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 ? |
9501 | 19857^4928+4928^19857 | 73326 | Hans Havermann | 12/2019 |
9502 | 2*5^104888-7 | 73314 | Lelio R Paula | 10/2014 |
9503 | (58695*2^243516-1)/983 | 73308 | Lelio R Paula | 04/2015 |
9504 | (3^153648-2)/33423138097 | 73299 | Lelio R Paula | 06/2012 |
9505 | (2^243467+470)/2 | 73291 | Stefano Morozzi | 01/2020 |
9506 | (2^243460+2410)/2 | 73289 | Stefano Morozzi | 01/2020 |
9507 | 18^58376-5255 | 73278 | Enzo Doria | 03/2013 |
9508 | 11^70363+32 | 73276 | Lelio R Paula | 02/2015 |
9509 | 2^243413+4257 | 73275 | Lelio R Paula | 05/2013 |
9510 | 104824^5+5^104824 | 73269 | Andrey Kulsha | 07/2010 |
9511 | 7^86689-6^86689 | 73261 | Predrag Minovic | 09/2004 |
9512 | (2^243355+3846)/2 | 73257 | Stefano Morozzi | 01/2020 |
9513 | (15954^17431-1)/15953 | 73257 | Serge Batalov | 08/2020 |
9514 | 2^243296+90195 | 73240 | Lelio R Paula | 05/2013 |
9515 | 2^243281-4843 | 73235 | Yves Minet | 05/2019 |
9517 | p(15433)#/2-4 | 73218 | Robert Price | 12/2016 |
9516 | (59*10^73217-77)/9 | 73218 | Robert Price | 11/2017 |
9518 | (18160^17191-1)/18159 | 73215 | Serge Batalov | 08/2020 |
9519 | 2*10^73199+23 | 73200 | Robert Price | 12/2020 |
9520 | 2^243127+71355 | 73189 | Lelio R Paula | 05/2013 |
9521 | 2^243116-249419 | 73186 | Paul Bourdelais | 03/2013 |
9522 | (58695*2^243085-1)/11 | 73180 | Lelio R Paula | 04/2015 |
9523 | (2^243084+3406)/2 | 73176 | Stefano Morozzi | 01/2020 |
9524 | 11^70262-98 | 73171 | Lelio R Paula | 02/2015 |
9525 | (2^243034+47)/722703303 | 73152 | Lelio R Paula | 07/2012 |
9526 | (2^242966+47)/46887 | 73136 | Lelio R Paula | 07/2012 |
9527 | (43^44773-1)/42 | 73134 | Robert Price | 04/2014 |
9528 | 6^93980-158359 | 73131 | Mike Oakes | 02/2011 |
9529 | 19953^4616+4616^19953 | 73114 | Hans Havermann | 11/2019 |
9530 | (2^242855+2546)/2 | 73107 | Stefano Morozzi | 01/2020 |
9531 | (2^242845+3926)/2 | 73104 | Stefano Morozzi | 01/2020 |
9532 | 6^93915+53 | 73081 | Lelio R Paula | 01/2015 |
9533 | (5*10^73079+211)/9 | 73079 | Robert Price | 12/2017 |
9534 | (74*10^73076+43)/9 | 73077 | Robert Price | 10/2015 |
9535 | (3^153125+2)/1144955 | 73054 | Lelio R Paula | 06/2012 |
9536 | 2^242632-112199 | 73040 | Enzo Doria | 06/2014 |
9537 | V(351646) | 73029 | Serge Batalov | 03/2015 |
9538 | (2^242598+1090)/2 | 73029 | Stefano Morozzi | 01/2020 |
9539 | 18^58175-85 | 73026 | Lelio R Paula | 01/2015 |
9540 | p(15395)#/2-4 | 73019 | Robert Price | 12/2016 |
9541 | (4^121271+3^121271)/7 | 73012 | Jean-Louis Charton | 03/2011 |
