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 | 2^382464+22741 | 115134 | Enzo Doria | 09/2023 |
5502 | 26752^20065+20065^26752 | 115099 | Hans Havermann | 02/2021 |
5503 | 26383^23008+23008^26383 | 115080 | Hans Havermann | 02/2021 |
5504 | 9^120590-120590 | 115073 | Robert Price | 03/2014 |
5505 | 28203^12016+12016^28203 | 115062 | Hans Havermann | 02/2021 |
5506 | 2^382198+9 | 115054 | Lelio R Paula | 03/2013 |
5507 | 30101^6630+6630^30101 | 115032 | Hans Havermann | 01/2021 |
5508 | (89*10^115030-179)/9 | 115031 | Robert Price | 12/2019 |
5509 | 25554^26089-26089^25554 | 114987 | Frank Doornink | 02/2024 |
5510 | 25142^26129-26129^25142 | 114979 | Frank Doornink | 12/2023 |
5511 | (2^381935+2774)/2 | 114974 | Stefano Morozzi | 02/2021 |
5512 | 27300^16271+16271^27300 | 114972 | Hans Havermann | 02/2021 |
5513 | (23*10^114962+7)/3 | 114963 | Robert Price | 05/2023 |
5514 | 2^381883-59197 | 114959 | Enzo Doria | 10/2015 |
5515 | 27423^15542+15542^27423 | 114944 | Hans Havermann | 02/2021 |
5516 | (2^381820+1626)/2 | 114939 | Stefano Morozzi | 09/2020 |
5518 | 2^381806-4481 | 114936 | Yves Minet | 07/2011 |
5517 | 2^381808-1851507 | 114936 | Enzo Doria | 08/2015 |
5519 | (2^381792+2502)/2 | 114931 | Stefano Morozzi | 09/2020 |
5520 | 2^381774-112373 | 114926 | Enzo Doria | 06/2014 |
5521 | 26341^22980+22980^26341 | 114883 | Hans Havermann | 01/2021 |
5522 | 2^381555-109051 | 114860 | Enzo Doria | 04/2014 |
5523 | 2^381551+213683 | 114859 | Enzo Doria | 09/2023 |
5524 | 2^381546-113501 | 114857 | Enzo Doria | 08/2014 |
5525 | (7*10^114845-19)/3 | 114846 | Robert Price | 10/2020 |
5526 | 26675^20178+20178^26675 | 114833 | Hans Havermann | 02/2021 |
5527 | (2^381436+1182)/2 | 114824 | Stefano Morozzi | 09/2020 |
5528 | 3^240644+16 | 114817 | Robert Price | 04/2020 |
5529 | 2^381360-110429 | 114801 | Enzo Doria | 05/2014 |
5530 | (17*10^114795+79)/3 | 114796 | Robert Price | 04/2019 |
5531 | (66^63079+1)/67 | 114774 | Paul Bourdelais | 09/2017 |
5532 | 2^381252-115079 | 114769 | Enzo Doria | 06/2015 |
5533 | (2^381251+3186)/2 | 114768 | Stefano Morozzi | 09/2020 |
5534 | 2^381209-29323 | 114756 | Enzo Doria | 07/2015 |
5535 | 2^381207-111546435 | 114755 | Paul Bourdelais | 10/2021 |
5536 | 569^41651-568^41651 | 114754 | Jean-Louis Charton | 11/2009 |
5537 | 2^381201+1873205 | 114753 | Enzo Doria | 06/2016 |
5538 | (85*10^114737+293)/9 | 114738 | Robert Price | 12/2019 |
5539 | (265*10^114736+17)/3 | 114738 | Robert Price | 03/2020 |
5540 | (2^381062+2170)/2 | 114711 | Stefano Morozzi | 09/2020 |
5541 | 2^380974-112991 | 114685 | Enzo Doria | 07/2014 |
5542 | 37097^1234+1234^37097 | 114679 | Hans Havermann | 02/2021 |
5543 | (26*10^114658-119)/3 | 114659 | Robert Price | 07/2023 |
5544 | 88^58949-87 | 114626 | Norbert Schneider | 11/2015 |
5545 | (2^380773+662)/2 | 114624 | Stefano Morozzi | 09/2020 |
5546 | Phi(155285,10)/242674820198201 | 114610 | Makoto Kamada | 02/2024 |
5547 | 29913^6782+6782^29913 | 114608 | Hans Havermann | 02/2021 |
5548 | (331*10^114605-7)/9 | 114607 | Robert Price | 04/2022 |
5549 | 2^380700+91 | 114603 | Lelio R Paula | 03/2013 |
