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 ? |
6501 | 3^242337-4042 | 115625 | Enzo Doria | 05/2014 |
6502 | (17699*31^77516-29)/30 | 115608 | Eric Chen | 05/2024 |
6503 | (2^383931+674)/2 | 115575 | Stefano Morozzi | 09/2020 |
6504 | 7*10^115561+79 | 115562 | Robert Price | 07/2019 |
6505 | (2^383872+3046)/2 | 115557 | Stefano Morozzi | 09/2020 |
6506 | (11*2^383859-13)/15 | 115553 | Renaud Lifchitz | 03/2011 |
6507 | 2^383780-3149 | 115530 | Yves Minet | 10/2014 |
6508 | (41*10^115523-383)/9 | 115524 | Robert Price | 11/2018 |
6509 | 27316^16923+16923^27316 | 115506 | Hans Havermann | 02/2021 |
6510 | (2^383618+618)/2 | 115481 | Stefano Morozzi | 09/2020 |
6511 | 2^383560-455 | 115464 | Yves Minet | 03/2012 |
6512 | (2^383544+5)/3 | 115458 | Lelio R Paula | 03/2014 |
6513 | (22*10^115454+119)/3 | 115455 | Robert Price | 04/2021 |
6514 | 89^59221-88^59221 | 115445 | Robert Price | 06/2015 |
6515 | (7^136559-5^136559)/2 | 115406 | Jon Grantham | 08/2023 |
6516 | 2^383347+1869413 | 115399 | Enzo Doria | 05/2016 |
6517 | (103*10^115366+17)/3 | 115368 | Robert Price | 08/2020 |
6518 | (2^383232+750)/2 | 115365 | Stefano Morozzi | 09/2020 |
6519 | (2^383130-7)/131793 | 115329 | Lelio R Paula | 03/2014 |
6520 | (12^106859+1)/13 | 115320 | Paul Bourdelais | 02/2010 |
6521 | 29961^7060+7060^29961 | 115315 | Hans Havermann | 01/2021 |
6523 | 2^383039+1869509 | 115307 | Enzo Doria | 05/2016 |
6522 | 32918^3183+3183^32918 | 115307 | Hans Havermann | 01/2021 |
6524 | (109^56597-1)/90183471012 | 115302 | Paul Vanderveen | 06/2025 |
6525 | 36964^1315+1315^36964 | 115288 | Hans Havermann | 02/2021 |
6526 | (13*2^382968+1)/7 | 115286 | Lelio R Paula | 04/2014 |
6527 | (22*10^115282+29)/3 | 115283 | Robert Price | 02/2021 |
6528 | (2^382945+2046)/2 | 115278 | Stefano Morozzi | 09/2020 |
6529 | (2^382789+3182)/2 | 115231 | Stefano Morozzi | 09/2020 |
6530 | 30416^6141+6141^30416 | 115224 | Hans Havermann | 02/2021 |
6531 | 6*10^115211+79 | 115212 | Robert Price | 09/2020 |
6532 | (2^382722+3598)/2 | 115211 | Stefano Morozzi | 09/2020 |
6533 | 27987^13078+13078^27987 | 115210 | Hans Havermann | 02/2021 |
6534 | (2^382699+2090)/2 | 115204 | Stefano Morozzi | 09/2020 |
6535 | (22*10^115188-193)/9 | 115189 | Robert Price | 07/2022 |
6536 | 2^382633+73697 | 115185 | Enzo Doria | 04/2016 |
6537 | 2^382509-113593 | 115147 | Enzo Doria | 08/2014 |
6538 | 2^382502-2791721 | 115145 | Enzo Doria | 03/2016 |
6539 | (2^382495+1274)/2 | 115143 | Stefano Morozzi | 01/2021 |
6540 | 2^382464+22741 | 115134 | Enzo Doria | 09/2023 |
6541 | 26752^20065+20065^26752 | 115099 | Hans Havermann | 02/2021 |
6542 | 26383^23008+23008^26383 | 115080 | Hans Havermann | 02/2021 |
6543 | 9^120590-120590 | 115073 | Robert Price | 03/2014 |
6544 | 28203^12016+12016^28203 | 115062 | Hans Havermann | 02/2021 |
6545 | 2^382198+9 | 115054 | Lelio R Paula | 03/2013 |
6546 | 18268*31^77131+22839 | 115035 | Eric Chen | 05/2024 |
6547 | 30101^6630+6630^30101 | 115032 | Hans Havermann | 01/2021 |
6548 | (89*10^115030-179)/9 | 115031 | Robert Price | 12/2019 |
6549 | 25554^26089-26089^25554 | 114987 | Frank Doornink | 02/2024 |
