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 | 22619^21286+21286^22619 | 97898 | Hans Havermann | 04/2020 |
8502 | (85*10^97876-49)/9 | 97877 | Robert Price | 12/2020 |
8503 | 7*10^97874-69 | 97875 | Robert Price | 01/2017 |
8504 | 5*10^97869-63 | 97870 | Robert Price | 04/2021 |
8506 | (17^79537+5^79537)/22 | 97865 | Robert Price | 06/2013 |
8505 | 15^83212+22 | 97865 | Lelio R Paula | 08/2016 |
8507 | 2^325087-109057 | 97861 | Enzo Doria | 04/2014 |
8508 | 2^325074+1870657 | 97858 | Enzo Doria | 05/2016 |
8510 | 2^325066-2111 | 97855 | Yves Minet | 08/2013 |
8509 | CopelandErdos(97855) | 97855 | Eric W. Weisstein | 10/2015 |
8511 | (8694^24841-1)/8693 | 97851 | Serge Batalov | 06/2022 |
8512 | (21159^22621-1)/21158 | 97843 | Serge Batalov | 06/2022 |
8513 | 26660^4677+4677^26660 | 97842 | Hans Havermann | 12/2019 |
8514 | 18^77944-89 | 97841 | Lelio R Paula | 07/2016 |
8515 | 119^47136+118 | 97833 | Norbert Schneider | 09/2018 |
8516 | 6^125720-17663 | 97830 | Enzo Doria | 02/2013 |
8517 | (73640^20101-1)/73639 | 97829 | Serge Batalov | 06/2022 |
8518 | 2^324972-2069 | 97827 | Yves Minet | 05/2013 |
8519 | 2^324958-4691 | 97823 | Yves Minet | 05/2019 |
8520 | (8087^25033-1)/8086 | 97820 | Serge Batalov | 06/2022 |
8521 | 2^324944-2788319 | 97818 | Enzo Doria | 03/2016 |
8522 | 22845^19112+19112^22845 | 97807 | Hans Havermann | 10/2019 |
8523 | (2^324898+1270)/2 | 97804 | Stefano Morozzi | 09/2020 |
8525 | 4*10^97799-9 | 97800 | Erik Branger | 11/2013 |
8524 | 2^324882+577294575 | 97800 | Lelio R Paula | 09/2016 |
8526 | 2^324863-1867921 | 97794 | Enzo Doria | 04/2016 |
8527 | 6^125662-18473 | 97785 | Enzo Doria | 03/2013 |
8528 | 12^90604+11 | 97779 | Lelio R Paula | 07/2016 |
8529 | (29*10^97751+49)/3 | 97752 | Robert Price | 09/2021 |
8530 | (37108^21391-1)/37107 | 97741 | Serge Batalov | 06/2022 |
8531 | 5^139830+24 | 97737 | Lelio R Paula | 04/2016 |
8532 | (2^324672+1198)/2 | 97736 | Stefano Morozzi | 09/2020 |
8533 | (101*10^97724+7)/9 | 97726 | Robert Price | 02/2022 |
8534 | 2^324612+1155 | 97718 | Lelio R Paula | 07/2016 |
8535 | (2^324597+3542)/2 | 97714 | Stefano Morozzi | 09/2020 |
8536 | 19^76404-44 | 97702 | Lelio R Paula | 07/2016 |
8537 | (5939^25889-1)/5938 | 97694 | Serge Batalov | 06/2022 |
8539 | 2^324505-2983 | 97686 | Yves Minet | 04/2011 |
8538 | (7740^25121-1)/7739 | 97686 | Serge Batalov | 06/2022 |
8540 | (144^45259+1)/145 | 97684 | Paul Bourdelais | 08/2019 |
8541 | (31*2^324496-1)/45 | 97683 | Lelio R Paula | 02/2014 |
8542 | (11234^24109-1)/11233 | 97651 | Serge Batalov | 06/2022 |
