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 ? |
5501 | (2^387073+2570)/2 | 116521 | Stefano Morozzi | 09/2020 |
5502 | (7*10^116514-1)/3 | 116515 | Robert Price | 08/2022 |
5503 | (26*32^77406-390071011)/31 | 116508 | Eric Chen | 11/2023 |
5504 | 28846^10931+10931^28846 | 116500 | Hans Havermann | 02/2021 |
5505 | 10^116496+3593037 | 116497 | Paul Bourdelais | 10/2021 |
5506 | (31*2^386968+1)/7 | 116490 | Lelio R Paula | 03/2014 |
5507 | 2^386947+1869143 | 116483 | Enzo Doria | 05/2016 |
5508 | (30^78857-1)/29 | 116480 | Robert Price | 12/2011 |
5509 | (2^386903-3)/5 | 116469 | Lelio R Paula | 03/2014 |
5510 | 28107^13892+13892^28107 | 116441 | Hans Havermann | 02/2021 |
5511 | 29110^9957+9957^29110 | 116386 | Hans Havermann | 02/2021 |
5512 | 26469^24838+24838^26469 | 116335 | Hans Havermann | 01/2021 |
5513 | 2^386389-1864783 | 116315 | Enzo Doria | 03/2016 |
5514 | (8*32^77250-163)/31 | 116273 | Eric Chen | 11/2023 |
5515 | (31*2^386231-1)/1711 | 116266 | Lelio R Paula | 03/2014 |
5516 | 297^47012+47012^297 | 116250 | Norbert Schneider | 07/2018 |
5517 | (35*10^116223-341)/9 | 116224 | Robert Price | 08/2021 |
5518 | (2^386087-5)/40959 | 116220 | Lelio R Paula | 03/2014 |
5519 | (2^386054+714)/2 | 116214 | Stefano Morozzi | 09/2020 |
5520 | (10^116201+7)/39623 | 116197 | Alfred Reich | 06/2021 |
5521 | 2^385969-823 | 116189 | Yves Minet | 05/2012 |
5522 | 57^66162+2 | 116173 | Lelio R Paula | 08/2013 |
5523 | 2^385854+75049 | 116154 | Enzo Doria | 04/2016 |
5524 | (2^385832+2590)/2 | 116147 | Stefano Morozzi | 09/2020 |
5525 | 2^385775-109587 | 116130 | Enzo Doria | 04/2014 |
5526 | (14*10^116112-101)/3 | 116113 | Robert Price | 02/2019 |
5527 | Phi(236354,10)/24817171 | 116109 | Makoto Kamada | 08/2019 |
5528 | 3*10^116107+23 | 116108 | Robert Price | 03/2022 |
5529 | 28729^10990+10990^28729 | 116094 | Hans Havermann | 02/2021 |
5530 | 2^385622+71839 | 116084 | Enzo Doria | 04/2016 |
5532 | 2^385564+781 | 116067 | Lelio R Paula | 01/2012 |
5531 | (2^385565+1562)/2 | 116067 | Stefano Morozzi | 09/2020 |
5533 | (2^385563+378)/2 | 116066 | Stefano Morozzi | 02/2021 |
5534 | (2^385557+2214)/2 | 116064 | Stefano Morozzi | 09/2020 |
5535 | 29431^8776+8776^29431 | 116056 | Hans Havermann | 01/2021 |
5536 | (266*10^116024-17)/3 | 116026 | Robert Price | 07/2020 |
5537 | 30263^6820+6820^30263 | 116022 | Hans Havermann | 02/2021 |
5538 | 2^385410+71893 | 116020 | Enzo Doria | 04/2016 |
5539 | 38225^1084+1084^38225 | 116014 | Hans Havermann | 02/2021 |
5540 | 26548^23433+23433^26548 | 116011 | Hans Havermann | 02/2021 |
5541 | 12^107422-11 | 115928 | Riley Fisher | 12/2023 |
5542 | (25*2^385091+1)/27 | 115924 | Lelio R Paula | 04/2014 |
5543 | (11*10^115910-131)/3 | 115911 | Robert Price | 09/2018 |
5544 | 2^384953-111943 | 115883 | Paul Bourdelais | 02/2012 |
5545 | 43^70942-2 | 115882 | Lelio R Paula | 04/2014 |
5546 | (94*10^115879-7)/3 | 115881 | Robert Price | 03/2020 |
5547 | 7*10^115863+3 | 115864 | Robert Price | 08/2015 |
5548 | 2^384834+93 | 115847 | Donovan Johnson | 07/2008 |
5549 | 2^384809+71297 | 115840 | Enzo Doria | 04/2016 |
