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 ? |
7501 | (19*10^84409-61)/3 | 84410 | Robert Price | 04/2017 |
7502 | 2^280388+117 | 84406 | Lelio R Paula | 08/2012 |
7503 | 2^280389-566533 | 84406 | Paul Bourdelais | 03/2013 |
7504 | 23423^4008+4008^23423 | 84392 | Hans Havermann | 07/2020 |
7505 | (2^280340+3322)/2 | 84391 | Stefano Morozzi | 01/2020 |
7506 | (141584^16384+1)/65537 | 84390 | Syoichiro Yamada | 06/2008 |
7507 | (1897^25741-1)/1896 | 84378 | Serge Batalov | 08/2020 |
7509 | 119^40652+118 | 84376 | Norbert Schneider | 09/2018 |
7508 | (2^280291+2274)/2 | 84376 | Stefano Morozzi | 01/2020 |
7510 | 2^280240-173 | 84361 | Lelio R Paula | 08/2012 |
7511 | (184*10^84348-1)/3 | 84350 | Robert Price | 02/2017 |
7512 | 21068^10083+10083^21068 | 84348 | Hans Havermann | 07/2020 |
7513 | 4*10^84346+93 | 84347 | Robert Price | 01/2017 |
7514 | 7*2^280156-5 | 84337 | Lelio R Paula | 08/2014 |
7515 | (8760^21391-1)/8759 | 84331 | Serge Batalov | 08/2020 |
7516 | 8*7^99782-3 | 84327 | Lelio R Paula | 10/2014 |
7517 | 2^280116+133 | 84324 | Lelio R Paula | 08/2012 |
7518 | 2^280118+54615 | 84324 | Lelio R Paula | 09/2013 |
7519 | (509203*2^280090-1)/3 | 84321 | Lelio R Paula | 04/2015 |
7520 | 6^108354+47 | 84316 | Lelio R Paula | 12/2015 |
7521 | (2^280069+462)/2 | 84309 | Stefano Morozzi | 01/2020 |
7522 | 20657^12058+12058^20657 | 84307 | Hans Havermann | 07/2020 |
7523 | 10^84305+141963 | 84306 | Patrick De Geest | 07/2013 |
7524 | 10^84305-57 | 84305 | Robert Price | 05/2011 |
7525 | 9^88339+3*8^88339-2*4^88339-3*2^88339 | 84297 | Mike Oakes | 01/2011 |
7526 | 75843!4-32 | 84295 | Robert Price | 11/2019 |
7527 | 3^176648-112804 | 84283 | Enzo Doria | 06/2014 |
7528 | 6^108310+115 | 84282 | Lelio R Paula | 12/2015 |
7529 | (11*2^279968-1)/5 | 84280 | Lelio R Paula | 02/2014 |
7530 | 6^108288-16643 | 84265 | Enzo Doria | 02/2013 |
7531 | 2^279917-58695 | 84264 | Lelio R Paula | 08/2013 |
7532 | 65^46477+36 | 84259 | Lelio R Paula | 01/2016 |
7533 | (2^279883+3230)/2 | 84253 | Stefano Morozzi | 01/2020 |
7534 | (2^279876+1422)/2 | 84251 | Stefano Morozzi | 01/2020 |
7535 | 2^279866-1907 | 84249 | Yves Minet | 05/2013 |
7536 | 23140^4373+4373^23140 | 84248 | Hans Havermann | 07/2020 |
7537 | 19736^18567+18567^19736 | 84248 | Hans Havermann | 07/2020 |
7539 | (3^176580-5)/28076 | 84246 | Lelio R Paula | 12/2012 |
7538 | (p(17530)#+4)/2 | 84246 | Robert Price | 09/2017 |
7540 | (2*10^84239+1)/3 | 84239 | Serge Batalov | 07/2009 |
7541 | 18^67103-41 | 84233 | Lelio R Paula | 11/2015 |