8543 | 6^125488-17843 | 97649 | Enzo Doria | 02/2013 |
8544 | (2^324350+3198)/2 | 97639 | Stefano Morozzi | 09/2020 |
8545 | (75638^20011-1)/75637 | 97624 | Serge Batalov | 06/2022 |
8546 | 7^115515-2 | 97622 | Lelio R Paula | 09/2013 |
8547 | (15672^23269-1)/15671 | 97613 | Serge Batalov | 06/2022 |
8548 | (89^50069-1)/88 | 97603 | Paul Bourdelais | 08/2017 |
8549 | 6^125418-5 | 97595 | Lelio R Paula | 11/2014 |
8550 | (2^324193+3522)/2 | 97592 | Stefano Morozzi | 09/2020 |
8551 | (2422^28837-1)/2421 | 97587 | Serge Batalov | 06/2022 |
8552 | (2^324172+666)/2 | 97586 | Stefano Morozzi | 09/2020 |
8553 | 2^324091+111546435 | 97562 | Paul Bourdelais | 10/2021 |
8554 | 2^324076+1155 | 97557 | Lelio R Paula | 07/2016 |
8555 | (2^324074+9)/13 | 97555 | Lelio R Paula | 10/2013 |
8556 | (11*10^97539-71)/3 | 97540 | Robert Price | 04/2022 |
8557 | (55459*2^323993+1)/21 | 97536 | Lelio R Paula | 07/2014 |
8558 | 13^87557-12^87557 | 97534 | Jean-Louis Charton | 12/2008 |
8559 | 2^323946+73939 | 97518 | Enzo Doria | 04/2016 |
8560 | 2^323937+1866947 | 97515 | Enzo Doria | 03/2016 |
8561 | (98339^19531-1)/98338 | 97508 | Serge Batalov | 06/2022 |
8563 | 6^125290-18533 | 97495 | Enzo Doria | 03/2013 |
8562 | (2^323870+1650)/2 | 97495 | Stefano Morozzi | 09/2020 |
8564 | 2^323845+101 | 97488 | Lelio R Paula | 12/2012 |
8565 | 18^77662+65 | 97487 | Lelio R Paula | 07/2016 |
8566 | 22634^20265+20265^22634 | 97479 | Hans Havermann | 04/2020 |
8567 | (5533^26041-1)/5532 | 97467 | Serge Batalov | 06/2022 |
8568 | 2^323772+76771 | 97466 | Enzo Doria | 05/2016 |
8569 | 2^323765-112213 | 97463 | Enzo Doria | 06/2014 |
8570 | 22730^19389+19389^22730 | 97457 | Hans Havermann | 02/2020 |
8571 | 2^323720-6555 | 97450 | Lelio R Paula | 07/2016 |
8572 | 2^323694-1860401 | 97442 | Enzo Doria | 03/2016 |
8573 | 20^74886+3 | 97429 | Lelio R Paula | 06/2016 |
8574 | 24476^9561+9561^24476 | 97427 | Hans Havermann | 01/2020 |
8575 | (2^323623+1766)/2 | 97420 | Stefano Morozzi | 09/2020 |
8576 | (3250^27739-1)/3249 | 97413 | Serge Batalov | 06/2022 |
8578 | 3*10^97399+73 | 97400 | Robert Price | 04/2016 |
8577 | 7^115252-50 | 97400 | Lelio R Paula | 07/2016 |
8579 | (4715^26513-1)/4714 | 97392 | Serge Batalov | 06/2022 |
8580 | 224737#+393581 | 97389 | Nicolas PALAU | 12/2016 |
8581 | (2^323468+3186)/2 | 97374 | Stefano Morozzi | 09/2020 |
8582 | (2^323456+538)/2 | 97370 | Stefano Morozzi | 09/2020 |
8583 | 23277^15236+15236^23277 | 97365 | Hans Havermann | 07/2020 |
8585 | (26*10^97353+31)/3 | 97354 | Robert Price | 07/2016 |