5550 | (55*10^115829+323)/9 | 115830 | Robert Price | 10/2021 |
5551 | (7*2^384759-1)/5 | 115825 | Lelio R Paula | 04/2014 |
5552 | 31348^4951+4951^31348 | 115822 | Hans Havermann | 02/2021 |
5553 | (793*10^115810+53)/9 | 115812 | Robert Price | 01/2022 |
5554 | (71*10^115810-17)/9 | 115811 | Ray Chandler | 08/2011 |
5555 | (97*10^115794-1)/3 | 115796 | Serge Batalov | 12/2014 |
5556 | 31115^5266+5266^31115 | 115794 | Hans Havermann | 02/2021 |
5557 | 2*10^115784-27 | 115785 | Robert Price | 05/2018 |
5558 | 10^115773+126739 | 115774 | Patrick De Geest | 04/2018 |
5560 | 10^115773-1000000001 | 115773 | Robert Price | 03/2018 |
5559 | 10^115773-396221 | 115773 | Patrick De Geest | 04/2018 |
5561 | (2^384582+778)/2 | 115771 | Stefano Morozzi | 09/2020 |
5562 | 30537^6178+6178^30537 | 115762 | Hans Havermann | 01/2021 |
5563 | 2^384530+1874305 | 115756 | Enzo Doria | 06/2016 |
5564 | (3*2^384524-1)/7631 | 115750 | Lelio R Paula | 03/2014 |
5565 | 28275^12406+12406^28275 | 115748 | Hans Havermann | 01/2021 |
5566 | (44*10^115697-413)/9 | 115698 | Robert Price | 03/2019 |
5567 | 31408^4825+4825^31408 | 115692 | Hans Havermann | 02/2021 |
5568 | (2^384317+1026)/2 | 115691 | Stefano Morozzi | 09/2020 |
5569 | 2^384312+81091 | 115690 | Yves Minet | 02/2010 |
5570 | (2^384265+2150)/2 | 115675 | Stefano Morozzi | 09/2020 |
5571 | (2^384246+2182)/2 | 115670 | Stefano Morozzi | 09/2020 |
5572 | (4*10^115668-79)/3 | 115669 | Robert Price | 02/2018 |
5573 | (2^384193+242)/2 | 115654 | Stefano Morozzi | 09/2020 |
5574 | (266947#+1267)/(7^2*181) | 115647 | Makoto Morimoto | 06/2019 |
5575 | (266947#+273808)/(2*109*157) | 115646 | Makoto Morimoto | 10/2017 |
5576 | 18^92125-92125^18 | 115642 | Dylan Delgado | 08/2019 |
5577 | 3^242337-4042 | 115625 | Enzo Doria | 05/2014 |
5578 | (2^383931+674)/2 | 115575 | Stefano Morozzi | 09/2020 |
5579 | 7*10^115561+79 | 115562 | Robert Price | 07/2019 |
5580 | (2^383872+3046)/2 | 115557 | Stefano Morozzi | 09/2020 |
5581 | (11*2^383859-13)/15 | 115553 | Renaud Lifchitz | 03/2011 |
5582 | 2^383780-3149 | 115530 | Yves Minet | 10/2014 |
5583 | (41*10^115523-383)/9 | 115524 | Robert Price | 11/2018 |
5584 | 27316^16923+16923^27316 | 115506 | Hans Havermann | 02/2021 |
5585 | (2^383618+618)/2 | 115481 | Stefano Morozzi | 09/2020 |
5586 | 2^383560-455 | 115464 | Yves Minet | 03/2012 |
5587 | (2^383544+5)/3 | 115458 | Lelio R Paula | 03/2014 |
5588 | (22*10^115454+119)/3 | 115455 | Robert Price | 04/2021 |
5589 | 89^59221-88^59221 | 115445 | Robert Price | 06/2015 |
5590 | (7^136559-5^136559)/2 | 115406 | Jon Grantham | 08/2023 |
5591 | 2^383347+1869413 | 115399 | Enzo Doria | 05/2016 |
5592 | (103*10^115366+17)/3 | 115368 | Robert Price | 08/2020 |
5593 | (2^383232+750)/2 | 115365 | Stefano Morozzi | 09/2020 |
5594 | (2^383130-7)/131793 | 115329 | Lelio R Paula | 03/2014 |
5595 | (12^106859+1)/13 | 115320 | Paul Bourdelais | 02/2010 |
5596 | 29961^7060+7060^29961 | 115315 | Hans Havermann | 01/2021 |
5598 | 2^383039+1869509 | 115307 | Enzo Doria | 05/2016 |
5597 | 32918^3183+3183^32918 | 115307 | Hans Havermann | 01/2021 |