7542 | 20^64743-43 | 84233 | Lelio R Paula | 01/2016 |
7543 | (2^279807+2454)/2 | 84230 | Stefano Morozzi | 01/2020 |
7544 | 19717^18672+18672^19717 | 84216 | Hans Havermann | 07/2020 |
7545 | (2^279754+3574)/2 | 84215 | Stefano Morozzi | 01/2020 |
7546 | 143^39070+39070^143 | 84209 | Serge Batalov | 05/2014 |
7547 | 11^80857-5400 | 84204 | Enzo Doria | 01/2013 |
7548 | (2^279642+3130)/2 | 84181 | Stefano Morozzi | 01/2020 |
7549 | (13*10^84179+161)/3 | 84180 | Robert Price | 04/2017 |
7550 | 2^279634-110405 | 84179 | Enzo Doria | 05/2014 |
7551 | (74*10^84170+313)/9 | 84171 | Robert Price | 11/2017 |
7552 | 2^279582-1017 | 84163 | Lelio R Paula | 08/2013 |
7553 | (22*10^84159-19)/3 | 84160 | Robert Price | 02/2017 |
7554 | p(17513)#/2+512 | 84156 | Robert Price | 05/2017 |
7555 | (19*10^84144-43)/3 | 84145 | Robert Price | 10/2017 |
7556 | (9*2^279507-1)/137 | 84139 | Lelio R Paula | 02/2014 |
7557 | 11^80792-810 | 84137 | Enzo Doria | 11/2012 |
7558 | (97*10^84134-1)/3 | 84136 | Serge Batalov | 12/2014 |
7559 | p(17507)#/2+32 | 84125 | Robert Price | 12/2016 |
7560 | (1750^25939-1)/1749 | 84118 | Serge Batalov | 08/2020 |
7561 | 21485^8224+8224^21485 | 84116 | Hans Havermann | 04/2017 |
7562 | 3^176294-340 | 84114 | Lelio R Paula | 12/2015 |
7563 | (3^176287+4)/2191 | 84107 | Lelio R Paula | 12/2012 |
7564 | 2^279388-719 | 84105 | Yves Minet | 05/2012 |
7565 | (22699*2^279378+1)/221 | 84104 | Lelio R Paula | 06/2014 |
7566 | (28*10^84095-67)/3 | 84096 | Robert Price | 10/2020 |
7567 | (2^279343+578)/2 | 84091 | Stefano Morozzi | 01/2020 |
7568 | 2^279332-110447 | 84088 | Enzo Doria | 05/2014 |
7569 | 2^279320+7 | 84084 | Lelio R Paula | 06/2012 |
7570 | 127^39966-2 | 84081 | Lelio R Paula | 10/2017 |
7571 | (2^279299+2490)/2 | 84078 | Stefano Morozzi | 01/2020 |
7572 | (2^279301+1694)/2 | 84078 | Stefano Morozzi | 01/2020 |
7573 | (2^279285+962)/2 | 84073 | Stefano Morozzi | 01/2020 |
7574 | (2^279269+1122)/2 | 84069 | Stefano Morozzi | 01/2020 |
7575 | (11*2^279240-1)/95 | 84059 | Lelio R Paula | 02/2014 |
7576 | 2^279218-111347 | 84053 | Enzo Doria | 05/2014 |
7577 | 35^54435+2 | 84052 | Lelio R Paula | 05/2013 |
7578 | (2^279176+2998)/2 | 84041 | Stefano Morozzi | 01/2020 |
7579 | (31*2^279191-1)/12511523 | 84040 | Lelio R Paula | 02/2014 |
7580 | 20996^10059+10059^20996 | 84038 | Hans Havermann | 05/2017 |
7581 | 97*10^84033-3 | 84035 | Robert Price | 01/2017 |
7582 | (2^279153+1326)/2 | 84034 | Stefano Morozzi | 01/2020 |