8584 | (2^323403+3510)/2 | 97354 | Stefano Morozzi | 09/2020 |
8586 | 23756^12519+12519^23756 | 97342 | Hans Havermann | 07/2020 |
8587 | 10^97335+271641 | 97336 | Patrick De Geest | 09/2022 |
8589 | 10^97335-4000001 | 97335 | Robert Price | 07/2021 |
8588 | 10^97335-98193 | 97335 | Patrick De Geest | 09/2022 |
8590 | (637*10^97331+11)/9 | 97333 | Robert Price | 10/2021 |
8591 | 23991^11402+11402^23991 | 97332 | Hans Havermann | 07/2020 |
8592 | 61^54517-60^54517 | 97331 | Jean-Louis Charton | 09/2005 |
8593 | (38*10^97329+187)/9 | 97330 | Robert Price | 11/2017 |
8594 | (31147^21661-1)/31146 | 97328 | Serge Batalov | 06/2022 |
8595 | 86455!4+8 | 97319 | Robert Price | 09/2019 |
8596 | 22591^20310+20310^22591 | 97316 | Hans Havermann | 06/2020 |
8597 | 2^323261-431173 | 97312 | Harrison Mesh | 06/2022 |
8598 | (3996^27017-1)/3995 | 97302 | Serge Batalov | 06/2022 |
8599 | 25101^7520+7520^25101 | 97297 | Hans Havermann | 05/2020 |
8600 | (233*10^97294+7)/3 | 97296 | Robert Price | 03/2022 |
8601 | p(19982)#/2-1024 | 97293 | Robert Price | 09/2017 |
8602 | (65*10^97288+277)/9 | 97289 | Robert Price | 08/2021 |
8603 | 6^125021-85 | 97286 | Lelio R Paula | 08/2016 |
8604 | 2^323164-47355 | 97283 | Lelio R Paula | 07/2016 |
8605 | (3^203902+4)/36051015157 | 97276 | Lelio R Paula | 09/2013 |
8606 | 2^323054-6555 | 97249 | Lelio R Paula | 07/2016 |
8607 | 22822^18247+18247^22822 | 97249 | Hans Havermann | 02/2020 |
8608 | 9*10^97247-29 | 97248 | Robert Price | 05/2017 |
8609 | 24382^9735+9735^24382 | 97244 | Hans Havermann | 02/2020 |
8610 | (31*10^97242+41)/9 | 97243 | Robert Price | 04/2015 |
8611 | GoldenRatioPrime[97241] | 97241 | Eric W. Weisstein | 06/2009 |
8612 | 12^90098-41 | 97233 | Lelio R Paula | 07/2016 |
8613 | 2^322928-216929 | 97212 | Enzo Doria | 05/2014 |
8614 | (16731^23017-1)/16730 | 97209 | Serge Batalov | 06/2022 |
8615 | 3*10^97205-61 | 97206 | Robert Price | 04/2021 |
8616 | (7*2^322896-1)/57 | 97201 | Lelio R Paula | 02/2014 |
8617 | (35*10^97194-341)/9 | 97195 | Robert Price | 08/2021 |
8618 | (4872^26357-1)/4871 | 97194 | Serge Batalov | 06/2022 |
8619 | (2^322897+5)/3146107489 | 97193 | Lelio R Paula | 10/2013 |
8620 | (796*10^97183-31)/9 | 97185 | Robert Price | 07/2022 |
8621 | (2^322789+2214)/2 | 97169 | Stefano Morozzi | 09/2020 |
8622 | 2^322725+75761 | 97150 | Enzo Doria | 04/2016 |
8623 | (13*2^322686+1)/119 | 97138 | Lelio R Paula | 11/2013 |
8624 | (2175^29101-1)/2174 | 97121 | Serge Batalov | 06/2022 |
8625 | 22402^21639+21639^22402 | 97118 | Hans Havermann | 07/2020 |