6550 | 25142^26129-26129^25142 | 114979 | Frank Doornink | 12/2023 |
6551 | (2^381935+2774)/2 | 114974 | Stefano Morozzi | 02/2021 |
6552 | 27300^16271+16271^27300 | 114972 | Hans Havermann | 02/2021 |
6553 | (23*10^114962+7)/3 | 114963 | Robert Price | 05/2023 |
6554 | 2^381883-59197 | 114959 | Enzo Doria | 10/2015 |
6555 | 27423^15542+15542^27423 | 114944 | Hans Havermann | 02/2021 |
6556 | (2^381820+1626)/2 | 114939 | Stefano Morozzi | 09/2020 |
6558 | 2^381806-4481 | 114936 | Yves Minet | 07/2011 |
6557 | 2^381808-1851507 | 114936 | Enzo Doria | 08/2015 |
6559 | (2^381792+2502)/2 | 114931 | Stefano Morozzi | 09/2020 |
6560 | 2^381774-112373 | 114926 | Enzo Doria | 06/2014 |
6561 | 26341^22980+22980^26341 | 114883 | Hans Havermann | 01/2021 |
6562 | (79^60539-1)/61747564999146 | 114867 | Paul Vanderveen | 05/2025 |
6563 | 2^381555-109051 | 114860 | Enzo Doria | 04/2014 |
6564 | 2^381551+213683 | 114859 | Enzo Doria | 09/2023 |
6565 | 2^381546-113501 | 114857 | Enzo Doria | 08/2014 |
6566 | (7*10^114845-19)/3 | 114846 | Robert Price | 10/2020 |
6567 | 26675^20178+20178^26675 | 114833 | Hans Havermann | 02/2021 |
6568 | (2^381436+1182)/2 | 114824 | Stefano Morozzi | 09/2020 |
6569 | 3^240644+16 | 114817 | Robert Price | 04/2020 |
6570 | (13^114814)%(10^114814) | 114814 | Henri Lifchitz | 05/2025 |
6571 | 2^381360-110429 | 114801 | Enzo Doria | 05/2014 |
6572 | (17*10^114795+79)/3 | 114796 | Robert Price | 04/2019 |
6573 | (66^63079+1)/67 | 114774 | Paul Bourdelais | 09/2017 |
6574 | 2^381252-115079 | 114769 | Enzo Doria | 06/2015 |
6575 | (2^381251+3186)/2 | 114768 | Stefano Morozzi | 09/2020 |
6576 | 2^381209-29323 | 114756 | Enzo Doria | 07/2015 |
6577 | 2^381207-111546435 | 114755 | Paul Bourdelais | 10/2021 |
6578 | 569^41651-568^41651 | 114754 | Jean-Louis Charton | 11/2009 |
6579 | 2^381201+1873205 | 114753 | Enzo Doria | 06/2016 |
6581 | (85*10^114737+293)/9 | 114738 | Robert Price | 12/2019 |
6582 | (265*10^114736+17)/3 | 114738 | Robert Price | 03/2020 |
6580 | 379*10^114735+9 | 114738 | Jason Yuen | 01/2025 |
6583 | (2^381062+2170)/2 | 114711 | Stefano Morozzi | 09/2020 |
6584 | 2^380974-112991 | 114685 | Enzo Doria | 07/2014 |
6585 | 37097^1234+1234^37097 | 114679 | Hans Havermann | 02/2021 |
6586 | (26*10^114658-119)/3 | 114659 | Robert Price | 07/2023 |
6587 | 88^58949-87 | 114626 | Norbert Schneider | 11/2015 |
6588 | (2^380773+662)/2 | 114624 | Stefano Morozzi | 09/2020 |
6589 | Phi(155285,10)/242674820198201 | 114610 | Makoto Kamada | 02/2024 |
6590 | (10^155285-1)*(10^2389-1)*(10^13-1)*11111/(10^31057-1)/(10^11945-1)/(10^65-1)/242674820198201 | 114610 | Anonymous | 04/2025 |
6591 | 29913^6782+6782^29913 | 114608 | Hans Havermann | 02/2021 |
6592 | (331*10^114605-7)/9 | 114607 | Robert Price | 04/2022 |
6593 | 2^380700+91 | 114603 | Lelio R Paula | 03/2013 |
6594 | (2^380632+2578)/2 | 114582 | Stefano Morozzi | 09/2020 |
6595 | 7*10^114577+27 | 114578 | Robert Price | 11/2020 |
6597 | 10^114575+3^114576 | 114576 | Mike Oakes | 09/2010 |
6596 | 10^114575+66561 | 114576 | Patrick De Geest | 02/2017 |
6598 | 10^114575-581081 | 114575 | Patrick De Geest | 02/2017 |
