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 | 5^154822-12866 | 108216 | Enzo Doria | 05/2014 |
6502 | (9*2^359480+1)/5 | 108215 | Lelio R Paula | 03/2014 |
6503 | 2^359454-177 | 108207 | Lelio R Paula | 12/2011 |
6504 | (19*10^108184+191)/3 | 108185 | Robert Price | 05/2019 |
6505 | 6^139022-59 | 108181 | Lelio R Paula | 02/2017 |
6506 | (19*10^108176-43)/3 | 108177 | Robert Price | 05/2019 |
6507 | 2^359352-179 | 108176 | Lelio R Paula | 12/2011 |
6508 | 2^359338-1858781 | 108172 | Enzo Doria | 04/2016 |
6509 | 12^100230-97 | 108167 | Lelio R Paula | 03/2017 |
6510 | 27028^10043+10043^27028 | 108163 | Hans Havermann | 11/2020 |
6511 | (25*2^359297-1)/137 | 108159 | Lelio R Paula | 03/2014 |
6512 | (2^359262-509203)/9 | 108148 | Lelio R Paula | 12/2013 |
6513 | 26583^11702+11702^26583 | 108147 | Hans Havermann | 11/2020 |
6514 | (3^226648-8)/352715633 | 108131 | Lelio R Paula | 12/2013 |
6515 | 6^138956-59 | 108129 | Lelio R Paula | 02/2017 |
6516 | 2^359184+30285 | 108126 | Lelio R Paula | 04/2017 |
6517 | 13^97059+10 | 108119 | Lelio R Paula | 03/2017 |
6518 | 2^359137-81 | 108112 | Lelio R Paula | 12/2011 |
6520 | (15185*2^359105-1)/567 | 108103 | Lelio R Paula | 05/2015 |
6519 | 2^359110-11655 | 108103 | Lelio R Paula | 04/2017 |
6521 | 2^359069-933075 | 108091 | Paul Bourdelais | 06/2021 |
6522 | 26227^13222+13222^26227 | 108090 | Hans Havermann | 10/2020 |
6523 | 8*6^138875+5 | 108067 | Lelio R Paula | 11/2014 |
6524 | 2^358987-216919 | 108066 | Enzo Doria | 05/2014 |
6525 | (2^358977+2714)/2 | 108063 | Stefano Morozzi | 09/2020 |
6526 | 39^67918-67918^39 | 108062 | Norbert Schneider | 02/2017 |
6527 | 24^78283-1*78283^24 | 108048 | Dylan Delgado | 12/2019 |
6528 | 25282^18767+18767^25282 | 108040 | Hans Havermann | 11/2020 |
6529 | 2^358785-16933 | 108006 | Enzo Doria | 05/2013 |
6530 | (2^358783+5)/33383 | 108000 | Lelio R Paula | 12/2013 |
6531 | 6^138770-77 | 107985 | Lelio R Paula | 02/2017 |
6532 | (2^358727-1017)/94178479 | 107980 | Lelio R Paula | 12/2013 |
6533 | 10^107976+19 | 107977 | Lelio R Paula | 02/2017 |
6534 | 10^107976-202103 | 107976 | Patrick De Geest | 03/2017 |
6535 | 2^358684+75781 | 107975 | Enzo Doria | 04/2016 |
6536 | 24928^21411+21411^24928 | 107955 | Hans Havermann | 10/2020 |
6537 | 99^54094+2 | 107952 | Lelio R Paula | 06/2013 |
6538 | (2^358583-9)/391 | 107942 | Lelio R Paula | 12/2013 |
6539 | (2^358562+838)/2 | 107938 | Stefano Morozzi | 02/2021 |
6540 | 2^358538+55 | 107931 | Lelio R Paula | 12/2011 |
6541 | 2^358517-109009 | 107925 | Enzo Doria | 03/2014 |
6542 | 26395^12258+12258^26395 | 107914 | Hans Havermann | 10/2020 |
6543 | 3^226159+104 | 107906 | Lelio R Paula | 03/2017 |
6544 | 7*10^107870+39 | 107871 | Robert Price | 01/2023 |
6545 | 2^358310-11655 | 107863 | Lelio R Paula | 03/2017 |
6546 | (68*10^107852-329)/9 | 107853 | Robert Price | 03/2021 |
6547 | 2^358275-112987 | 107852 | Enzo Doria | 07/2014 |
6548 | 3^226038+4 | 107848 | Tyler NeSmith | 01/2021 |
6549 | 2^358233-20005 | 107839 | Enzo Doria | 06/2013 |