8626 | 23399^14128+14128^23399 | 97108 | Hans Havermann | 06/2020 |
8627 | (2892^28057-1)/2891 | 97108 | Serge Batalov | 06/2022 |
8628 | (9578^24391-1)/9577 | 97104 | Serge Batalov | 06/2022 |
8629 | (2^322525+2606)/2 | 97090 | Stefano Morozzi | 09/2020 |
8630 | (9563^24391-1)/9562 | 97087 | Serge Batalov | 06/2022 |
8631 | (41*10^97085+373)/9 | 97086 | Robert Price | 04/2017 |
8632 | (2^322473+854)/2 | 97074 | Stefano Morozzi | 02/2021 |
8633 | 2^322413+77 | 97056 | Lelio R Paula | 12/2012 |
8634 | 2^322406+177 | 97054 | Lelio R Paula | 12/2012 |
8635 | (349^38167-1)/348 | 97050 | Serge Batalov | 06/2022 |
8636 | (14*10^97044+31)/9 | 97045 | Robert Price | 02/2015 |
8637 | 2^322302+255255 | 97023 | Lelio R Paula | 07/2016 |
8638 | (1302^31147-1)/1301 | 97008 | Serge Batalov | 06/2022 |
8639 | 2^322196-113093 | 96991 | Enzo Doria | 07/2014 |
8640 | (2^322169+2690)/2 | 96983 | Stefano Morozzi | 09/2020 |
8641 | (49*10^96980+221)/9 | 96981 | Robert Price | 02/2022 |
8642 | 9*10^96968-47 | 96969 | Robert Price | 11/2021 |
8643 | (5895^25717-1)/5894 | 96962 | Serge Batalov | 06/2022 |
8644 | (27473^21841-1)/27472 | 96946 | Serge Batalov | 06/2022 |
8645 | 23109^15668+15668^23109 | 96943 | Hans Havermann | 07/2020 |
8646 | 2^322000-1861469 | 96932 | Enzo Doria | 03/2016 |
8647 | 2^321971-112747 | 96923 | Enzo Doria | 06/2014 |
8648 | 23754^12023+12023^23754 | 96917 | Hans Havermann | 07/2020 |
8649 | 11^93062-92 | 96915 | Lelio R Paula | 06/2016 |
8650 | (37871^21169-1)/37870 | 96914 | Serge Batalov | 06/2022 |
8651 | (2^321922+2214)/2 | 96908 | Stefano Morozzi | 09/2020 |
8652 | (2^321900+2086)/2 | 96902 | Stefano Morozzi | 09/2020 |
8653 | (11*2^321841-7)/15 | 96884 | Renaud Lifchitz | 03/2011 |
8654 | 3^203051+34 | 96880 | Lelio R Paula | 05/2016 |
8655 | 2*10^96867-51 | 96868 | Robert Price | 11/2017 |
8656 | (2^321779+2618)/2 | 96865 | Stefano Morozzi | 09/2020 |
8657 | (64*10^96863-1)/27 | 96864 | Serge Batalov | 05/2010 |
8658 | 2^321716-2786939 | 96847 | Enzo Doria | 03/2016 |
8659 | 22496^20181+20181^22496 | 96844 | Hans Havermann | 03/2020 |
8660 | 2^321701-113053 | 96842 | Enzo Doria | 07/2014 |
8661 | 2^321700-28635 | 96842 | Lelio R Paula | 06/2016 |
8662 | (5*2^321691-1)/27 | 96838 | Lelio R Paula | 02/2014 |
8663 | 6*10^96835+13 | 96836 | Robert Price | 10/2021 |
8664 | 6^124433-124433 | 96828 | Robert Price | 01/2014 |
8665 | 2^321631+69 | 96821 | Lelio R Paula | 12/2012 |
8666 | (27109^21841-1)/27108 | 96820 | Serge Batalov | 06/2022 |
8667 | (13*2^321606+1)/119 | 96813 | Lelio R Paula | 11/2013 |