6599 | 2^380587-115315 | 114569 | Enzo Doria | 06/2015 |
6600 | 31874^3927+3927^31874 | 114558 | Hans Havermann | 02/2021 |
6601 | 2^380513-216925 | 114546 | Enzo Doria | 05/2014 |
6602 | 2^380451-1840087 | 114528 | Enzo Doria | 06/2015 |
6603 | (2^380426+1510)/2 | 114520 | Stefano Morozzi | 09/2020 |
6604 | (15*2^380388-1)/539 | 114507 | Lelio R Paula | 04/2014 |
6605 | 2^380279-461347 | 114476 | Enzo Doria | 07/2015 |
6606 | (331*10^114454-7)/9 | 114456 | Robert Price | 04/2022 |
6607 | 27121^16590+16590^27121 | 114447 | Hans Havermann | 01/2021 |
6608 | 25966^25555+25555^25966 | 114445 | Hans Havermann | 01/2021 |
6609 | (611*32^76029-53)/31 | 114437 | Eric Chen | 11/2023 |
6610 | (11*10^114400+529)/9 | 114401 | Robert Price | 09/2021 |
6611 | 2^380021+1871147 | 114398 | Enzo Doria | 05/2016 |
6612 | 2^380005+1869707 | 114393 | Enzo Doria | 05/2016 |
6613 | 2^379958-113297 | 114379 | Enzo Doria | 07/2014 |
6614 | (2^379897+1274)/2 | 114361 | Stefano Morozzi | 01/2021 |
6615 | 2^379881-109105 | 114356 | Enzo Doria | 04/2014 |
6616 | (21*2^379857-1)/13 | 114349 | Lelio R Paula | 03/2014 |
6617 | (4*10^114340+83)/3 | 114341 | Robert Price | 02/2018 |
6618 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
6619 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
6620 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
6621 | 27764^13065+13065^27764 | 114280 | Hans Havermann | 01/2021 |
6622 | (132^53891-1)/131 | 114278 | Anonymous | 06/2024 |
6623 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
6624 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
6625 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
6626 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
6627 | (22*10^114178+47)/3 | 114179 | Robert Price | 06/2021 |
6628 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
6629 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
6630 | (228^48407-1)/227 | 114139 | Alexander Zhirkov | 03/2023 |
6631 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
6632 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
6633 | 332*25^81584+12357 | 114053 | Eric Chen | 10/2023 |
6634 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
6635 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
6636 | 13643^27572-27572^13643 | 114008 | Norbert Schneider | 12/2022 |
6638 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
6637 | (83*10^113969+133)/9 | 113970 | Robert Price | 08/2021 |
6639 | 32226^3439+3439^32226 | 113966 | Hans Havermann | 02/2021 |
6640 | (77*10^113963-149)/9 | 113964 | Robert Price | 03/2022 |
6641 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
6642 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
6643 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
6646 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
6644 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
6645 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
6647 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
6648 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
6649 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