5599 | 36964^1315+1315^36964 | 115288 | Hans Havermann | 02/2021 |
5600 | (13*2^382968+1)/7 | 115286 | Lelio R Paula | 04/2014 |
5601 | (22*10^115282+29)/3 | 115283 | Robert Price | 02/2021 |
5602 | (2^382945+2046)/2 | 115278 | Stefano Morozzi | 09/2020 |
5603 | (2^382789+3182)/2 | 115231 | Stefano Morozzi | 09/2020 |
5604 | 30416^6141+6141^30416 | 115224 | Hans Havermann | 02/2021 |
5605 | 6*10^115211+79 | 115212 | Robert Price | 09/2020 |
5606 | (2^382722+3598)/2 | 115211 | Stefano Morozzi | 09/2020 |
5607 | 27987^13078+13078^27987 | 115210 | Hans Havermann | 02/2021 |
5608 | (2^382699+2090)/2 | 115204 | Stefano Morozzi | 09/2020 |
5609 | (22*10^115188-193)/9 | 115189 | Robert Price | 07/2022 |
5610 | 2^382633+73697 | 115185 | Enzo Doria | 04/2016 |
5611 | 2^382509-113593 | 115147 | Enzo Doria | 08/2014 |
5612 | 2^382502-2791721 | 115145 | Enzo Doria | 03/2016 |
5613 | (2^382495+1274)/2 | 115143 | Stefano Morozzi | 01/2021 |
5614 | 2^382464+22741 | 115134 | Enzo Doria | 09/2023 |
5615 | 26752^20065+20065^26752 | 115099 | Hans Havermann | 02/2021 |
5616 | 26383^23008+23008^26383 | 115080 | Hans Havermann | 02/2021 |
5617 | 9^120590-120590 | 115073 | Robert Price | 03/2014 |
5618 | 28203^12016+12016^28203 | 115062 | Hans Havermann | 02/2021 |
5619 | 2^382198+9 | 115054 | Lelio R Paula | 03/2013 |
5620 | 30101^6630+6630^30101 | 115032 | Hans Havermann | 01/2021 |
5621 | (89*10^115030-179)/9 | 115031 | Robert Price | 12/2019 |
5622 | 25554^26089-26089^25554 | 114987 | Frank Doornink | 02/2024 |
5623 | 25142^26129-26129^25142 | 114979 | Frank Doornink | 12/2023 |
5624 | (2^381935+2774)/2 | 114974 | Stefano Morozzi | 02/2021 |
5625 | 27300^16271+16271^27300 | 114972 | Hans Havermann | 02/2021 |
5626 | (23*10^114962+7)/3 | 114963 | Robert Price | 05/2023 |
5627 | 2^381883-59197 | 114959 | Enzo Doria | 10/2015 |
5628 | 27423^15542+15542^27423 | 114944 | Hans Havermann | 02/2021 |
5629 | (2^381820+1626)/2 | 114939 | Stefano Morozzi | 09/2020 |
5631 | 2^381806-4481 | 114936 | Yves Minet | 07/2011 |
5630 | 2^381808-1851507 | 114936 | Enzo Doria | 08/2015 |
5632 | (2^381792+2502)/2 | 114931 | Stefano Morozzi | 09/2020 |
5633 | 2^381774-112373 | 114926 | Enzo Doria | 06/2014 |
5634 | 26341^22980+22980^26341 | 114883 | Hans Havermann | 01/2021 |
5635 | 2^381555-109051 | 114860 | Enzo Doria | 04/2014 |
5636 | 2^381551+213683 | 114859 | Enzo Doria | 09/2023 |
5637 | 2^381546-113501 | 114857 | Enzo Doria | 08/2014 |
5638 | (7*10^114845-19)/3 | 114846 | Robert Price | 10/2020 |
5639 | 26675^20178+20178^26675 | 114833 | Hans Havermann | 02/2021 |
5640 | (2^381436+1182)/2 | 114824 | Stefano Morozzi | 09/2020 |
5641 | 3^240644+16 | 114817 | Robert Price | 04/2020 |
5642 | 2^381360-110429 | 114801 | Enzo Doria | 05/2014 |
5643 | (17*10^114795+79)/3 | 114796 | Robert Price | 04/2019 |
5644 | (66^63079+1)/67 | 114774 | Paul Bourdelais | 09/2017 |
5645 | 2^381252-115079 | 114769 | Enzo Doria | 06/2015 |
5646 | (2^381251+3186)/2 | 114768 | Stefano Morozzi | 09/2020 |
5647 | 2^381209-29323 | 114756 | Enzo Doria | 07/2015 |
5648 | 2^381207-111546435 | 114755 | Paul Bourdelais | 10/2021 |