7583 | (2^279155+3534)/2 | 84034 | Stefano Morozzi | 01/2020 |
7584 | 21234^9067+9067^21234 | 84033 | Hans Havermann | 05/2017 |
7585 | (134569^16384+1)/2 | 84033 | Richard Fischer | 06/2018 |
7586 | (5858^22303-1)/5857 | 84029 | Serge Batalov | 08/2020 |
7587 | 2^279095-109717 | 84016 | Enzo Doria | 04/2014 |
7588 | (2^279054+1438)/2 | 84004 | Stefano Morozzi | 01/2020 |
7589 | 10^83997-40001 | 83997 | Robert Price | 01/2021 |
7590 | 14^73282-99 | 83991 | Lelio R Paula | 11/2015 |
7591 | 21457^8200+8200^21457 | 83979 | Hans Havermann | 06/2017 |
7592 | 89*10^83976-7 | 83978 | Robert Price | 03/2017 |
7593 | 2^278955+173 | 83974 | Lelio R Paula | 06/2012 |
7594 | (7*2^278851+1)/39 | 83942 | Lelio R Paula | 12/2012 |
7595 | (9913*2^278844-1)/40803 | 83940 | Lelio R Paula | 04/2015 |
7596 | 5^120076-14846 | 83930 | Enzo Doria | 02/2013 |
7597 | (85*10^83921-1)/3 | 83923 | Robert Price | 07/2017 |
7598 | (2^278770-3)/23 | 83917 | Lelio R Paula | 12/2012 |
7599 | (10^83916+449)/9 | 83916 | Serge Batalov | 12/2014 |
7601 | 2^278715+99 | 83902 | Lelio R Paula | 06/2012 |
7600 | (132109^16384+1)/2 | 83902 | Richard Fischer | 05/2018 |
7602 | 40235!2-2^40235 | 83899 | Robert Price | 09/2015 |
7603 | 2^278698-171 | 83897 | Lelio R Paula | 06/2012 |
7604 | (19*10^83894-43)/3 | 83895 | Robert Price | 10/2017 |
7605 | 6^107802+25 | 83887 | Lelio R Paula | 12/2015 |
7606 | 8*10^83880+39 | 83881 | Robert Price | 01/2017 |
7607 | (58*10^83878+23)/9 | 83879 | Robert Price | 09/2015 |
7608 | 7^99244-90 | 83871 | Lelio R Paula | 10/2015 |
7609 | (2^278601+1674)/2 | 83867 | Stefano Morozzi | 01/2020 |
7610 | 2^278554+4257 | 83854 | Lelio R Paula | 08/2013 |
7611 | 2^278504-109619 | 83839 | Enzo Doria | 04/2014 |
7612 | 2^278499+1870433 | 83837 | Enzo Doria | 05/2016 |
7613 | (2^278487+570)/2 | 83833 | Stefano Morozzi | 01/2020 |
7614 | 105^41475+104 | 83829 | Norbert Schneider | 11/2016 |
7615 | 15^71276-68 | 83828 | Lelio R Paula | 11/2015 |
7616 | 2^278464-216413 | 83827 | Enzo Doria | 04/2013 |
7617 | (2^278440+1602)/2 | 83819 | Stefano Morozzi | 01/2020 |
7618 | (76*10^83816-211)/9 | 83817 | Robert Price | 10/2017 |
7619 | (2^278429+1950)/2 | 83816 | Stefano Morozzi | 01/2020 |
7620 | (2^278428+1750)/2 | 83815 | Stefano Morozzi | 01/2020 |
7621 | (8*10^83811-101)/3 | 83812 | Robert Price | 10/2017 |
7622 | 2^278400+40755 | 83807 | Lelio R Paula | 08/2013 |
7623 | (2^278394+3718)/2 | 83805 | Stefano Morozzi | 01/2020 |
7624 | (2^278388+3586)/2 | 83803 | Stefano Morozzi | 01/2020 |