5550 | (2^380632+2578)/2 | 114582 | Stefano Morozzi | 09/2020 |
5551 | 7*10^114577+27 | 114578 | Robert Price | 11/2020 |
5553 | 10^114575+3^114576 | 114576 | Mike Oakes | 09/2010 |
5552 | 10^114575+66561 | 114576 | Patrick De Geest | 02/2017 |
5554 | 10^114575-581081 | 114575 | Patrick De Geest | 02/2017 |
5555 | 2^380587-115315 | 114569 | Enzo Doria | 06/2015 |
5556 | 31874^3927+3927^31874 | 114558 | Hans Havermann | 02/2021 |
5557 | 2^380513-216925 | 114546 | Enzo Doria | 05/2014 |
5558 | 2^380451-1840087 | 114528 | Enzo Doria | 06/2015 |
5559 | (2^380426+1510)/2 | 114520 | Stefano Morozzi | 09/2020 |
5560 | (15*2^380388-1)/539 | 114507 | Lelio R Paula | 04/2014 |
5561 | 2^380279-461347 | 114476 | Enzo Doria | 07/2015 |
5562 | (331*10^114454-7)/9 | 114456 | Robert Price | 04/2022 |
5563 | 27121^16590+16590^27121 | 114447 | Hans Havermann | 01/2021 |
5564 | 25966^25555+25555^25966 | 114445 | Hans Havermann | 01/2021 |
5565 | (611*32^76029-53)/31 | 114437 | Eric Chen | 11/2023 |
5566 | (11*10^114400+529)/9 | 114401 | Robert Price | 09/2021 |
5567 | 2^380021+1871147 | 114398 | Enzo Doria | 05/2016 |
5568 | 2^380005+1869707 | 114393 | Enzo Doria | 05/2016 |
5569 | 2^379958-113297 | 114379 | Enzo Doria | 07/2014 |
5570 | (2^379897+1274)/2 | 114361 | Stefano Morozzi | 01/2021 |
5571 | 2^379881-109105 | 114356 | Enzo Doria | 04/2014 |
5572 | (21*2^379857-1)/13 | 114349 | Lelio R Paula | 03/2014 |
5573 | (4*10^114340+83)/3 | 114341 | Robert Price | 02/2018 |
5574 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
5575 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
5576 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
5577 | 27764^13065+13065^27764 | 114280 | Hans Havermann | 01/2021 |
5578 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
5579 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
5580 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
5581 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
5582 | (22*10^114178+47)/3 | 114179 | Robert Price | 06/2021 |
5583 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
5584 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
5585 | (228^48407-1)/227 | 114139 | Alexander Zhirkov | 03/2023 |
5586 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
5587 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
5588 | 332*25^81584+12357 | 114053 | Eric Chen | 10/2023 |
5589 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
5590 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
5591 | 13643^27572-27572^13643 | 114008 | Norbert Schneider | 12/2022 |
5593 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
5592 | (83*10^113969+133)/9 | 113970 | Robert Price | 08/2021 |
5594 | 32226^3439+3439^32226 | 113966 | Hans Havermann | 02/2021 |
5595 | (77*10^113963-149)/9 | 113964 | Robert Price | 03/2022 |
5596 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
5597 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
5598 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
5601 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
5599 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