6651 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
6650 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
6652 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
6653 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
6654 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
6655 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
6656 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
6657 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
6658 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
6659 | 2^378137+1334025 | 113831 | Henri Lifchitz | 10/2024 |
6660 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
6661 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
6662 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
6663 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
6664 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
6665 | 34596^1949+1949^34596 | 113815 | Hans Havermann | 02/2021 |
6666 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
6667 | (28*10^113810+449)/9 | 113811 | Robert Price | 11/2021 |
6668 | 26273^21448+21448^26273 | 113799 | Hans Havermann | 02/2021 |
6669 | 2^377966+255255 | 113780 | Paul Bourdelais | 10/2021 |
6670 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
6671 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
6672 | (7*10^113757+11)/9 | 113757 | Tyler Busby | 01/2023 |
6673 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
6674 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
6675 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
6676 | (2^377791+3258)/2 | 113727 | Stefano Morozzi | 02/2021 |
6677 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
6678 | 28206^10739+10739^28206 | 113698 | Hans Havermann | 01/2021 |
6679 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
6680 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
6681 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
6682 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
6683 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |
6684 | (10^113656+7)/40231/6439001 | 113645 | Alfred Reich | 06/2021 |
6685 | (2*10^113631+529)/9 | 113631 | Robert Price | 01/2018 |
6686 | 2^377408-2798279 | 113612 | Enzo Doria | 04/2016 |
6687 | (292^46336+1)/(292^256+1) | 113605 | Martin Hopf | 12/2023 |
6688 | 13706^27459+27459^13706 | 113596 | Norbert Schneider | 12/2020 |
6689 | 5*10^113586+57 | 113587 | Robert Price | 12/2018 |
6690 | (2^377242+2794)/2 | 113561 | Stefano Morozzi | 02/2021 |
6691 | 2^377222+93679 | 113556 | Yves Minet | 02/2011 |
6693 | 262147#+316847 | 113552 | Makoto Morimoto | 02/2016 |
6692 | 27172^15101+15101^27172 | 113552 | Hans Havermann | 01/2021 |
6694 | (2^377131+2018)/2 | 113528 | Stefano Morozzi | 09/2020 |
6695 | 2^377026-458903 | 113497 | Enzo Doria | 04/2015 |
6696 | (2^377014+2202)/2 | 113493 | Stefano Morozzi | 09/2020 |
6697 | 34791^1828+1828^34791 | 113488 | Hans Havermann | 02/2021 |
6698 | (2^376967+258)/2 | 113479 | Stefano Morozzi | 09/2020 |
6699 | (29*10^113468-17)/3 | 113469 | Robert Price | 09/2019 |
6700 | 35^73477-34^73477 | 113454 | Robert Price | 05/2012 |