7625 | (9*2^278381-1)/7 | 83802 | Lelio R Paula | 02/2014 |
7626 | 3*9^87813+6^87813+2*4^87813 | 83796 | Mike Oakes | 12/2010 |
7627 | 19551^19268+19268^19551 | 83773 | Hans Havermann | 05/2017 |
7628 | 20625^11522+11522^20625 | 83770 | Hans Havermann | 05/2017 |
7629 | (44*10^83768-791)/9 | 83769 | Robert Price | 10/2017 |
7630 | (87^43189+1)/88 | 83764 | Paul Bourdelais | 08/2017 |
7631 | 2^278235-49 | 83758 | Lelio R Paula | 06/2012 |
7632 | 23543^3610+3610^23543 | 83755 | Hans Havermann | 05/2017 |
7634 | 18^66711-91 | 83741 | Lelio R Paula | 11/2015 |
7633 | 4*10^83740-89 | 83741 | Robert Price | 08/2017 |
7635 | (3*2^278172-1)/11 | 83738 | Lelio R Paula | 02/2014 |
7636 | (15*29^57251+2561)/28 | 83724 | Curtis Bright | 05/2015 |
7637 | 4*7^99059-9 | 83716 | Lelio R Paula | 10/2014 |
7638 | (2^278128-3)/126844724149 | 83714 | Lelio R Paula | 12/2012 |
7639 | 2^278077+1867211 | 83710 | Enzo Doria | 03/2016 |
7640 | (85*10^83708+203)/9 | 83709 | Robert Price | 09/2020 |
7641 | 2^278059-27 | 83705 | Lelio R Paula | 06/2012 |
7642 | 214^35917+35917^214 | 83702 | Norbert Schneider | 12/2014 |
7643 | (23*10^83693+7)/3 | 83694 | Robert Price | 10/2015 |
7644 | 85^43375-2 | 83689 | Lelio R Paula | 09/2013 |
7645 | 5*3^175360+4 | 83669 | Lelio R Paula | 07/2014 |
7646 | (23^61441-1)/22 | 83665 | Robert Price | 01/2012 |
7647 | 20091^14596+14596^20091 | 83664 | Hans Havermann | 03/2017 |
7648 | 2^277907-110629 | 83659 | Enzo Doria | 05/2014 |
7649 | 19522^19283+19283^19522 | 83656 | Hans Havermann | 04/2017 |
7650 | (2^277881+2570)/2 | 83651 | Stefano Morozzi | 01/2020 |
7651 | 8*5^119664-3 | 83643 | Lelio R Paula | 10/2014 |
7652 | 8*3^175299-7 | 83640 | Lelio R Paula | 07/2014 |
7653 | (26*10^83638-413)/9 | 83639 | Robert Price | 10/2017 |
7654 | (197*10^83636+7)/3 | 83638 | Robert Price | 12/2020 |
7655 | 3^175250+22 | 83616 | Robert Price | 12/2013 |
7656 | 11^80285+30 | 83609 | Lelio R Paula | 10/2015 |
7657 | 26870^1293+1293^26870 | 83609 | Hans Havermann | 03/2017 |
7658 | (6730^21841-1)/6729 | 83604 | Serge Batalov | 08/2020 |
7659 | (2828^24223-1)/2827 | 83602 | Serge Batalov | 08/2020 |
7660 | 2^277691+1874423 | 83594 | Enzo Doria | 06/2016 |
7661 | 13^75022-60 | 83571 | Lelio R Paula | 11/2015 |
7662 | (2^277593+2622)/2 | 83564 | Stefano Morozzi | 01/2020 |
7663 | 14^72905-51 | 83559 | Lelio R Paula | 11/2015 |
7664 | (17*2^277565-1)/543 | 83554 | Lelio R Paula | 02/2014 |
7665 | 5^119532-12 | 83550 | Lelio R Paula | 10/2015 |
7666 | 6^107358-16253 | 83541 | Enzo Doria | 02/2013 |