9542 | 8*6^93825-5 | 73011 | Lelio R Paula | 10/2014 |
9543 | (28123^16411-1)/28122 | 73010 | Serge Batalov | 08/2020 |
9544 | 19090^6673+6673^19090 | 73007 | Hans Havermann | 11/2019 |
9546 | (F(220)^1601-F(219)^1601)/F(218) | 73005 | Wayne Halsdorf | 08/2013 |
9545 | (73^39181+1)/74 | 73005 | Paul Bourdelais | 08/2017 |
9547 | 6^93815+143 | 73003 | Lelio R Paula | 01/2015 |
9549 | 6^93812+5 | 73000 | Wojciech Florek | 03/2006 |
9548 | 17874^12137+12137^17874 | 73000 | Hans Havermann | 11/2019 |
9550 | 19^57078-62 | 72989 | Lelio R Paula | 01/2015 |
9551 | (29*10^72985+91)/3 | 72986 | Robert Price | 03/2016 |
9552 | 20691^3368+3368^20691 | 72985 | Hans Havermann | 10/2019 |
9553 | (2^242451+558)/2 | 72985 | Stefano Morozzi | 01/2020 |
9554 | 11^70082-2520 | 72983 | Enzo Doria | 12/2012 |
9555 | (2^242429+2142)/2 | 72979 | Stefano Morozzi | 01/2020 |
9556 | (27992^16411-1)/27991 | 72976 | Serge Batalov | 08/2020 |
9557 | 6^93773+133 | 72970 | Lelio R Paula | 01/2015 |
9558 | (2^242392+3762)/2 | 72967 | Stefano Morozzi | 01/2020 |
9559 | 2^242328-59 | 72948 | Lelio R Paula | 03/2012 |
9560 | 5^104358-13016 | 72944 | Enzo Doria | 02/2013 |
9561 | (75*10^72941-57)/99 | 72941 | Ray Chandler | 03/2011 |
9562 | 12^67585-95 | 72937 | Lelio R Paula | 02/2015 |
9563 | (2^242261+3834)/2 | 72928 | Stefano Morozzi | 01/2020 |
9564 | 2^242230-113285 | 72919 | Enzo Doria | 07/2014 |
9565 | (2^242229+2742)/2 | 72918 | Stefano Morozzi | 01/2020 |
9566 | (5*10^72916-11)/3 | 72917 | Erik Branger | 12/2013 |
9567 | 2^242220-110399 | 72916 | Enzo Doria | 05/2014 |
9568 | 2^242205-113413 | 72911 | Enzo Doria | 07/2014 |
9569 | (3^152809+3^76405+1)/7 | 72908 | Serge Batalov | 03/2014 |
9570 | 2^242175+74243 | 72902 | Yves Minet | 03/2011 |
9571 | (2^242168+3406)/2 | 72900 | Stefano Morozzi | 01/2020 |
9572 | 2^242150+17659 | 72895 | Lelio R Paula | 11/2009 |
9573 | 18902^7179+7179^18902 | 72888 | Hans Havermann | 11/2019 |
9574 | (17366^17191-1)/17365 | 72881 | Serge Batalov | 08/2020 |
9575 | 11^69982-2340 | 72879 | Enzo Doria | 12/2012 |
9576 | (23*2^242093+1)/10117 | 72875 | Lelio R Paula | 08/2012 |
9577 | (67607*2^242043+1)/33263 | 72863 | Lelio R Paula | 06/2014 |
9578 | 2^242032+87 | 72859 | Lelio R Paula | 03/2012 |
9579 | (194*10^72852-17)/3 | 72854 | Robert Price | 09/2020 |
9580 | (2^241983+2690)/2 | 72844 | Stefano Morozzi | 01/2020 |
9582 | 2^241968-110595 | 72840 | Enzo Doria | 05/2014 |
9581 | (2^241968+598)/2 | 72840 | Stefano Morozzi | 01/2020 |
9583 | 18850^7313+7313^18850 | 72839 | Hans Havermann | 11/2019 |