6701 | (14*10^113451-89)/3 | 113452 | Robert Price | 05/2020 |
6702 | 2^376790+27 | 113426 | Lelio R Paula | 03/2013 |
6703 | 2^376786-112701 | 113424 | Enzo Doria | 06/2014 |
6704 | 2^376731+21 | 113408 | Lelio R Paula | 03/2013 |
6705 | 9*10^113394-73 | 113395 | Robert Price | 01/2021 |
6706 | 29660^6651+6651^29660 | 113387 | Hans Havermann | 01/2021 |
6708 | 2^376634+1871245 | 113379 | Enzo Doria | 05/2016 |
6707 | (4^188318+17)/3 | 113379 | Robert Price | 02/2017 |
6709 | E(29532)/492842728006750534663978950979381235 | 113367 | Thomas Ritschel | 01/2021 |
6710 | 2^376576-344369 | 113361 | Enzo Doria | 07/2015 |
6711 | 2^376549-4543 | 113353 | Yves Minet | 03/2019 |
6712 | (17*2^376526-1)/223 | 113345 | Lelio R Paula | 03/2014 |
6713 | (27*2^376441-1)/811 | 113319 | Lelio R Paula | 03/2014 |
6714 | (2^376350+3538)/2 | 113293 | Stefano Morozzi | 09/2020 |
6715 | 27265^14272+14272^27265 | 113273 | Hans Havermann | 02/2021 |
6716 | 32174^3315+3315^32174 | 113269 | Hans Havermann | 02/2021 |
6717 | (2*10^113248+73)/3 | 113248 | Robert Price | 09/2020 |
6718 | 2^376128-21237 | 113226 | Enzo Doria | 06/2013 |
6719 | 5*10^113224-81 | 113225 | Robert Price | 01/2019 |
6720 | 10^113220+51409 | 113221 | Patrick De Geest | 02/2021 |
6722 | 10^113220-1000001 | 113220 | Robert Price | 08/2020 |
6721 | 10^113220-61779 | 113220 | Patrick De Geest | 02/2021 |
6723 | (2^376102+1578)/2 | 113218 | Stefano Morozzi | 09/2020 |
6724 | (2^375996+3102)/2 | 113186 | Stefano Morozzi | 09/2020 |
6725 | 37258^1091+1091^37258 | 113184 | Hans Havermann | 02/2021 |
6726 | (94*10^113176+11)/3 | 113178 | Robert Price | 04/2020 |
6727 | (46*10^113162+521)/9 | 113163 | Robert Price | 02/2019 |
6728 | 14*29^77375+17875 | 113155 | Eric Chen | 06/2025 |
6729 | (2^375885+3614)/2 | 113153 | Stefano Morozzi | 09/2020 |
6730 | (2^375873+2006)/2 | 113149 | Stefano Morozzi | 09/2020 |
6731 | 3*10^113137+23 | 113138 | Robert Price | 03/2022 |
6732 | (8*10^113135-269)/9 | 113135 | Robert Price | 09/2021 |
6733 | (2^375726+694)/2 | 113105 | Stefano Morozzi | 09/2020 |
6734 | 26050^21959+21959^26050 | 113100 | Hans Havermann | 02/2021 |
6735 | 2^375638-1787171 | 113079 | Enzo Doria | 03/2016 |
6736 | (2^375563+2178)/2 | 113056 | Stefano Morozzi | 09/2020 |
6737 | 2^375537+4847 | 113048 | Yves Minet | 02/2011 |
6738 | (17^91837+16^91837)/33 | 113000 | Robert Price | 08/2013 |
6739 | 5*F(270343)*(F(270343)+1)+1 | 112997 | Serge Batalov | 04/2015 |
6740 | 2^375350-3221 | 112992 | Yves Minet | 07/2011 |
6741 | (13*2^375306+1)/7 | 112979 | Lelio R Paula | 03/2014 |
6742 | 2^375305+76757 | 112979 | Enzo Doria | 05/2016 |
6743 | 2^375258+1874989 | 112964 | Enzo Doria | 07/2016 |
6744 | (29^77237-1)/28 | 112950 | Robert Price | 04/2012 |
6745 | 26620^17483+17483^26620 | 112939 | Hans Havermann | 01/2021 |
6746 | 2^375154+74959 | 112933 | Enzo Doria | 04/2016 |
6747 | (266*10^112930-17)/3 | 112932 | Robert Price | 07/2020 |
6748 | (2^375134+778)/2 | 112927 | Stefano Morozzi | 09/2020 |
6749 | (85*10^112919+293)/9 | 112920 | Robert Price | 12/2019 |
6750 | 29412^6899+6899^29412 | 112907 | Hans Havermann | 02/2021 |