6550 | 2^358194+175 | 107828 | Lelio R Paula | 12/2011 |
6551 | 107^53125-106 | 107812 | Norbert Schneider | 12/2016 |
6552 | (25*10^107804+161)/3 | 107805 | Robert Price | 09/2019 |
6553 | 3^225922-2 | 107793 | Lelio R Paula | 10/2013 |
6554 | 12^99874-95 | 107783 | Lelio R Paula | 03/2017 |
6555 | (2^358042+3042)/2 | 107782 | Stefano Morozzi | 09/2020 |
6556 | 2^358030-171 | 107778 | Lelio R Paula | 12/2011 |
6557 | 11^103488-4170 | 107772 | Enzo Doria | 12/2012 |
6558 | 2^357930-4547 | 107748 | Yves Minet | 03/2019 |
6559 | 2^357911-20089 | 107742 | Enzo Doria | 06/2013 |
6560 | 2^357885+12857 | 107735 | Yves Minet | 07/2010 |
6561 | (248587#+9289)/(7*1327*111512899) | 107733 | Makoto Morimoto | 06/2019 |
6562 | 7^127473+40 | 107728 | Lelio R Paula | 03/2017 |
6563 | 5*2^357850+9 | 107725 | Lelio R Paula | 09/2014 |
6564 | 2^357786-291191 | 107705 | Enzo Doria | 06/2015 |
6565 | 28907^5320+5320^28907 | 107705 | Hans Havermann | 10/2020 |
6566 | 18^85798-23 | 107700 | Lelio R Paula | 04/2017 |
6567 | 3*10^107697+61 | 107698 | Robert Price | 09/2020 |
6568 | (14*10^107694-83)/3 | 107695 | Robert Price | 12/2018 |
6569 | 2^357688-1854269 | 107675 | Enzo Doria | 09/2015 |
6570 | 2^357671-112225 | 107670 | Enzo Doria | 06/2014 |
6571 | 34323^1370+1370^34323 | 107662 | Hans Havermann | 11/2020 |
6572 | 2^357618-21005 | 107654 | Enzo Doria | 06/2013 |
6573 | 2^357615-17505 | 107653 | Enzo Doria | 05/2013 |
6574 | 26350^12171+12171^26350 | 107649 | Hans Havermann | 11/2020 |
6575 | (30^72871-1)/29 | 107638 | Robert Price | 12/2011 |
6576 | 25021^20026+20026^25021 | 107631 | Hans Havermann | 11/2020 |
6577 | 2^357521+1870217 | 107625 | Enzo Doria | 05/2016 |
6578 | 2^357477-1203 | 107612 | Lelio R Paula | 03/2017 |
6579 | 26008^13693+13693^26008 | 107583 | Hans Havermann | 10/2020 |
6580 | 25432^16991+16991^25432 | 107583 | Hans Havermann | 10/2020 |
6581 | (2^357371+3726)/2 | 107580 | Stefano Morozzi | 09/2020 |
6582 | 5^153894-18 | 107568 | Lelio R Paula | 02/2017 |
6583 | 2^357324-69059 | 107566 | Enzo Doria | 11/2015 |
6584 | 467^40296+40296^467 | 107563 | Norbert Schneider | 03/2019 |
6585 | 29446^4495+4495^29446 | 107559 | Hans Havermann | 10/2020 |
6586 | (8*10^107556-149)/3 | 107557 | Robert Price | 12/2021 |
6587 | 2^357283-869688105 | 107553 | Lelio R Paula | 01/2017 |
6588 | 2^357280-16805 | 107552 | Enzo Doria | 05/2013 |
6589 | 10^107547+481719 | 107548 | Patrick De Geest | 03/2017 |
6590 | 10^107547-47 | 107547 | Lelio R Paula | 02/2017 |
6591 | (23*10^107540+37)/3 | 107541 | Robert Price | 02/2022 |
6592 | 17^87396-32 | 107537 | Lelio R Paula | 04/2017 |
6593 | (746*10^107505+1)/9 | 107507 | Robert Price | 03/2020 |
6594 | (2^357125+1730)/2 | 107506 | Stefano Morozzi | 09/2020 |
6595 | 2^357081+107 | 107493 | Lelio R Paula | 12/2011 |
6596 | 2^357058-3911 | 107486 | Yves Minet | 04/2017 |
6597 | 3^225274+64 | 107484 | Lelio R Paula | 03/2017 |
6598 | 5*10^107482-57 | 107483 | Robert Price | 03/2021 |
6599 | 18^85619-95 | 107476 | Lelio R Paula | 04/2017 |
6600 | (2^357013+2526)/2 | 107472 | Stefano Morozzi | 09/2020 |