9584 | (2^241921+3290)/2 | 72826 | Stefano Morozzi | 01/2020 |
9585 | (10^72821+71)/9 | 72821 | Robert Price | 11/2014 |
9586 | 2^241852-64875 | 72805 | Lelio R Paula | 05/2013 |
9587 | F(348307)/20562890865908093 | 72776 | Donovan Johnson | 07/2007 |
9588 | 13952^17559+17559^13952 | 72776 | Norbert Schneider | 09/2019 |
9589 | (67*10^72772+23)/9 | 72773 | Ray Chandler | 11/2010 |
9590 | (8*10^72759+67)/3 | 72760 | Robert Price | 01/2017 |
9591 | 2^241670+145 | 72750 | Lelio R Paula | 03/2012 |
9592 | 5^104074+52 | 72745 | Lelio R Paula | 08/2014 |
9593 | (3^152466-5)/724 | 72742 | Lelio R Paula | 06/2012 |
9594 | 20^55908-41 | 72738 | Lelio R Paula | 01/2015 |
9595 | (73*10^72735+53)/9 | 72736 | Robert Price | 10/2015 |
9596 | 10^72730+224011 | 72731 | Patrick De Geest | 05/2013 |
9597 | 10^72730-99 | 72730 | Robert Price | 05/2011 |
9598 | 3*10^72723+67 | 72724 | Robert Price | 11/2017 |
9600 | (11^69833+3^69833)/14 | 72723 | Robert Price | 03/2013 |
9599 | (10^72723-79)/3 | 72723 | Robert Price | 10/2017 |
9601 | 18706^7719+7719^18706 | 72721 | Hans Havermann | 11/2019 |
9602 | (88*10^72711-43)/9 | 72712 | Robert Price | 11/2015 |
9603 | 3^152379+280 | 72704 | Lelio R Paula | 01/2015 |
9605 | (25*10^72703-241)/9 | 72704 | Robert Price | 01/2017 |
9604 | (2^241517+3774)/2 | 72704 | Stefano Morozzi | 01/2020 |
9606 | (2^241515+3518)/2 | 72703 | Stefano Morozzi | 01/2020 |
9607 | (47*10^72696+133)/9 | 72697 | Robert Price | 04/2017 |
9608 | 17^59077-90 | 72692 | Lelio R Paula | 01/2015 |
9609 | 65^40096+58 | 72691 | Lelio R Paula | 01/2016 |
9610 | (29*2^241446-1)/245 | 72682 | Lelio R Paula | 01/2014 |
9612 | 11^69783-38400000 | 72672 | Enzo Doria | 01/2013 |
9611 | 2^241409-112903 | 72672 | Enzo Doria | 06/2014 |
9613 | 18895^7014+7014^18895 | 72670 | Hans Havermann | 11/2019 |
9614 | (3^152287+1)/4 | 72659 | Paul Bourdelais | 04/2008 |
9615 | (2^241353+3186)/2 | 72655 | Stefano Morozzi | 01/2020 |
9616 | 17638^13137+13137^17638 | 72643 | Hans Havermann | 11/2019 |
9617 | 3^152249-56 | 72642 | Lelio R Paula | 01/2015 |
9618 | 10^72631+37 | 72632 | Robert Price | 01/2011 |
9619 | 10^72631-8813 | 72631 | Patrick De Geest | 05/2013 |
9620 | (19*2^241254+1)/101 | 72624 | Lelio R Paula | 08/2012 |
9621 | (32202^16111-1)/32201 | 72622 | Serge Batalov | 08/2020 |
9622 | (9^76103+8^76103)/17 | 72620 | Robert Price | 12/2012 |
9624 | 2^241220-112337 | 72615 | Enzo Doria | 06/2014 |
9623 | 7*10^72614+33 | 72615 | Robert Price | 06/2020 |
9625 | (82*10^72612+449)/9 | 72613 | Robert Price | 02/2017 |