5649 | 569^41651-568^41651 | 114754 | Jean-Louis Charton | 11/2009 |
5650 | 2^381201+1873205 | 114753 | Enzo Doria | 06/2016 |
5651 | (85*10^114737+293)/9 | 114738 | Robert Price | 12/2019 |
5652 | (265*10^114736+17)/3 | 114738 | Robert Price | 03/2020 |
5653 | (2^381062+2170)/2 | 114711 | Stefano Morozzi | 09/2020 |
5654 | 2^380974-112991 | 114685 | Enzo Doria | 07/2014 |
5655 | 37097^1234+1234^37097 | 114679 | Hans Havermann | 02/2021 |
5656 | (26*10^114658-119)/3 | 114659 | Robert Price | 07/2023 |
5657 | 88^58949-87 | 114626 | Norbert Schneider | 11/2015 |
5658 | (2^380773+662)/2 | 114624 | Stefano Morozzi | 09/2020 |
5659 | Phi(155285,10)/242674820198201 | 114610 | Makoto Kamada | 02/2024 |
5660 | 29913^6782+6782^29913 | 114608 | Hans Havermann | 02/2021 |
5661 | (331*10^114605-7)/9 | 114607 | Robert Price | 04/2022 |
5662 | 2^380700+91 | 114603 | Lelio R Paula | 03/2013 |
5663 | (2^380632+2578)/2 | 114582 | Stefano Morozzi | 09/2020 |
5664 | 7*10^114577+27 | 114578 | Robert Price | 11/2020 |
5666 | 10^114575+3^114576 | 114576 | Mike Oakes | 09/2010 |
5665 | 10^114575+66561 | 114576 | Patrick De Geest | 02/2017 |
5667 | 10^114575-581081 | 114575 | Patrick De Geest | 02/2017 |
5668 | 2^380587-115315 | 114569 | Enzo Doria | 06/2015 |
5669 | 31874^3927+3927^31874 | 114558 | Hans Havermann | 02/2021 |
5670 | 2^380513-216925 | 114546 | Enzo Doria | 05/2014 |
5671 | 2^380451-1840087 | 114528 | Enzo Doria | 06/2015 |
5672 | (2^380426+1510)/2 | 114520 | Stefano Morozzi | 09/2020 |
5673 | (15*2^380388-1)/539 | 114507 | Lelio R Paula | 04/2014 |
5674 | 2^380279-461347 | 114476 | Enzo Doria | 07/2015 |
5675 | (331*10^114454-7)/9 | 114456 | Robert Price | 04/2022 |
5676 | 27121^16590+16590^27121 | 114447 | Hans Havermann | 01/2021 |
5677 | 25966^25555+25555^25966 | 114445 | Hans Havermann | 01/2021 |
5678 | (611*32^76029-53)/31 | 114437 | Eric Chen | 11/2023 |
5679 | (11*10^114400+529)/9 | 114401 | Robert Price | 09/2021 |
5680 | 2^380021+1871147 | 114398 | Enzo Doria | 05/2016 |
5681 | 2^380005+1869707 | 114393 | Enzo Doria | 05/2016 |
5682 | 2^379958-113297 | 114379 | Enzo Doria | 07/2014 |
5683 | (2^379897+1274)/2 | 114361 | Stefano Morozzi | 01/2021 |
5684 | 2^379881-109105 | 114356 | Enzo Doria | 04/2014 |
5685 | (21*2^379857-1)/13 | 114349 | Lelio R Paula | 03/2014 |
5686 | (4*10^114340+83)/3 | 114341 | Robert Price | 02/2018 |
5687 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
5688 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
5689 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
5690 | 27764^13065+13065^27764 | 114280 | Hans Havermann | 01/2021 |
5691 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
5692 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
5693 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
5694 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
5695 | (22*10^114178+47)/3 | 114179 | Robert Price | 06/2021 |
5696 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
5697 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
5698 | (228^48407-1)/227 | 114139 | Alexander Zhirkov | 03/2023 |
5699 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