8668 | (5052^26141-1)/5051 | 96809 | Serge Batalov | 06/2022 |
8669 | (46*10^96798-1)/9 | 96799 | Serge Batalov | 10/2014 |
8670 | (59*10^96798+31)/9 | 96799 | Robert Price | 09/2015 |
8671 | (2^321536+2062)/2 | 96792 | Stefano Morozzi | 09/2020 |
8672 | 223319#+231223 | 96790 | Makoto Morimoto | 02/2016 |
8673 | 2^321519-28635 | 96787 | Lelio R Paula | 06/2016 |
8674 | 2^321487-1867237 | 96778 | Enzo Doria | 03/2016 |
8675 | 2^321461-63 | 96770 | Lelio R Paula | 12/2012 |
8676 | (16053^23011-1)/16052 | 96770 | Serge Batalov | 06/2022 |
8677 | 2^321439+123 | 96763 | Lelio R Paula | 12/2012 |
8678 | 2^321434+709 | 96762 | Lelio R Paula | 02/2013 |
8679 | (2^321415+554)/2 | 96756 | Stefano Morozzi | 09/2020 |
8680 | 14^84413+89 | 96749 | Lelio R Paula | 07/2016 |
8681 | (28*10^96743+71)/9 | 96744 | Serge Batalov | 12/2008 |
8682 | 13^86847+96 | 96743 | Lelio R Paula | 06/2016 |
8683 | 9*2^321336-7 | 96733 | Lelio R Paula | 08/2014 |
8684 | (209*10^96722-17)/3 | 96724 | Robert Price | 05/2017 |
8685 | (54163^20431-1)/54162 | 96710 | Serge Batalov | 06/2022 |
8686 | 2^321237-4615 | 96702 | Yves Minet | 05/2019 |
8687 | (27273^21799-1)/27272 | 96691 | Serge Batalov | 06/2022 |
8688 | 2^321187-1159 | 96687 | Yves Minet | 11/2012 |
8689 | 2^321182-1850081 | 96686 | Enzo Doria | 08/2015 |
8691 | 23359^13768+13768^23359 | 96680 | Hans Havermann | 06/2020 |
8690 | (65630^20071-1)/65629 | 96680 | Serge Batalov | 06/2022 |
8692 | 2^321116-112829 | 96666 | Enzo Doria | 06/2014 |
8693 | 2^321108-109499 | 96664 | Enzo Doria | 04/2014 |
8694 | 2^321098+2793859 | 96661 | Enzo Doria | 03/2016 |
8695 | (2^321100+1170)/2 | 96661 | Stefano Morozzi | 09/2020 |
8696 | 2^321057+4097 | 96648 | Lelio R Paula | 06/2016 |
8697 | (4435*29^66081-4)/7 | 96640 | Curtis Bright | 05/2015 |
8698 | 22743^17750+17750^22743 | 96640 | Hans Havermann | 04/2020 |
8699 | 22882^16679+16679^22882 | 96612 | Hans Havermann | 02/2020 |
8700 | 23452^13163+13163^23452 | 96608 | Hans Havermann | 02/2020 |
8701 | (1042^32009-1)/1041 | 96596 | Serge Batalov | 06/2022 |
8702 | (85*10^96594-373)/9 | 96595 | Robert Price | 05/2021 |
8703 | (2^320863+666)/2 | 96590 | Stefano Morozzi | 09/2020 |
8704 | p(19849)#/2-64 | 96581 | Robert Price | 04/2017 |
8705 | 2^320817+189 | 96576 | Lelio R Paula | 12/2012 |
8706 | 2^320816-4685 | 96576 | Yves Minet | 05/2019 |
8708 | (361*10^96567-163)/9 | 96569 | Robert Price | 10/2017 |
8707 | (19*10^96568+137)/3 | 96569 | Robert Price | 03/2022 |
8709 | 2^320788-110327 | 96567 | Enzo Doria | 05/2014 |