7667 | 2^277478-460211 | 83530 | Enzo Doria | 06/2015 |
7668 | (2^277474+814)/2 | 83528 | Stefano Morozzi | 01/2020 |
7669 | (22*10^83525-7)/3 | 83526 | Erik Branger | 09/2014 |
7670 | (30^56543+1)/31 | 83520 | Robert Price | 07/2013 |
7671 | 2^277417-460783 | 83511 | Paul Bourdelais | 04/2013 |
7672 | (2^277404+3582)/2 | 83507 | Stefano Morozzi | 01/2020 |
7673 | (22699*2^277387+1)/219 | 83504 | Lelio R Paula | 06/2014 |
7674 | (31*2^277383-1)/247 | 83500 | Lelio R Paula | 02/2014 |
7675 | 21368^8085+8085^21368 | 83500 | Hans Havermann | 04/2017 |
7676 | (2^277353+3750)/2 | 83492 | Stefano Morozzi | 01/2020 |
7677 | (2^277348+3538)/2 | 83490 | Stefano Morozzi | 01/2020 |
7678 | (9*2^277342+1)/13 | 83489 | Lelio R Paula | 12/2012 |
7679 | 5^119439-66 | 83485 | Lelio R Paula | 10/2015 |
7680 | (14*10^83471-101)/3 | 83472 | Robert Price | 04/2017 |
7681 | 5*6^107263+7 | 83468 | Lelio R Paula | 10/2014 |
7682 | (64*10^83461-1)/9 | 83462 | Serge Batalov | 11/2010 |
7683 | 2^277244-175709 | 83459 | Hans Havermann | 10/2018 |
7684 | (2^277230+1450)/2 | 83455 | Stefano Morozzi | 01/2020 |
7685 | (2^277232+3658)/2 | 83455 | Stefano Morozzi | 01/2020 |
7686 | 17^67796-92 | 83420 | Lelio R Paula | 11/2015 |
7687 | 2^277106-110057 | 83418 | Enzo Doria | 04/2014 |
7688 | 6^107200-19 | 83418 | Lelio R Paula | 12/2015 |
7689 | 11^80096-1830 | 83412 | Enzo Doria | 12/2012 |
7690 | 3^174801+56 | 83402 | Lelio R Paula | 12/2015 |
7691 | 2^277033-111973 | 83396 | Enzo Doria | 05/2014 |
7692 | 120^40105-119 | 83386 | Norbert Schneider | 09/2018 |
7693 | (2^276992+2046)/2 | 83383 | Stefano Morozzi | 01/2020 |
7694 | (2^276990+474)/2 | 83382 | Stefano Morozzi | 01/2020 |
7695 | 20519^11572+11572^20519 | 83378 | Hans Havermann | 03/2017 |
7696 | 20850^9971+9971^20850 | 83374 | Hans Havermann | 03/2017 |
7699 | 6^107122-18863 | 83358 | Enzo Doria | 03/2013 |
7697 | (2^276908+2322)/2 | 83358 | Stefano Morozzi | 01/2020 |
7698 | (2^276910+2734)/2 | 83358 | Stefano Morozzi | 01/2020 |
7700 | 4*10^83354-89 | 83355 | Robert Price | 08/2017 |
7701 | 3^174697-34 | 83352 | Robert Price | 11/2013 |
7703 | (9787*2^276855-1)/2345 | 83343 | Lelio R Paula | 04/2015 |
7702 | (2^276860+1678)/2 | 83343 | Stefano Morozzi | 01/2020 |
7704 | (2^276841+3354)/2 | 83338 | Stefano Morozzi | 01/2020 |
7706 | 2^276825+75227 | 83333 | Enzo Doria | 04/2016 |
7705 | 34*10^83331-9 | 83333 | Robert Price | 06/2020 |
7707 | 5^119214+84 | 83328 | Lelio R Paula | 10/2015 |
7708 | 2^276803-159 | 83327 | Lelio R Paula | 06/2012 |
