | 10/2010 |
8608 | 12^94713-59 | 102213 | Lelio R Paula | 10/2016 |
8609 | 12^94705+89 | 102204 | Lelio R Paula | 10/2016 |
8610 | 2^339407-177 | 102172 | Lelio R Paula | 08/2011 |
8611 | 4*10^102163-57 | 102164 | Robert Price | 10/2018 |
8612 | 2^339355+65 | 102157 | Lelio R Paula | 07/2011 |
8613 | (28*10^102142-31)/3 | 102143 | Robert Price | 12/2019 |
8614 | (2^339323+47)/880715 | 102141 | Lelio R Paula | 12/2013 |
8615 | (2^339268+1758)/2 | 102130 | Stefano Morozzi | 09/2020 |
8616 | U(610865) | 102130 | Serge Batalov | 07/2024 |
8617 | s(178551) | 102122 | Robert Price | 05/2020 |
8618 | (797*10^102112-23)/9 | 102114 | Robert Price | 01/2022 |
8619 | (5*10^102107+7)/3 | 102108 | Tyler Busby | 02/2023 |
8620 | (2^339175+2790)/2 | 102102 | Stefano Morozzi | 02/2021 |
8622 | 2^339168+21 | 102100 | Lelio R Paula | 07/2011 |
8623 | 2^339167-108807 | 102100 | Enzo Doria | 03/2014 |
8621 | (2^339169+42)/2 | 102100 | Stefano Morozzi | 09/2020 |
8624 | 2^339156+163 | 102097 | Lelio R Paula | 08/2011 |
8625 | (2^339149+746)/2 | 102094 | Stefano Morozzi | 09/2020 |
8626 | (25*10^102076-67)/3 | 102077 | Robert Price | 06/2019 |
8627 | 2^339081-4543 | 102074 | Yves Minet | 03/2019 |
8628 | (3435^28867-1)/3434 | 102069 | Serge Batalov | 06/2022 |
8629 | 2*10^102062+77 | 102063 | Robert Price | 08/2020 |
8631 | 24327^15668+15668^24327 | 102053 | Hans Havermann | 02/2019 |
8630 | 10^102052+66793 | 102053 | Patrick De Geest | 09/2022 |
8632 | 10^102052-7000001 | 102052 | Robert Price | 06/2021 |
8633 | 10^102052-618623 | 102052 | Patrick De Geest | 09/2022 |
8634 | 2^339000-175635 | 102050 | Stefano Morozzi | 12/2021 |
8635 | 2^339000-151667 | 102050 | Stefano Morozzi | 12/2021 |
8636 | 2^338999+79289 | 102049 | Stefano Morozzi | 01/2022 |
8637 | 2^338998-63897 | 102049 | Stefano Morozzi | 12/2022 |
8638 | 2^338997+99659 | 102049 | Stefano Morozzi | 02/2023 |
8639 | 2^338994+188757 | 102048 | Stefano Morozzi | 05/2022 |
8640 | 2^338995+70785 | 102048 | Stefano Morozzi | 11/2022 |
8641 | 2^338994-12183 | 102048 | Stefano Morozzi | 12/2022 |
8642 | 2^338992+181591 | 102047 | Stefano Morozzi | 01/2022 |
8643 | 2^338992+136171 | 102047 | Stefano Morozzi | 03/2022 |
8644 | 2^338991-109435 | 102047 | Stefano Morozzi | 04/2022 |
8645 | 2^338991+193611 | 102047 | Stefano Morozzi | 09/2022 |
8646 | 2^338988-134619 | 102046 | Stefano Morozzi | 04/2022 |
8647 | 2^338987-62557 | 102046 | Stefano Morozzi | 12/2022 |
8648 | 2^338986-60141 | 102045 | Stefano Morozzi | 06/2022 |
8649 | 2^338986+98209 | 102045 | Stefano Morozzi | 11/2022 |
8650 | 2^338981+106391 | 102044 | Stefano Morozzi | 01/2022 |
8651 | 2^338982+34069 | 102044 | Stefano Morozzi | 03/2022 |
8652 | 2^338982-58781 | 102044 | Stefano Morozzi | 04/2022 |
8653 | 2^338980+70833 | 102044 | Stefano Morozzi | 11/2022 |
8654 | 2^338982-92111 | 102044 | Stefano Morozzi | 02/2023 |
8655 | 2^338977+83007 | 102043 | Stefano Morozzi | 02/2022 |