9626 | 2^241169-15793 | 72600 | Enzo Doria | 04/2013 |
9627 | 9*10^72595+11 | 72596 | Robert Kinser | 01/2017 |
9628 | (2^241152+2478)/2 | 72594 | Stefano Morozzi | 01/2020 |
9629 | (2^241143+2438)/2 | 72591 | Stefano Morozzi | 01/2020 |
9630 | (2^241115+1598)/2 | 72583 | Stefano Morozzi | 01/2020 |
9631 | (1003^24179-1)/1002 | 72566 | Serge Batalov | 07/2020 |
9632 | (25*2^241048+1)/89 | 72563 | Lelio R Paula | 08/2012 |
9633 | (2^241045+1550)/2 | 72562 | Stefano Morozzi | 01/2020 |
9634 | (22*10^72560+17)/3 | 72561 | Robert Price | 09/2015 |
9635 | 2^241032-23 | 72558 | Lelio R Paula | 02/2012 |
9636 | 34485*10^72550+2^241004-3 | 72555 | Gilbert Mozzo | 12/2012 |
9637 | 19460^5349+5349^19460 | 72553 | Hans Havermann | 11/2019 |
9638 | (2^241016+3850)/2 | 72553 | Stefano Morozzi | 01/2020 |
9639 | 2^241004-3 | 72550 | M. Frind & P. Underwood | 07/2002 |
9640 | 22061^1942+1942^22061 | 72543 | Hans Havermann | 11/2019 |
9641 | 9*10^72528-89 | 72529 | Robert Price | 04/2020 |
9643 | 97*10^72522-3 | 72524 | Robert Price | 01/2017 |
9642 | (29*10^72523-107)/3 | 72524 | Robert Price | 08/2020 |
9644 | (68*10^72516+7)/3 | 72518 | Robert Price | 03/2016 |
9645 | (13993^17491-1)/13992 | 72512 | Serge Batalov | 08/2020 |
9646 | (78557*2^240844+1)/63 | 72505 | Lelio R Paula | 06/2014 |
9647 | 67^39701-66 | 72497 | Norbert Schneider | 11/2014 |
9648 | (185*10^72484+7)/3 | 72486 | Robert Price | 02/2017 |
9649 | (2^240780+2686)/2 | 72482 | Stefano Morozzi | 01/2020 |
9650 | (2^240759+1106)/2 | 72476 | Stefano Morozzi | 01/2020 |
9652 | (293*27^50631-1736)/13 | 72473 | Curtis Bright | 05/2015 |
9651 | 103^36005-102 | 72473 | Norbert Schneider | 10/2016 |
9653 | 15^61616+88 | 72467 | Lelio R Paula | 01/2015 |
9654 | 45^43831+2 | 72462 | Lelio R Paula | 05/2013 |
9655 | 2^240692-492389 | 72456 | Paul Bourdelais | 03/2013 |
9656 | (646*10^72446-7)/9 | 72448 | Robert Price | 07/2016 |
9657 | 9*6^93083+7 | 72434 | Lelio R Paula | 10/2014 |
9658 | 2^240617-3645 | 72433 | Lelio R Paula | 05/2013 |
9659 | (9391^18229-1)/9390 | 72415 | Serge Batalov | 08/2020 |
9660 | 11^69531-24000 | 72410 | Enzo Doria | 12/2012 |
9661 | (259*10^72400-1)/3 | 72402 | Robert Price | 12/2020 |
9662 | 18714^7367+7367^18714 | 72373 | Hans Havermann | 11/2019 |
9663 | (22699*2^240396+1)/55 | 72370 | Lelio R Paula | 06/2014 |
9664 | 6^92995+13 | 72365 | Lelio R Paula | 01/2015 |
9665 | (11*10^72362-149)/3 | 72363 | Robert Price | 01/2017 |
9666 | 18567^7892+7892^18567 | 72360 | Hans Havermann | 11/2019 |
9667 | (2^240381+3)/51655 | 72358 | Lelio R Paula | 05/2012 |