5600 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
5602 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
5603 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
5604 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
5606 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
5605 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
5607 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
5608 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
5609 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
5610 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
5611 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
5612 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
5613 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
5614 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
5615 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
5616 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
5617 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
5618 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
5619 | 34596^1949+1949^34596 | 113815 | Hans Havermann | 02/2021 |
5620 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
5621 | (28*10^113810+449)/9 | 113811 | Robert Price | 11/2021 |
5622 | 26273^21448+21448^26273 | 113799 | Hans Havermann | 02/2021 |
5623 | 2^377966+255255 | 113780 | Paul Bourdelais | 10/2021 |
5624 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
5625 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
5626 | (7*10^113757+11)/9 | 113757 | Tyler Busby | 01/2023 |
5627 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
5628 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
5629 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
5630 | (2^377791+3258)/2 | 113727 | Stefano Morozzi | 02/2021 |
5631 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
5632 | 28206^10739+10739^28206 | 113698 | Hans Havermann | 01/2021 |
5633 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
5634 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
5635 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
5636 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
5637 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |
5638 | (10^113656+7)/40231/6439001 | 113645 | Alfred Reich | 06/2021 |
5639 | (2*10^113631+529)/9 | 113631 | Robert Price | 01/2018 |
5640 | 2^377408-2798279 | 113612 | Enzo Doria | 04/2016 |
5641 | (292^46336+1)/(292^256+1) | 113605 | Martin Hopf | 12/2023 |
5642 | 13706^27459+27459^13706 | 113596 | Norbert Schneider | 12/2020 |
5643 | 5*10^113586+57 | 113587 | Robert Price | 12/2018 |
5644 | (2^377242+2794)/2 | 113561 | Stefano Morozzi | 02/2021 |
5645 | 2^377222+93679 | 113556 | Yves Minet | 02/2011 |
5647 | 262147#+316847 | 113552 | Makoto Morimoto | 02/2016 |
5646 | 27172^15101+15101^27172 | 113552 | Hans Havermann | 01/2021 |
5648 | (2^377131+2018)/2 | 113528 | Stefano Morozzi | 09/2020 |
5649 | 2^377026-458903 | 113497 | Enzo Doria | 04/2015 |
5650 | (2^377014+2202)/2 | 113493 | Stefano Morozzi | 09/2020 |
5651 | 34791^1828+1828^34791 | 113488 | Hans Havermann | 02/2021 |