6601 | 3^225248-130 | 107471 | Lelio R Paula | 03/2017 |
6602 | (5*2^356961+1)/143 | 107455 | Lelio R Paula | 03/2014 |
6603 | 27029^9448+9448^27029 | 107450 | Hans Havermann | 11/2020 |
6604 | 2^356935-18085 | 107449 | Enzo Doria | 05/2013 |
6605 | 55396^87+87^55396 | 107442 | Hans Havermann | 10/2020 |
6606 | (14*10^107439-137)/3 | 107440 | Robert Price | 04/2021 |
6607 | 10^107409+37 | 107410 | Lelio R Paula | 02/2017 |
6608 | 10^107409-329633 | 107409 | Patrick De Geest | 03/2017 |
6609 | (2^356799+3134)/2 | 107407 | Stefano Morozzi | 09/2020 |
6610 | 2^356787-109027 | 107404 | Enzo Doria | 04/2014 |
6611 | (2^356767+78557)/5 | 107397 | Lelio R Paula | 12/2013 |
6612 | 3*10^107389+77 | 107390 | Robert Price | 09/2018 |
6613 | (11*10^107385+61)/9 | 107386 | Tyler Busby | 01/2023 |
6614 | 5^153622-42 | 107378 | Lelio R Paula | 02/2017 |
6615 | 2^356699-97305 | 107378 | Lelio R Paula | 03/2017 |
6616 | 13129^26072-26072^13129 | 107371 | Norbert Schneider | 09/2022 |
6617 | 2^356654+85 | 107364 | Lelio R Paula | 11/2011 |
6618 | 2^356641-19921 | 107360 | Enzo Doria | 06/2013 |
6619 | 5*10^107355+11 | 107356 | Robert Price | 08/2019 |
6620 | 2^356623-465 | 107355 | Lelio R Paula | 03/2017 |
6621 | 19^83945-92 | 107345 | Lelio R Paula | 02/2017 |
6622 | 25491^16246+16246^25491 | 107337 | Hans Havermann | 11/2020 |
6623 | 10^107334+28149 | 107335 | Patrick De Geest | 03/2017 |
6624 | 10^107334-83 | 107334 | Lelio R Paula | 02/2017 |
6625 | 2^356518+885 | 107323 | Lelio R Paula | 03/2017 |
6626 | (59*10^107322+391)/9 | 107323 | Robert Price | 03/2021 |
6627 | (3*2^356506-1)/467 | 107317 | Lelio R Paula | 03/2014 |
6628 | (2^356468+1162)/2 | 107308 | Stefano Morozzi | 09/2020 |
6629 | 2^356442-114713 | 107300 | Enzo Doria | 04/2015 |
6630 | 2^356425+72257 | 107295 | Enzo Doria | 04/2016 |
6631 | 2^356351-112237 | 107273 | Enzo Doria | 06/2014 |
6633 | (6^137845-11)/5 | 107264 | Paul Bourdelais | 11/2011 |
6632 | 2^356322+74539 | 107264 | Enzo Doria | 04/2016 |
6634 | 2^356274+1874479 | 107250 | Enzo Doria | 06/2016 |
6635 | 24606^22819+22819^24606 | 107241 | Hans Havermann | 10/2020 |
6636 | (2^356243+3686)/2 | 107240 | Stefano Morozzi | 09/2020 |
6637 | 20^82422-93 | 107234 | Lelio R Paula | 02/2017 |
6638 | 2^356157-20365 | 107214 | Enzo Doria | 06/2013 |
6639 | 8*10^107209+99 | 107210 | Robert Price | 03/2021 |
6640 | 2^356129-18949 | 107206 | Enzo Doria | 05/2013 |
6641 | 2^356100-109587 | 107197 | Enzo Doria | 04/2014 |
6642 | 2*10^107173-37 | 107174 | Robert Price | 11/2021 |
6643 | (2^355978+3262)/2 | 107160 | Stefano Morozzi | 02/2021 |
6644 | (31*2^355970+1)/5 | 107159 | Lelio R Paula | 03/2014 |
6645 | 2^355968-64875 | 107158 | Lelio R Paula | 01/2017 |
6646 | 2^355952+1870321 | 107153 | Enzo Doria | 05/2016 |
6647 | 27069^9088+9088^27069 | 107152 | Hans Havermann | 11/2020 |
6648 | 2^355915-117 | 107142 | Lelio R Paula | 11/2011 |
6649 | (16*10^107140+77)/3 | 107141 | Robert Price | 03/2019 |
6650 | 24628^22369+22369^24628 | 107124 | Hans Havermann | 11/2020 |