9668 | 6^92974-55 | 72348 | Lelio R Paula | 01/2015 |
9669 | 2^240295+11 | 72337 | Lelio R Paula | 02/2012 |
9670 | (2^240294+1342)/2 | 72336 | Stefano Morozzi | 01/2020 |
9671 | (14*10^72326-53)/3 | 72327 | Robert Price | 12/2016 |
9672 | 15^61493-46 | 72322 | Lelio R Paula | 01/2015 |
9673 | (2^240245+2094)/2 | 72321 | Stefano Morozzi | 01/2020 |
9674 | (2^240237+1554)/2 | 72319 | Stefano Morozzi | 01/2020 |
9675 | lucasU(46,1,43497)-lucasU(46,1,43496) | 72315 | Mike Oakes | 04/2009 |
9676 | (2^240211+1514)/2 | 72311 | Stefano Morozzi | 01/2020 |
9678 | 5*10^72302+9 | 72303 | Dmitry Domanov | 11/2010 |
9677 | (9787*2^240179-1)/265 | 72303 | Lelio R Paula | 04/2015 |
9679 | (2^240171+3246)/2 | 72299 | Stefano Morozzi | 01/2020 |
9680 | 11^69419-3120 | 72293 | Enzo Doria | 12/2012 |
9681 | 18069^10016+10016^18069 | 72289 | Hans Havermann | 11/2019 |
9682 | (25883^16381-1)/25882 | 72286 | Serge Batalov | 08/2020 |
9683 | (2^240095+594)/2 | 72276 | Stefano Morozzi | 01/2020 |
9684 | (25848^16381-1)/25847 | 72276 | Serge Batalov | 08/2020 |
9685 | 4*10^72274-83 | 72275 | Robert Price | 10/2017 |
9686 | (2^240075+2790)/2 | 72270 | Stefano Morozzi | 01/2020 |
9687 | 10^72267+33 | 72268 | Robert Price | 01/2011 |
9688 | 10^72267-553947 | 72267 | Patrick De Geest | 05/2013 |
9689 | (2^240066+1030)/2 | 72267 | Stefano Morozzi | 01/2020 |
9691 | 11^69389-1380 | 72262 | Enzo Doria | 12/2012 |
9692 | (16^60013-5^60013)/11 | 72262 | Robert Price | 07/2013 |
9690 | 7^85507-30 | 72262 | Lelio R Paula | 08/2014 |
9693 | (19*2^240026-1)/15 | 72256 | Lelio R Paula | 02/2014 |
9694 | (3^151426-2)/661337 | 72243 | Lelio R Paula | 06/2012 |
9695 | (2^239940+3850)/2 | 72229 | Stefano Morozzi | 01/2020 |
9696 | gcd(V(1296375),5*F(259275)*(F(259275)-1)+1) | 72227 | Serge Batalov | 04/2015 |
9697 | (2789^20959-1)/2788 | 72210 | Serge Batalov | 08/2020 |
9698 | 2^239863-112267 | 72206 | Enzo Doria | 06/2014 |
9699 | 2^239853-114853 | 72203 | Enzo Doria | 05/2015 |
9700 | (14068^17401-1)/14067 | 72180 | Serge Batalov | 08/2020 |
9701 | (5669^19231-1)/5668 | 72180 | Serge Batalov | 08/2020 |
9702 | (4*10^72174+149)/9 | 72174 | Robert Price | 03/2017 |
9703 | 4*10^72166+67 | 72167 | Robert Price | 01/2021 |
9705 | (2^239671+9)/296131 | 72143 | Lelio R Paula | 05/2012 |
9704 | 2^239651-110665 | 72143 | Enzo Doria | 05/2014 |
9706 | (29*2^239650-1)/3305 | 72140 | Lelio R Paula | 01/2014 |
9707 | (17099^17041-1)/17098 | 72130 | Serge Batalov | 08/2020 |
9709 | 241^30280+30280^241 | 72128 | Norbert Schneider | 11/2014 |