8710 | (3065^27697-1)/3064 | 96561 | Serge Batalov | 06/2022 |
8711 | 2^320764+793 | 96560 | Lelio R Paula | 02/2013 |
8712 | (2^320752+1842)/2 | 96556 | Stefano Morozzi | 02/2021 |
8713 | (48331^20611-1)/48330 | 96542 | Serge Batalov | 06/2022 |
8714 | (50595^20521-1)/50594 | 96529 | Serge Batalov | 06/2022 |
8715 | (9179*2^320652-1)/221 | 96528 | Lelio R Paula | 05/2015 |
8716 | 23288^13947+13947^23288 | 96517 | Hans Havermann | 07/2020 |
8718 | 2^320616-78795 | 96516 | Lelio R Paula | 03/2016 |
8717 | (19*10^96520+287)/9/109/409 | 96516 | Alfred Reich | 11/2022 |
8719 | 2^320612-461939 | 96514 | Enzo Doria | 07/2015 |
8720 | (21245^22303-1)/21244 | 96507 | Serge Batalov | 06/2022 |
8721 | 10^96504+987654321 | 96505 | Robert Price | 12/2019 |
8722 | 10^96504+71887 | 96505 | Patrick De Geest | 02/2020 |
8723 | 22654^18195+18195^22654 | 96505 | Hans Havermann | 07/2020 |
8724 | 10^96504-139277 | 96504 | Patrick De Geest | 02/2020 |
8725 | (218^41263-1)/217 | 96490 | Serge Batalov | 06/2022 |
8726 | (47220^20641-1)/47219 | 96474 | Serge Batalov | 06/2022 |
8727 | 5*2^320431+7 | 96461 | Lelio R Paula | 08/2014 |
8728 | 19^75431-90 | 96458 | Lelio R Paula | 06/2016 |
8729 | (11853^23677-1)/11852 | 96452 | Serge Batalov | 06/2022 |
8730 | (2^320386+2814)/2 | 96446 | Stefano Morozzi | 02/2021 |
8731 | (2^320370+3)/67 | 96440 | Lelio R Paula | 10/2013 |
8732 | 3*10^96431-17 | 96432 | Robert Price | 03/2021 |
8733 | 2^320309+1871477 | 96423 | Enzo Doria | 05/2016 |
8734 | (10412^24001-1)/10411 | 96421 | Serge Batalov | 06/2022 |
8735 | tau(809^6028) | 96410 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
8736 | (29*10^96407-23)/3 | 96408 | Robert Price | 03/2022 |
8737 | (21*2^320252-1)/1895 | 96404 | Lelio R Paula | 02/2014 |
8738 | 2^320232-173 | 96400 | Lelio R Paula | 12/2012 |
8739 | 175^42975-2 | 96395 | Lelio R Paula | 10/2017 |
8740 | (2^320170+2718)/2 | 96381 | Stefano Morozzi | 09/2020 |
8741 | 10^96377+199911 | 96378 | Patrick De Geest | 12/2016 |
8742 | 10^96377-20001 | 96377 | Robert Price | 11/2016 |
8743 | (18079^22639-1)/18078 | 96374 | Serge Batalov | 06/2022 |
8745 | 2^320124-2790449 | 96367 | Enzo Doria | 03/2016 |
8744 | 10^96366+19197 | 96367 | Patrick De Geest | 08/2021 |
8747 | 10^96366-10000000001 | 96366 | Robert Price | 04/2021 |
8746 | 10^96366-287367 | 96366 | Patrick De Geest | 08/2021 |
8748 | (26351^21799-1)/26350 | 96365 | Serge Batalov | 06/2022 |
8749 | 22488^19255+19255^22488 | 96351 | Hans Havermann | 11/2019 |
8750 | 23248^13939+13939^23248 | 96346 | Hans Havermann | 11/2019 |