8656 | 2^338979-83601 | 102043 | Stefano Morozzi | 04/2022 |
8657 | 2^338975+28355 | 102042 | Stefano Morozzi | 01/2022 |
8658 | 2^338973+134619 | 102042 | Stefano Morozzi | 03/2022 |
8659 | 2^338974+183777 | 102042 | Stefano Morozzi | 03/2022 |
8660 | 2^338975+69293 | 102042 | Stefano Morozzi | 09/2022 |
8668 | 25080^11711+11711^25080 | 102041 | Hans Havermann | 02/2019 |
8661 | 2^338972+152343 | 102041 | Stefano Morozzi | 12/2021 |
8662 | 2^338972-180623 | 102041 | Stefano Morozzi | 02/2022 |
8663 | 2^338972+84441 | 102041 | Stefano Morozzi | 03/2022 |
8664 | 2^338971+64875 | 102041 | Stefano Morozzi | 05/2022 |
8665 | 2^338971+44985 | 102041 | Stefano Morozzi | 11/2022 |
8666 | 2^338971+145065 | 102041 | Stefano Morozzi | 11/2022 |
8667 | 2^338972-16635 | 102041 | Stefano Morozzi | 01/2023 |
8669 | 2^338968+53461 | 102040 | Stefano Morozzi | 01/2022 |
8670 | 2^338969+160175 | 102040 | Stefano Morozzi | 03/2022 |
8671 | 2^338967+13409 | 102040 | Stefano Morozzi | 04/2022 |
8672 | 2^338969+67635 | 102040 | Stefano Morozzi | 09/2022 |
8673 | 2^338969-15913 | 102040 | Stefano Morozzi | 01/2023 |
8674 | 2^338963-127455 | 102039 | Stefano Morozzi | 12/2021 |
8675 | 2^338963+132195 | 102039 | Stefano Morozzi | 02/2022 |
8676 | 2^338966+137919 | 102039 | Stefano Morozzi | 04/2022 |
8677 | 2^338963-87529 | 102039 | Stefano Morozzi | 12/2022 |
8678 | 2^338963-116859 | 102039 | Stefano Morozzi | 02/2023 |
8685 | (2^338982-7)/1754517 | 102038 | Lelio R Paula | 12/2013 |
8679 | 2^338960+53935 | 102038 | Stefano Morozzi | 01/2022 |
8680 | 2^338962+83157 | 102038 | Stefano Morozzi | 02/2022 |
8681 | 2^338961+12341 | 102038 | Stefano Morozzi | 04/2022 |
8682 | 2^338960+63933 | 102038 | Stefano Morozzi | 05/2022 |
8683 | 2^338960+170791 | 102038 | Stefano Morozzi | 11/2022 |
8684 | 2^338962+21679 | 102038 | Stefano Morozzi | 11/2022 |
8686 | 2^338959-30919 | 102037 | Stefano Morozzi | 02/2022 |
8687 | 2^338959+164769 | 102037 | Stefano Morozzi | 05/2022 |
8688 | 2^338958-43161 | 102037 | Stefano Morozzi | 02/2023 |
8689 | 2^338954-177003 | 102036 | Stefano Morozzi | 12/2021 |
8690 | 2^338954+154207 | 102036 | Stefano Morozzi | 01/2022 |
8691 | 2^338956+182493 | 102036 | Stefano Morozzi | 02/2022 |
8692 | 2^338954+162715 | 102036 | Stefano Morozzi | 04/2022 |
8693 | 2^338955-158241 | 102036 | Stefano Morozzi | 04/2022 |
8694 | 2^338956-33395 | 102036 | Stefano Morozzi | 04/2022 |
8695 | 2^338954+190915 | 102036 | Stefano Morozzi | 06/2022 |
8696 | 2^338955+41643 | 102036 | Stefano Morozzi | 09/2022 |
8697 | 2^338954+194427 | 102036 | Stefano Morozzi | 09/2022 |
8698 | 2^338956+117967 | 102036 | Stefano Morozzi | 09/2022 |
8699 | 2^338956+195495 | 102036 | Stefano Morozzi | 11/2022 |
8700 | 2^338954-186371 | 102036 | Stefano Morozzi | 12/2022 |
8701 | 2^338950+175077 | 102035 | Stefano Morozzi | 12/2021 |
8702 | 2^338951-51687 | 102035 | Stefano Morozzi | 12/2021 |