9708 | (2^239603+950)/2 | 72128 | Stefano Morozzi | 01/2020 |
9710 | (2^239594+3058)/2 | 72125 | Stefano Morozzi | 01/2020 |
9711 | (68*10^72120-23)/9 | 72121 | Robert Price | 10/2015 |
9712 | (9^75571+8^75571)/17 | 72112 | Robert Price | 12/2012 |
9713 | (15661^17191-1)/15660 | 72109 | Serge Batalov | 08/2020 |
9714 | (3508^20341-1)/3507 | 72107 | Serge Batalov | 08/2020 |
9715 | (15*2^239530-1)/173 | 72105 | Lelio R Paula | 01/2014 |
9716 | (2^239523+2378)/2 | 72104 | Stefano Morozzi | 01/2020 |
9717 | 2^239513-112519 | 72101 | Enzo Doria | 06/2014 |
9718 | (10^72097+149)/3 | 72097 | Serge Batalov | 12/2014 |
9719 | 2^239442-6555 | 72080 | Lelio R Paula | 05/2012 |
9720 | 2*5^103106+3 | 72069 | Lelio R Paula | 10/2014 |
9721 | 2^239402-197 | 72068 | Lelio R Paula | 02/2012 |
9722 | 2^239356-111875 | 72054 | Enzo Doria | 05/2014 |
9723 | (38*10^72050-119)/9 | 72051 | Robert Price | 11/2016 |
9724 | 2^239339+15015 | 72049 | Lelio R Paula | 01/2013 |
9725 | 159^32728+2 | 72048 | Lelio R Paula | 05/2013 |
9726 | 15^61259-38 | 72047 | Lelio R Paula | 01/2015 |
9727 | 92^36683-91^36683 | 72038 | Robert Price | 08/2015 |
9728 | 2^239280+54615 | 72031 | Lelio R Paula | 05/2012 |
9729 | (47*2^239267+1)/77 | 72027 | Lelio R Paula | 04/2015 |
9730 | (85*10^72025-1)/3 | 72027 | Robert Price | 07/2017 |
9731 | (2^239269+1730)/2 | 72027 | Stefano Morozzi | 01/2020 |
9732 | 2^239236-97305 | 72018 | Lelio R Paula | 05/2012 |
9734 | 144^33361-143 | 72006 | Norbert Schneider | 11/2018 |
9733 | (2^239198+1422)/2 | 72006 | Stefano Morozzi | 01/2020 |
9735 | 2^239169+59 | 71998 | Lelio R Paula | 02/2012 |
9736 | lucasU(2010,1,21796)-lucasU(2010,1,21795) | 71994 | Mike Oakes | 05/2009 |
9737 | (15*2^239171-1)/106301 | 71994 | Lelio R Paula | 01/2014 |
9738 | (2^239121+1926)/2 | 71983 | Stefano Morozzi | 01/2020 |
9740 | (29^49211-1)/28 | 71965 | Robert Price | 04/2012 |
9739 | (299*10^71963+1)/3 | 71965 | Robert Price | 11/2017 |
9741 | (58729^15091-1)/58728 | 71962 | Anonymous | 11/2020 |
9743 | L(344293) | 71953 | Renaud Lifchitz | 09/2003 |
9742 | V(344293) | 71953 | Serge Batalov | 03/2015 |
9744 | 6*10^71951+7 | 71952 | Dmitry Domanov | 11/2010 |
9745 | 3^150798+40 | 71949 | Lelio R Paula | 01/2015 |
9746 | 2^238961+95 | 71935 | Lelio R Paula | 02/2012 |
9747 | 12511^17556+17556^12511 | 71933 | Norbert Schneider | 09/2019 |
9748 | 2^238936-407 | 71927 | Yves Minet | 02/2012 |
9749 | (2^238930+2982)/2 | 71925 | Stefano Morozzi | 01/2020 |
9750 | 2^238923-28635 | 71923 | Lelio R Paula | 05/2012 |