5700 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
5701 | 332*25^81584+12357 | 114053 | Eric Chen | 10/2023 |
5702 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
5703 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
5704 | 13643^27572-27572^13643 | 114008 | Norbert Schneider | 12/2022 |
5706 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
5705 | (83*10^113969+133)/9 | 113970 | Robert Price | 08/2021 |
5707 | 32226^3439+3439^32226 | 113966 | Hans Havermann | 02/2021 |
5708 | (77*10^113963-149)/9 | 113964 | Robert Price | 03/2022 |
5709 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
5710 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
5711 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
5714 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
5712 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
5713 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
5715 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
5716 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
5717 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
5719 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
5718 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
5720 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
5721 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
5722 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
5723 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
5724 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
5725 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
5726 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
5727 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
5728 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
5729 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
5730 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
5731 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
5732 | 34596^1949+1949^34596 | 113815 | Hans Havermann | 02/2021 |
5733 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
5734 | (28*10^113810+449)/9 | 113811 | Robert Price | 11/2021 |
5735 | 26273^21448+21448^26273 | 113799 | Hans Havermann | 02/2021 |
5736 | 2^377966+255255 | 113780 | Paul Bourdelais | 10/2021 |
5737 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
5738 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
5739 | (7*10^113757+11)/9 | 113757 | Tyler Busby | 01/2023 |
5740 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
5741 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
5742 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
5743 | (2^377791+3258)/2 | 113727 | Stefano Morozzi | 02/2021 |
5744 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
5745 | 28206^10739+10739^28206 | 113698 | Hans Havermann | 01/2021 |
5746 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
5747 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
5748 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
5749 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
5750 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |