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 | 8^97731-7^(97731-1) | 88260 | Robert Price | 04/2016 |
9502 | 5*10^88254+59 | 88255 | Robert Price | 09/2016 |
9503 | 4*3^184968-7 | 88253 | Lelio R Paula | 08/2014 |
9504 | 34*10^88251-9 | 88253 | Robert Price | 06/2020 |
9505 | 7^104426+88 | 88251 | Lelio R Paula | 10/2015 |
9506 | 2^293142+99 | 88245 | Lelio R Paula | 10/2012 |
9507 | 11^84736-2130 | 88244 | Enzo Doria | 12/2012 |
9508 | 23884^4945+4945^23884 | 88232 | Hans Havermann | 08/2020 |
9509 | 22080^9907+9907^22080 | 88231 | Hans Havermann | 08/2020 |
9510 | (10^110315+1)/(10^22063+1)/5824812682167648578460281641 | 88225 | Makoto Kamada | 04/2019 |
9511 | 2^293068-89 | 88223 | Lelio R Paula | 10/2012 |
9512 | 2126^26511+26511^2126 | 88218 | Norbert Schneider | 02/2020 |
9513 | 2^293014-109655 | 88207 | Enzo Doria | 04/2014 |
9514 | 8*7^104371-3 | 88205 | Lelio R Paula | 10/2014 |
9515 | 2^292995-2145 | 88201 | Lelio R Paula | 10/2013 |
9516 | 28534^1233+1233^28534 | 88198 | Hans Havermann | 08/2020 |
9517 | 5*10^88182+77 | 88183 | Robert Price | 06/2016 |
9519 | 2^292929+43877 | 88181 | Donovan Johnson | 04/2005 |
9518 | 6^113321+131 | 88181 | Lelio R Paula | 12/2015 |
9520 | (2^292919+3366)/2 | 88178 | Stefano Morozzi | 01/2020 |
9521 | 2^292904-583949 | 88173 | Paul Bourdelais | 04/2013 |
9522 | 2^292894-111851 | 88170 | Enzo Doria | 05/2014 |
9523 | 2^292872-2797349 | 88164 | Enzo Doria | 03/2016 |
9524 | (2^292864+1042)/2 | 88161 | Stefano Morozzi | 01/2020 |
9525 | 2^292860-109685 | 88160 | Enzo Doria | 04/2014 |
9526 | 9*6^113288-5 | 88157 | Lelio R Paula | 10/2014 |
9527 | (2^292735+2166)/2 | 88122 | Stefano Morozzi | 01/2020 |
9528 | (2^292735+2346)/2 | 88122 | Stefano Morozzi | 01/2020 |
9529 | (2^292705+1934)/2 | 88113 | Stefano Morozzi | 01/2020 |
9530 | (2*10^88112-23)/3 | 88112 | Robert Price | 09/2017 |
9531 | 29622^943+943^29622 | 88111 | Hans Havermann | 08/2020 |
9532 | (73*10^88105+17)/9 | 88106 | Robert Price | 10/2015 |
9533 | (2^292673+3026)/2 | 88104 | Stefano Morozzi | 01/2020 |
9534 | 11^84588+78 | 88090 | Lelio R Paula | 11/2015 |
9535 | (265*10^88077+11)/3 | 88079 | Robert Price | 03/2017 |
9536 | 91^44959-90^44959 | 88077 | Robert Price | 07/2015 |
9537 | 6^113177-5^113177 | 88069 | Jean-Louis Charton | 01/2009 |
9538 | 1316^28233+28233^1316 | 88066 | Norbert Schneider | 07/2020 |
9539 | (2^292543+2234)/2 | 88064 | Stefano Morozzi | 01/2020 |
9540 | (10^88020+101)/3 | 88020 | Robert Price | 04/2016 |
9541 | 52^51285-51 | 88006 | Norbert Schneider | 09/2018 |
9542 | 21002^15501+15501^21002 | 88006 | Hans Havermann | 08/2020 |
9543 | 20466^19951+19951^20466 | 88004 | Hans Havermann | 08/2020 |
9544 | (2^292324+2346)/2 | 87998 | Stefano Morozzi | 01/2020 |
9545 | 21634^11683+11683^21634 | 87998 | Hans Havermann | 08/2020 |
9546 | (6340^23143-1)/6339 | 87988 | Serge Batalov | 08/2020 |
9547 | 20674^18013+18013^20674 | 87980 | Hans Havermann | 08/2020 |
9548 | (10^(2*43982+1)-6*10^43982-1)/3 | 87965 | Robert Price | 02/2016 |
9549 | (11*10^87960-41)/3 | 87961 | Robert Price | 12/2021 |
9550 | (424*10^87930-1)/9 | 87932 | Robert Price | 11/2017 |
9551 | (2^292100+3070)/2 | 87931 | Stefano Morozzi | 01/2020 |
9552 | p(18225)#/2+128 | 87929 | Robert Price | 04/2017 |
9553 | (10607^21841-1)/10606 | 87919 | Serge Batalov | 08/2020 |
9554 | (2^292047+1550)/2 | 87915 | Stefano Morozzi | 01/2020 |
9555 | 4*6^112977+7 | 87914 | Lelio R Paula | 10/2014 |
9556 | (2^292016+1062)/2 | 87906 | Stefano Morozzi | 01/2020 |
9557 | (11359^21673-1)/11358 | 87888 | Serge Batalov | 08/2020 |
9558 | 14^76677+69 | 87882 | Lelio R Paula | 11/2015 |
9559 | (22*10^87863-1)/3 | 87864 | Robert Price | 12/2014 |
9560 | 2^291822-2797811 | 87848 | Enzo Doria | 04/2016 |
9561 | (22*10^87821-43)/3 | 87822 | Robert Price | 10/2021 |
9562 | (2^291730+1614)/2 | 87820 | Stefano Morozzi | 01/2020 |
9563 | (277*10^87817-7)/9 | 87819 | Robert Price | 02/2022 |
9564 | 2^291716-111389 | 87816 | Enzo Doria | 05/2014 |
9565 | (782*10^87809-71)/9 | 87811 | Robert Price | 06/2017 |
9566 | 3^184039+1820 | 87809 | Lelio R Paula | 12/2015 |
9567 | 12^81356+41 | 87798 | Lelio R Paula | 11/2015 |
9568 | 21307^13188+13188^21307 | 87789 | Hans Havermann | 08/2020 |
9569 | (2527^25801-1)/2526 | 87788 | Serge Batalov | 08/2020 |
9570 | (2^291596+1986)/2 | 87779 | Stefano Morozzi | 01/2020 |
9572 | 6^112792-18983 | 87770 | Enzo Doria | 03/2013 |
9571 | (3123^25117-1)/3122 | 87770 | Serge Batalov | 08/2020 |
9573 | 3*6^112787-5 | 87766 | Lelio R Paula | 10/2014 |
9574 | 7^103848+82 | 87762 | Lelio R Paula | 10/2015 |
9575 | (19*2^291534-1)/178696395 | 87754 | Lelio R Paula | 02/2014 |
9576 | (2^291506+918)/2 | 87752 | Stefano Morozzi | 01/2020 |
9577 | 24411^3932+3932^24411 | 87749 | Hans Havermann | 08/2020 |
9578 | (3^183915+5)/128 | 87748 | Lelio R Paula | 01/2013 |
9579 | 19^68619+24 | 87747 | Lelio R Paula | 12/2015 |
9581 | 2^291480+1185 | 87745 | Lelio R Paula | 10/2013 |
9580 | 2^291482+1867369 | 87745 | Enzo Doria | 03/2016 |
9582 | 2^291467-937 | 87741 | Yves Minet | 05/2013 |
9583 | (25*10^87726+161)/3 | 87727 | Robert Price | 01/2017 |
9584 | 11^84239-2850 | 87726 | Enzo Doria | 12/2012 |
9585 | 10^87720+215541 | 87721 | Patrick De Geest | 01/2011 |
9586 | 10^87720-3 | 87720 | Markus Tervooren | 07/2010 |
9587 | (2^291383+3078)/2 | 87715 | Stefano Morozzi | 01/2020 |
9589 | (5*10^87712-473)/9 | 87712 | Robert Price | 12/2017 |
9588 | (2^291373+2946)/2 | 87712 | Stefano Morozzi | 01/2020 |
9590 | 22591^7630+7630^22591 | 87711 | Hans Havermann | 08/2020 |
9591 | 22081^9378+9378^22081 | 87709 | Hans Havermann | 08/2020 |
9592 | (2^291357+5)/6847 | 87704 | Lelio R Paula | 01/2013 |
9593 | (2^291280+462)/2 | 87684 | Stefano Morozzi | 01/2020 |
9594 | 20749^16816+16816^20749 | 87680 | Hans Havermann | 08/2020 |
9595 | 93^44541-2 | 87679 | Lelio R Paula | 09/2013 |
9596 | 2^291261-109561 | 87679 | Enzo Doria | 04/2014 |
9597 | (4*10^87669+149)/9 | 87669 | Robert Price | 03/2017 |
9598 | 2^291217+4257 | 87666 | Lelio R Paula | 10/2013 |
9599 | 22748^7137+7137^22748 | 87660 | Hans Havermann | 08/2020 |
9600 | 2*12^81214-5^81214 | 87645 | Mike Oakes | 12/2010 |
9601 | (85*10^87637-139)/9 | 87638 | Robert Price | 08/2021 |
9602 | (5005^23689-1)/5004 | 87632 | Serge Batalov | 08/2020 |
9603 | (5*10^87622-473)/9 | 87622 | Robert Price | 12/2017 |
9604 | 12^81188+85 | 87617 | Lelio R Paula | 11/2015 |
9605 | (2^291030+3462)/2 | 87609 | Stefano Morozzi | 01/2020 |
9606 | (21*2^290980-1)/5 | 87595 | Lelio R Paula | 02/2014 |
9607 | (2^290980+1150)/2 | 87594 | Stefano Morozzi | 01/2020 |
9608 | 2^290958+127 | 87588 | Lelio R Paula | 09/2012 |
9609 | 78531!4-8 | 87580 | Robert Price | 11/2019 |
9610 | 2^290923-67 | 87577 | Lelio R Paula | 09/2012 |
9611 | 3*10^87573-97 | 87574 | Robert Price | 08/2017 |
9612 | (13*10^87565+11)/3 | 87566 | Robert Price | 05/2015 |
9613 | (2^290881+1022)/2 | 87564 | Stefano Morozzi | 01/2020 |
9614 | (2^290857-7)/55 | 87555 | Lelio R Paula | 01/2013 |
9615 | (2^290808+1498)/2 | 87542 | Stefano Morozzi | 01/2020 |
9616 | 2^290802+82875 | 87541 | Lelio R Paula | 10/2013 |
9617 | (509203*2^290792-1)/7560987 | 87536 | Lelio R Paula | 04/2015 |
9618 | 3^183441+236 | 87524 | Lelio R Paula | 12/2015 |
9619 | 4*10^87513-69 | 87514 | Robert Price | 01/2022 |
9620 | 2^290618-33 | 87485 | Lelio R Paula | 09/2012 |
9621 | (184*10^87473-1)/3 | 87475 | Robert Price | 02/2017 |
9622 | 2^290534-115151 | 87460 | Enzo Doria | 06/2015 |
9623 | 2^290512-109757 | 87453 | Enzo Doria | 04/2014 |
9624 | (32*10^87448-41)/9 | 87449 | Robert Price | 05/2015 |
9625 | (2^290493+3074)/2 | 87447 | Stefano Morozzi | 01/2020 |
9626 | (2^290491+9)/3179 | 87444 | Lelio R Paula | 01/2013 |
9627 | (25*2^290489-1)/51809 | 87443 | Lelio R Paula | 02/2014 |
9628 | 2^290426+82875 | 87427 | Lelio R Paula | 11/2013 |
9629 | 3*2^290415-7 | 87425 | Lelio R Paula | 08/2014 |
9631 | 10^87408+57 | 87409 | Robert Price | 03/2011 |
9630 | 2^290366+74779 | 87409 | Enzo Doria | 04/2016 |
9632 | 22284^8365+8365^22284 | 87409 | Hans Havermann | 08/2020 |
9633 | 10^87408-123569 | 87408 | Patrick De Geest | 07/2013 |
9634 | 21517^11536+11536^21517 | 87404 | Hans Havermann | 08/2020 |
9635 | 6*10^87396+77 | 87397 | Robert Price | 06/2020 |
9636 | (2^290268+3882)/2 | 87380 | Stefano Morozzi | 01/2020 |
9637 | 2^290265+58905 | 87379 | Lelio R Paula | 10/2013 |
9638 | 7^103391+16 | 87376 | Lelio R Paula | 10/2015 |
9639 | (115*10^87368-7)/9 | 87370 | Robert Price | 11/2017 |
9640 | (92^44491-1)/91 | 87369 | Paul Bourdelais | 08/2017 |
9641 | 8*7^103379+3 | 87367 | Lelio R Paula | 10/2014 |
9642 | (2^290226+7)/71 | 87365 | Lelio R Paula | 01/2013 |
9643 | (29*10^87355-89)/3 | 87356 | Robert Price | 01/2022 |
9644 | (3216^24907-1)/3215 | 87354 | Serge Batalov | 08/2020 |
9645 | (2^290160+2686)/2 | 87347 | Stefano Morozzi | 01/2020 |
9646 | 13^78407-30 | 87341 | Lelio R Paula | 11/2015 |
9647 | 4*10^87337-99 | 87338 | Robert Price | 12/2016 |
9648 | 7*10^87319+39 | 87320 | Robert Price | 07/2016 |
9649 | 3*5^124921+2 | 87317 | Lelio R Paula | 10/2014 |
9650 | 78315!4-16 | 87316 | Robert Price | 11/2019 |
9651 | 3^182984+88 | 87306 | Lelio R Paula | 12/2015 |
9652 | (10^87323-1)/(9*33909262597306826117) | 87303 | Alfred Reich | 02/2019 |
9653 | (10^87323-1)/305183363375761435053 | 87303 | Anonymous | 10/2020 |
9654 | (2^290008+958)/2 | 87301 | Stefano Morozzi | 01/2020 |
9655 | (10536^21701-1)/10535 | 87293 | Serge Batalov | 08/2020 |
9656 | 4*10^87287-99 | 87288 | Robert Price | 12/2016 |
9658 | 11^83813-1860 | 87283 | Enzo Doria | 12/2012 |
9657 | (3068^25033-1)/3067 | 87283 | Serge Batalov | 08/2020 |
9659 | 734^30453+30453^734 | 87270 | Norbert Schneider | 04/2017 |
9660 | 12^80855+29 | 87258 | Lelio R Paula | 11/2015 |
9661 | 22209^8482+8482^22209 | 87249 | Hans Havermann | 08/2020 |
9662 | 3*6^112119-5 | 87247 | Lelio R Paula | 10/2014 |
9663 | 12^80806+79 | 87205 | Lelio R Paula | 11/2015 |
9664 | 91382^9+9^91382 | 87201 | Andrey Kulsha | 11/2004 |
9665 | (21181*2^289657+1)/81 | 87198 | Lelio R Paula | 06/2014 |
9666 | 6^112054-47 | 87195 | Lelio R Paula | 12/2015 |
9667 | (55459*2^289632+1)/55 | 87191 | Lelio R Paula | 06/2014 |
9668 | 15^74131-68 | 87185 | Lelio R Paula | 11/2015 |
9669 | 2^289583-741 | 87174 | Lelio R Paula | 10/2013 |
9670 | 7^103146+12 | 87169 | Lelio R Paula | 10/2015 |
9671 | (2^289562+2182)/2 | 87167 | Stefano Morozzi | 01/2020 |
9672 | 5*2^289556-3 | 87166 | Lelio R Paula | 08/2014 |
9673 | 2^289543-112735 | 87162 | Enzo Doria | 06/2014 |
9674 | 10890^21589-21589^10890 | 87156 | Norbert Schneider | 01/2022 |
9675 | 20204^20243-20243^20204 | 87155 | Norbert Schneider | 04/2020 |
9676 | 8*3^182655-7 | 87150 | Lelio R Paula | 08/2014 |
9677 | 2*10^87144+81 | 87145 | Robert Price | 01/2017 |
9678 | 20541^17468+17468^20541 | 87140 | Hans Havermann | 07/2020 |
9680 | 7^103106+2*3^103106 | 87135 | Mike Oakes | 08/2010 |
9679 | (65*10^87134-29)/9 | 87135 | Robert Price | 01/2021 |
9681 | 8*3^182614+7 | 87130 | Lelio R Paula | 08/2014 |
9682 | 3*10^87121-73 | 87122 | Robert Price | 02/2021 |
9683 | (4982^23563-1)/4981 | 87119 | Serge Batalov | 08/2020 |
9684 | (2^289368-509203)/9 | 87108 | Lelio R Paula | 01/2013 |
9685 | 3^182566+14 | 87107 | Robert Price | 09/2013 |
9686 | 1738^26881+26881^1738 | 87096 | Norbert Schneider | 04/2020 |
9687 | 24290^3849+3849^24290 | 87089 | Hans Havermann | 08/2020 |
9688 | (10^87038+7)/43/53/751 | 87032 | Alfred Reich | 06/2021 |
9689 | (3^182385-5)/2 | 87020 | Lelio R Paula | 01/2013 |
9690 | 11^83534-8 | 86992 | Lelio R Paula | 11/2015 |
9691 | 4^144479-14267 | 86986 | Mike Oakes | 03/2011 |
9692 | (145*10^86982-1)/9 | 86984 | Serge Batalov | 12/2014 |
9693 | (151*10^86963-1)/3 | 86965 | Robert Price | 04/2017 |
9694 | (2^288892+826)/2 | 86965 | Stefano Morozzi | 01/2020 |
9695 | 15^73940-58 | 86961 | Lelio R Paula | 11/2015 |
9696 | (11*10^86951-71)/3 | 86952 | Robert Price | 04/2022 |
9697 | 2^288761-113143 | 86926 | Enzo Doria | 07/2014 |
9698 | 24440^3603+3603^24440 | 86925 | Hans Havermann | 08/2020 |
9700 | 2^288747+3083 | 86922 | Yves Minet | 03/2011 |
9699 | (2^288749+3102)/2 | 86922 | Stefano Morozzi | 01/2020 |
9701 | 30^58842+29 | 86917 | Norbert Schneider | 03/2017 |
9702 | (2^288705+2186)/2 | 86909 | Stefano Morozzi | 01/2020 |
9703 | (10^86906-13)/3 | 86906 | Robert Price | 02/2017 |
9704 | 2^288692-110027 | 86905 | Enzo Doria | 04/2014 |
9705 | (2^288682+7)/259895449 | 86894 | Lelio R Paula | 01/2013 |
9706 | (26*10^86885+79)/3 | 86886 | Robert Price | 09/2020 |
9707 | (2^288636+47)/1293 | 86885 | Lelio R Paula | 01/2013 |