7709 | (3*2^276753-1)/25 | 83311 | Lelio R Paula | 02/2014 |
7710 | 7*10^83297-39 | 83298 | Robert Price | 07/2020 |
7711 | 114^40495+40495^114 | 83295 | Norbert Schneider | 12/2014 |
7712 | 39967!2+1024 | 83282 | Robert Price | 02/2015 |
7713 | (1894^25411-1)/1893 | 83279 | Serge Batalov | 08/2020 |
7715 | 2^276641+57 | 83278 | Lelio R Paula | 06/2012 |
7716 | 35^53934-3414 | 83278 | Enzo Doria | 03/2013 |
7714 | 21777^6670+6670^21777 | 83278 | Hans Havermann | 06/2020 |
7717 | (2^276655+78557)/365425 | 83276 | Lelio R Paula | 12/2012 |
7718 | 2^276615-4035 | 83270 | Lelio R Paula | 08/2013 |
7719 | 2^276588+1155 | 83262 | Lelio R Paula | 08/2013 |
7720 | 2^276587-937 | 83261 | Yves Minet | 10/2009 |
7721 | 2^276584-113183 | 83261 | Enzo Doria | 07/2014 |
7722 | (2^276592-1017)/13843739 | 83256 | Lelio R Paula | 12/2012 |
7723 | 2^276547-109591 | 83249 | Enzo Doria | 04/2014 |
7724 | 13^74733-32 | 83249 | Lelio R Paula | 11/2015 |
7725 | 2^276523-2587 | 83242 | Yves Minet | 07/2014 |
7726 | 18^66311-43 | 83239 | Lelio R Paula | 11/2015 |
7727 | 11^79919-1530 | 83228 | Enzo Doria | 12/2012 |
7728 | 2^276468-109745 | 83226 | Enzo Doria | 04/2014 |
7729 | 2^276462+5385 | 83224 | Lelio R Paula | 08/2013 |
7730 | (9787*2^276422-1)/3 | 83215 | Lelio R Paula | 04/2015 |
7732 | 2^276391-15 | 83202 | Lelio R Paula | 06/2012 |
7731 | (4*10^83201+173)/3 | 83202 | Robert Price | 01/2017 |
7733 | (259*10^83186-1)/3 | 83188 | Robert Price | 12/2020 |
7734 | (2^276327+534)/2 | 83183 | Stefano Morozzi | 01/2020 |
7735 | 2^276316-109655 | 83180 | Enzo Doria | 04/2014 |
7736 | 2^276305-112063 | 83177 | Enzo Doria | 06/2014 |
7737 | (2^276296+3406)/2 | 83174 | Stefano Morozzi | 01/2020 |
7738 | (10^(2*41585+1)+54*10^41585-1)/9 | 83171 | Robert Price | 04/2017 |
7739 | (2^276277+3254)/2 | 83168 | Stefano Morozzi | 01/2020 |
7740 | 74913!4-2 | 83161 | Robert Price | 07/2017 |
7741 | (3023^23893-1)/3022 | 83155 | Serge Batalov | 08/2020 |
7742 | 2^276227-1837207 | 83153 | Enzo Doria | 05/2015 |
7744 | (2^276227+3)/7 | 83152 | Lelio R Paula | 12/2012 |
7743 | (2^276229+12)/28 | 83152 | Stefano Morozzi | 03/2020 |
7745 | (2^276205+1190)/2 | 83146 | Stefano Morozzi | 01/2020 |
7746 | 2^276184-63389 | 83140 | Paul Bourdelais | 02/2012 |
7747 | (2^276165+2046)/2 | 83134 | Stefano Morozzi | 01/2020 |
7748 | (2^276105+1274)/2 | 83116 | Stefano Morozzi | 01/2020 |
7749 | 2^276066-111603 | 83105 | Enzo Doria | 05/2014 |
7750 | 19438^18837+18837^19438 | 83098 | Hans Havermann | 07/2020 |