5652 | (2^376967+258)/2 | 113479 | Stefano Morozzi | 09/2020 |
5653 | (29*10^113468-17)/3 | 113469 | Robert Price | 09/2019 |
5654 | 35^73477-34^73477 | 113454 | Robert Price | 05/2012 |
5655 | (14*10^113451-89)/3 | 113452 | Robert Price | 05/2020 |
5656 | 2^376790+27 | 113426 | Lelio R Paula | 03/2013 |
5657 | 2^376786-112701 | 113424 | Enzo Doria | 06/2014 |
5658 | 2^376731+21 | 113408 | Lelio R Paula | 03/2013 |
5659 | 9*10^113394-73 | 113395 | Robert Price | 01/2021 |
5660 | 29660^6651+6651^29660 | 113387 | Hans Havermann | 01/2021 |
5662 | 2^376634+1871245 | 113379 | Enzo Doria | 05/2016 |
5661 | (4^188318+17)/3 | 113379 | Robert Price | 02/2017 |
5663 | E(29532)/492842728006750534663978950979381235 | 113367 | Thomas Ritschel | 01/2021 |
5664 | 2^376576-344369 | 113361 | Enzo Doria | 07/2015 |
5665 | 2^376549-4543 | 113353 | Yves Minet | 03/2019 |
5666 | (17*2^376526-1)/223 | 113345 | Lelio R Paula | 03/2014 |
5667 | (27*2^376441-1)/811 | 113319 | Lelio R Paula | 03/2014 |
5668 | (2^376350+3538)/2 | 113293 | Stefano Morozzi | 09/2020 |
5669 | 27265^14272+14272^27265 | 113273 | Hans Havermann | 02/2021 |
5670 | 32174^3315+3315^32174 | 113269 | Hans Havermann | 02/2021 |
5671 | (2*10^113248+73)/3 | 113248 | Robert Price | 09/2020 |
5672 | 2^376128-21237 | 113226 | Enzo Doria | 06/2013 |
5673 | 5*10^113224-81 | 113225 | Robert Price | 01/2019 |
5674 | 10^113220+51409 | 113221 | Patrick De Geest | 02/2021 |
5676 | 10^113220-1000001 | 113220 | Robert Price | 08/2020 |
5675 | 10^113220-61779 | 113220 | Patrick De Geest | 02/2021 |
5677 | (2^376102+1578)/2 | 113218 | Stefano Morozzi | 09/2020 |
5678 | (2^375996+3102)/2 | 113186 | Stefano Morozzi | 09/2020 |
5679 | 37258^1091+1091^37258 | 113184 | Hans Havermann | 02/2021 |
5680 | (94*10^113176+11)/3 | 113178 | Robert Price | 04/2020 |
5681 | (46*10^113162+521)/9 | 113163 | Robert Price | 02/2019 |
5682 | (2^375885+3614)/2 | 113153 | Stefano Morozzi | 09/2020 |
5683 | (2^375873+2006)/2 | 113149 | Stefano Morozzi | 09/2020 |
5684 | 3*10^113137+23 | 113138 | Robert Price | 03/2022 |
5685 | (8*10^113135-269)/9 | 113135 | Robert Price | 09/2021 |
5686 | (2^375726+694)/2 | 113105 | Stefano Morozzi | 09/2020 |
5687 | 26050^21959+21959^26050 | 113100 | Hans Havermann | 02/2021 |
5688 | 2^375638-1787171 | 113079 | Enzo Doria | 03/2016 |
5689 | (2^375563+2178)/2 | 113056 | Stefano Morozzi | 09/2020 |
5690 | 2^375537+4847 | 113048 | Yves Minet | 02/2011 |
5691 | (17^91837+16^91837)/33 | 113000 | Robert Price | 08/2013 |
5692 | 5*F(270343)*(F(270343)+1)+1 | 112997 | Serge Batalov | 04/2015 |
5693 | 2^375350-3221 | 112992 | Yves Minet | 07/2011 |
5694 | (13*2^375306+1)/7 | 112979 | Lelio R Paula | 03/2014 |
5695 | 2^375305+76757 | 112979 | Enzo Doria | 05/2016 |
5696 | 2^375258+1874989 | 112964 | Enzo Doria | 07/2016 |
5697 | (29^77237-1)/28 | 112950 | Robert Price | 04/2012 |
5698 | 26620^17483+17483^26620 | 112939 | Hans Havermann | 01/2021 |
5699 | 2^375154+74959 | 112933 | Enzo Doria | 04/2016 |
5700 | (266*10^112930-17)/3 | 112932 | Robert Price | 07/2020 |