6651 | (19*2^355838-1)/15 | 107119 | Lelio R Paula | 03/2014 |
6652 | 3^224495-8 | 107112 | Lelio R Paula | 12/2013 |
6653 | 3*10^107096+23 | 107097 | Robert Price | 03/2022 |
6654 | 25077^18632+18632^25077 | 107086 | Hans Havermann | 10/2020 |
6655 | 7^126691-44 | 107067 | Lelio R Paula | 03/2017 |
6656 | 4^177830-8243 | 107065 | Mike Oakes | 02/2011 |
6657 | (509*10^107057+13)/9 | 107059 | Robert Price | 05/2022 |
6658 | 2^355604+214051 | 107048 | Enzo Doria | 10/2023 |
6659 | (2^355562+1702)/2 | 107035 | Stefano Morozzi | 09/2020 |
6660 | 10^107028+3593037 | 107029 | Paul Bourdelais | 10/2021 |
6661 | 2^355524-21237 | 107024 | Enzo Doria | 06/2013 |
6662 | (10223*2^355515+1)/4855 | 107022 | Lelio R Paula | 07/2014 |
6663 | 24843^20320+20320^24843 | 107022 | Hans Havermann | 10/2020 |
6664 | 2^355494-19901 | 107015 | Enzo Doria | 06/2013 |
6665 | 2^355454-869688105 | 107003 | Lelio R Paula | 01/2017 |
6666 | 4*10^106997+13 | 106998 | Robert Price | 02/2021 |
6667 | (8*10^106985-629)/9 | 106985 | Robert Price | 01/2022 |
6668 | 2^355366+54615 | 106976 | Lelio R Paula | 01/2017 |
6669 | (31*2^355346-1)/1563 | 106969 | Lelio R Paula | 03/2014 |
6671 | 2^355275-19861 | 106949 | Enzo Doria | 06/2013 |
6670 | 27503^7738+7738^27503 | 106949 | Hans Havermann | 11/2020 |
6672 | (85*10^106943+77)/9 | 106944 | Robert Price | 07/2022 |
6673 | (9179*2^355234-1)/1915 | 106937 | Lelio R Paula | 05/2015 |
6674 | 4*10^106914-93 | 106915 | Robert Price | 07/2020 |
6675 | 4*10^106892-3 | 106893 | Robert Price | 10/2015 |
6676 | (66^58741+1)/67 | 106881 | Paul Bourdelais | 08/2017 |
6677 | 2^354946-3 | 106850 | M. Frind & P. Underwood | 08/2002 |
6678 | (2^354945+3134)/2 | 106849 | Stefano Morozzi | 09/2020 |
6679 | (46^64384+1)/(46^128+1) | 106843 | Martin Hopf | 12/2023 |
6680 | 2^354873-18693 | 106828 | Enzo Doria | 05/2013 |
6681 | (2^354831+3186)/2 | 106815 | Stefano Morozzi | 09/2020 |
6683 | 2^354828+4257 | 106814 | Lelio R Paula | 01/2017 |
6682 | 25216^17217+17217^25216 | 106814 | Hans Havermann | 10/2020 |
6684 | (2^354799+1218)/2 | 106805 | Stefano Morozzi | 09/2020 |
6685 | 436^40459+40459^436 | 106791 | Norbert Schneider | 03/2019 |
6686 | (21^80761+1)/22 | 106783 | Paul Bourdelais | 03/2010 |
6687 | (115*10^106772-7)/9 | 106774 | Robert Price | 02/2020 |
6688 | 2^354640-354640+1 | 106758 | Robert Price | 10/2014 |
6689 | (2^354603+2606)/2 | 106746 | Stefano Morozzi | 09/2020 |
6690 | 2^354585+95 | 106741 | Lelio R Paula | 10/2011 |
6691 | (206*10^106737+1)/9 | 106739 | Robert Price | 04/2022 |
6692 | 2^354553+189 | 106732 | Lelio R Paula | 10/2011 |
6694 | (2^354546+1338)/2 | 106729 | Stefano Morozzi | 02/2021 |
6693 | (8*10^106729-71)/9 | 106729 | Serge Batalov | 11/2022 |
6695 | 2^354514-109031 | 106720 | Enzo Doria | 04/2014 |
6696 | 2^354509-109465 | 106718 | Enzo Doria | 04/2014 |
6697 | fibonacci(510628)+2 | 106715 | Jon Grantham | 08/2023 |
6698 | (15*2^354478-1)/26983 | 106706 | Lelio R Paula | 03/2014 |
6699 | Phi(217618,10)/899798636917 | 106693 | Makoto Kamada | 06/2019 |