9708 | 3^182089-34 | 86879 | Robert Price | 11/2013 |
9709 | (2^288571+1698)/2 | 86869 | Stefano Morozzi | 01/2020 |
9710 | 2^288543-109801 | 86861 | Enzo Doria | 04/2014 |
9711 | 60^48847-59 | 86858 | Norbert Schneider | 10/2014 |
9712 | (2^288484+3186)/2 | 86843 | Stefano Morozzi | 01/2020 |
9713 | 2^288470-64875 | 86839 | Lelio R Paula | 10/2013 |
9714 | 10^86816+32997 | 86817 | Patrick De Geest | 07/2013 |
9715 | 10^86816-99 | 86816 | Robert Price | 05/2011 |
9716 | (89*10^86802-539)/9 | 86803 | Robert Price | 11/2017 |
9717 | 20272^19123+19123^20272 | 86796 | Hans Havermann | 07/2020 |
9718 | (2^288316+1530)/2 | 86792 | Stefano Morozzi | 01/2020 |
9719 | (2^288316+2218)/2 | 86792 | Stefano Morozzi | 01/2020 |
9720 | 41487!2+256 | 86786 | Robert Price | 01/2015 |
9721 | (5*2^288301-1)/999 | 86785 | Lelio R Paula | 02/2014 |
9722 | (2^288278+1750)/2 | 86781 | Stefano Morozzi | 01/2020 |
9723 | 2^288243+76043 | 86770 | Enzo Doria | 04/2016 |
9724 | (29*2^288232+1)/15 | 86767 | Lelio R Paula | 01/2013 |
9725 | 9*6^111495+5 | 86761 | Lelio R Paula | 10/2014 |
9726 | (5*2^288202+1)/81 | 86757 | Lelio R Paula | 01/2013 |
9727 | 9*7^102654-2 | 86754 | Lelio R Paula | 10/2014 |
9728 | (2^288155+1154)/2 | 86743 | Stefano Morozzi | 01/2020 |
9729 | 2^288122-112841 | 86734 | Enzo Doria | 06/2014 |
9730 | 5^124062-68 | 86716 | Lelio R Paula | 10/2015 |
9731 | (2^288007-3)/125 | 86697 | Lelio R Paula | 01/2013 |
9732 | 379*10^86691+9 | 86694 | Arkadiusz Wesolowski | 10/2011 |
9733 | 7^102555+48 | 86670 | Lelio R Paula | 10/2015 |
9734 | 2^287902-4961 | 86668 | Yves Minet | 05/2019 |
9735 | 77779!4+2 | 86660 | Robert Price | 08/2017 |
9736 | (2^287842+3250)/2 | 86649 | Stefano Morozzi | 01/2020 |
9737 | 2^287809-111355 | 86640 | Enzo Doria | 05/2014 |
9738 | (47*2^287795+1)/7 | 86636 | Lelio R Paula | 04/2015 |
9739 | 9*7^102504+8 | 86627 | Lelio R Paula | 10/2014 |
9740 | 3^181551+32 | 86622 | Robert Price | 11/2013 |
9741 | 2^287735-521347 | 86617 | Paul Bourdelais | 05/2013 |
9742 | 21912^8953+8953^21912 | 86596 | Hans Havermann | 08/2020 |
9743 | (1017*2^287600-1)/91 | 86578 | Lelio R Paula | 04/2015 |
9744 | (2^287552+1302)/2 | 86562 | Stefano Morozzi | 01/2020 |
9745 | 2^287540-112967 | 86559 | Enzo Doria | 06/2014 |
9746 | 3^181407+20 | 86554 | Robert Price | 10/2013 |
9747 | (28*10^86550+101)/3 | 86551 | Robert Price | 09/2020 |
9749 | (13*10^86546+83)/3 | 86547 | Robert Price | 09/2016 |
9748 | (2^287502+2238)/2 | 86547 | Stefano Morozzi | 01/2020 |
9750 | 2^287482-4715 | 86541 | Yves Minet | 05/2019 |