5701 | (2^375134+778)/2 | 112927 | Stefano Morozzi | 09/2020 |
5702 | (85*10^112919+293)/9 | 112920 | Robert Price | 12/2019 |
5703 | 29412^6899+6899^29412 | 112907 | Hans Havermann | 02/2021 |
5704 | (191*10^112903-17)/3 | 112905 | Robert Price | 11/2021 |
5705 | (2^375022+1210)/2 | 112893 | Stefano Morozzi | 09/2020 |
5706 | 25992^22039+22039^25992 | 112889 | Hans Havermann | 02/2021 |
5707 | 2^374993-2787163 | 112885 | Enzo Doria | 11/2015 |
5708 | (697*10^112869+23)/9 | 112871 | Robert Price | 04/2021 |
5709 | (19*10^112851+377)/9 | 112852 | Alfred Reich | 11/2022 |
5710 | 30420^5111+5111^30420 | 112813 | Hans Havermann | 02/2021 |
5711 | 13699^27268+27268^13699 | 112800 | Norbert Schneider | 12/2020 |
5712 | 123^53970+2 | 112793 | Lelio R Paula | 06/2013 |
5713 | 12308*27^78796+23 | 112790 | Eric Chen | 07/2023 |
5714 | 28942^7887+7887^28942 | 112785 | Hans Havermann | 01/2021 |
5715 | 2^374640-4685 | 112778 | Yves Minet | 05/2019 |
5716 | 12^104501-11 | 112776 | Riley Fisher | 12/2023 |
5717 | 57290^93+93^57290 | 112775 | Hans Havermann | 01/2021 |
5718 | 13608^27277-27277^13608 | 112758 | Norbert Schneider | 12/2022 |
5719 | (2^374556+682)/2 | 112753 | Stefano Morozzi | 09/2020 |
5720 | p(22852)#/2+512 | 112745 | Robert Price | 05/2017 |
5721 | (11*10^112742-131)/3 | 112743 | Robert Price | 09/2018 |
5722 | 2^374441-21049 | 112718 | Enzo Doria | 06/2013 |
5723 | 4*10^112706-93 | 112707 | Robert Price | 07/2020 |
5724 | (2^374321+1)/3 | 112682 | H.&R. Lifchitz | 12/2000 |
5725 | (2^374213+1766)/2 | 112650 | Stefano Morozzi | 09/2020 |
5726 | (2^374094+2674)/2 | 112614 | Stefano Morozzi | 09/2020 |
5727 | (13^101089-1)/12 | 112607 | Paul Bourdelais | 03/2010 |
5728 | 3^236011-8828 | 112606 | Enzo Doria | 05/2014 |
5729 | 31854^3427+3427^31854 | 112602 | Hans Havermann | 02/2021 |
5730 | (85*10^112530-373)/9 | 112531 | Robert Price | 05/2021 |
5731 | (2^373748+1090)/2 | 112510 | Stefano Morozzi | 09/2020 |
5732 | (2^373727+1806)/2 | 112503 | Stefano Morozzi | 01/2021 |
5733 | 27618^11821+11821^27618 | 112479 | Hans Havermann | 01/2021 |
5734 | 26343^18580+18580^26343 | 112460 | Hans Havermann | 01/2021 |
5735 | 39^70666+2 | 112435 | Lelio R Paula | 11/2013 |
5736 | (134*10^112400+7)/3 | 112402 | Robert Price | 02/2020 |
5737 | (2^373365+326)/2 | 112394 | Stefano Morozzi | 09/2020 |
5738 | (2^373350-7)/3 | 112390 | Lelio R Paula | 03/2014 |
5739 | 2^373306-1840931 | 112377 | Enzo Doria | 06/2015 |
5740 | 8*10^112371-37 | 112372 | Robert Price | 03/2022 |
5741 | (2^373262+270)/2 | 112363 | Stefano Morozzi | 09/2020 |
5742 | 6*10^112359-79 | 112360 | Robert Price | 03/2022 |
5743 | 34409^1842+1842^34409 | 112356 | Hans Havermann | 02/2021 |
5744 | (142*10^112352-7)/9 | 112354 | Robert Price | 12/2019 |
5745 | 25580^24633+24633^25580 | 112336 | Hans Havermann | 01/2021 |
5746 | 2^373117+1872107 | 112320 | Enzo Doria | 05/2016 |
5747 | (2^373083+2226)/2 | 112309 | Stefano Morozzi | 01/2021 |
5748 | (35*10^112308+37)/9 | 112309 | Robert Price | 05/2022 |
5749 | 27963^10376+10376^27963 | 112301 | Hans Havermann | 01/2021 |
5750 | 2^373040+2792761 | 112297 | Enzo Doria | 03/2016 |