6700 | 15^90710+28 | 106684 | Lelio R Paula | 04/2017 |
6701 | 7^126237+34 | 106683 | Lelio R Paula | 03/2017 |
6702 | 2^354385-115315 | 106681 | Enzo Doria | 06/2015 |
6703 | 2^354327-19305 | 106664 | Enzo Doria | 06/2013 |
6704 | (35*10^106649+1)/9 | 106650 | Serge Batalov | 11/2010 |
6705 | 10^106636+97 | 106637 | Lelio R Paula | 02/2017 |
6706 | 10^106636-22923 | 106636 | Patrick De Geest | 03/2017 |
6707 | 2^354220-57 | 106631 | Lelio R Paula | 10/2011 |
6708 | (68*10^106628-329)/9 | 106629 | Robert Price | 03/2021 |
6709 | (29*10^106620-77)/3 | 106621 | Robert Price | 02/2020 |
6710 | (746*10^106618+1)/9 | 106620 | Robert Price | 03/2020 |
6711 | 2*10^106613+29 | 106614 | Robert Price | 02/2021 |
6712 | 2^354157-17041 | 106612 | Enzo Doria | 05/2013 |
6713 | 12^98778+73 | 106600 | Lelio R Paula | 02/2017 |
6714 | 33333^1576+1576^33333 | 106585 | Hans Havermann | 11/2020 |
6715 | 2^354064+1870651 | 106584 | Enzo Doria | 05/2016 |
6716 | 24568^21779+21779^24568 | 106577 | Hans Havermann | 11/2020 |
6717 | 2^353992-57 | 106563 | Lelio R Paula | 10/2011 |
6718 | 9*10^106559+59 | 106560 | Robert Price | 01/2020 |
6719 | 7*10^106553+27 | 106554 | Robert Price | 11/2020 |
6720 | 28847^4938+4938^28847 | 106548 | Hans Havermann | 10/2020 |
6721 | NSW(139160) | 106535 | Robert Price | 12/2018 |
6722 | f(278321) | 106535 | Robert Price | 07/2023 |
6723 | (7*10^106516+113)/3 | 106517 | Robert Price | 02/2018 |
6724 | 2^353832-1853849 | 106515 | Enzo Doria | 09/2015 |
6725 | (7^126037-1)/6 | 106513 | Paul Bourdelais | 09/2008 |
6726 | 2^353794-18453 | 106503 | Enzo Doria | 05/2013 |
6727 | (2^353761+2094)/2 | 106493 | Stefano Morozzi | 09/2020 |
6728 | 2^353728+7695 | 106483 | Lelio R Paula | 01/2017 |
6729 | 26476^10509+10509^26476 | 106475 | Hans Havermann | 10/2020 |
6730 | 2^353672-303 | 106466 | Yves Minet | 02/2012 |
6731 | 6^136811+101 | 106460 | Lelio R Paula | 02/2017 |
6732 | 2^353641+1871981 | 106457 | Enzo Doria | 05/2016 |
6733 | (8*10^106454-29)/3 | 106455 | Robert Price | 07/2018 |
6734 | 27845^6654+6654^27845 | 106454 | Hans Havermann | 11/2020 |
6735 | 10^106449-700001 | 106449 | Robert Price | 08/2021 |
6736 | 25316^16017+16017^25316 | 106444 | Hans Havermann | 11/2020 |
6737 | 2^353553+54615 | 106431 | Lelio R Paula | 01/2017 |
6738 | 2^353544-18897 | 106428 | Enzo Doria | 05/2013 |
6739 | 28894^4823+4823^28894 | 106426 | Hans Havermann | 11/2020 |
6740 | (19*10^106422+29)/3 | 106423 | Robert Price | 12/2021 |
6741 | (694*10^106418+53)/9 | 106420 | Robert Price | 02/2022 |
6742 | 6*10^106412-13 | 106413 | Robert Price | 07/2022 |
6743 | 25696^13829+13829^25696 | 106402 | Hans Havermann | 11/2020 |
6744 | 6^136729-133 | 106396 | Lelio R Paula | 02/2017 |
6745 | 2^353418+1605 | 106390 | Lelio R Paula | 01/2017 |
6746 | 2^353387+65 | 106381 | Lelio R Paula | 10/2011 |
6747 | 2^353372+74311 | 106376 | Enzo Doria | 04/2016 |
6748 | 2^353344-3693 | 106368 | Yves Minet | 03/2017 |
6749 | (772*10^106365-7)/9 | 106367 | Robert Price | 06/2022 |
6750 | (56*10^106351+619)/9 | 106352 | Robert Price | 05/2021 |