8703 | 2^338951+66123 | 102035 | Stefano Morozzi | 06/2022 |
8704 | 2^338950+19317 | 102035 | Stefano Morozzi | 09/2022 |
8705 | 2^338951-161479 | 102035 | Stefano Morozzi | 12/2022 |
8706 | 2^338950-66335 | 102035 | Stefano Morozzi | 01/2023 |
8707 | 2^338951-41611 | 102035 | Stefano Morozzi | 02/2023 |
8708 | 2^338952-142887 | 102035 | Stefano Morozzi | 02/2023 |
8709 | 2^338949+85449 | 102034 | Stefano Morozzi | 03/2022 |
8710 | 2^338949-184585 | 102034 | Stefano Morozzi | 04/2022 |
8711 | 2^338949-88555 | 102034 | Stefano Morozzi | 12/2022 |
8712 | 2^338948+49393 | 102034 | Stefano Morozzi | 02/2023 |
8713 | 2^338949-17601 | 102034 | Stefano Morozzi | 02/2023 |
8714 | 2^338943+127103 | 102033 | Stefano Morozzi | 12/2021 |
8715 | 2^338944-84003 | 102033 | Stefano Morozzi | 04/2022 |
8716 | 2^338945+138605 | 102033 | Stefano Morozzi | 05/2022 |
8717 | 2^338943+95133 | 102033 | Stefano Morozzi | 11/2022 |
8718 | 2^338943+147435 | 102033 | Stefano Morozzi | 11/2022 |
8719 | 2^338943-116067 | 102033 | Stefano Morozzi | 01/2023 |
8720 | 2^338940+78607 | 102032 | Stefano Morozzi | 01/2022 |
8721 | 2^338942-86687 | 102032 | Stefano Morozzi | 12/2022 |
8722 | 2^338939+151569 | 102031 | Stefano Morozzi | 12/2021 |
8723 | 2^338938+183975 | 102031 | Stefano Morozzi | 03/2022 |
8724 | 2^338937-33429 | 102031 | Stefano Morozzi | 04/2022 |
8725 | 2^338939-183259 | 102031 | Stefano Morozzi | 04/2022 |
8726 | 2^338937-139951 | 102031 | Stefano Morozzi | 01/2023 |
8727 | 2^338938-40151 | 102031 | Stefano Morozzi | 01/2023 |
8728 | 2^338938+74449 | 102031 | Stefano Morozzi | 02/2023 |
8729 | 2^338935+179999 | 102030 | Stefano Morozzi | 01/2022 |
8730 | 2^338936+84147 | 102030 | Stefano Morozzi | 03/2022 |
8731 | 2^338935+171261 | 102030 | Stefano Morozzi | 11/2022 |
8732 | 2^338936-41355 | 102030 | Stefano Morozzi | 01/2023 |
8733 | 2^338931+2349 | 102029 | Stefano Morozzi | 12/2021 |
8734 | 2^338933+79935 | 102029 | Stefano Morozzi | 01/2022 |
8735 | 2^338930+33819 | 102029 | Stefano Morozzi | 03/2022 |
8736 | 2^338931-31555 | 102029 | Stefano Morozzi | 03/2022 |
8737 | 2^338933+64169 | 102029 | Stefano Morozzi | 05/2022 |
8738 | 2^338932-184965 | 102029 | Stefano Morozzi | 06/2022 |
8739 | 2^338932+66943 | 102029 | Stefano Morozzi | 09/2022 |
8740 | 2^338932+92995 | 102029 | Stefano Morozzi | 09/2022 |
8741 | 2^338931+97001 | 102029 | Stefano Morozzi | 11/2022 |
8742 | 2^338932-165453 | 102029 | Stefano Morozzi | 01/2023 |
8743 | 2^338933-40899 | 102029 | Stefano Morozzi | 01/2023 |
8744 | 2^338928+2367 | 102028 | Stefano Morozzi | 12/2021 |
8745 | 2^338927+52779 | 102028 | Stefano Morozzi | 01/2022 |
8746 | 2^338927-79539 | 102028 | Stefano Morozzi | 02/2022 |
8747 | 2^338929+60819 | 102028 | Stefano Morozzi | 03/2022 |
8748 | 2^338928-156959 | 102028 | Stefano Morozzi | 03/2022 |
8749 | 2^338927-63919 | 102028 | Stefano Morozzi | 12/2022 |
8750 | 2^338925+106631 | 102027 | Stefano Morozzi | 01/2022 |