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 ? |
7751 | 1036^27529+27529^1036 | 83010 | Norbert Schneider | 03/2020 |
7752 | 11^79701-3150 | 83001 | Enzo Doria | 12/2012 |
7753 | (2^275721+1554)/2 | 83000 | Stefano Morozzi | 01/2020 |
7754 | (2^275704+1858)/2 | 82995 | Stefano Morozzi | 01/2020 |
7755 | (2^275696+1726)/2 | 82993 | Stefano Morozzi | 01/2020 |
7756 | (55459*2^275657+1)/21 | 82985 | Lelio R Paula | 06/2014 |
7757 | (7*10^82960-457)/9 | 82960 | Robert Price | 09/2017 |
7758 | 2^275535-6855 | 82945 | Lelio R Paula | 08/2013 |
7759 | 3^173828-14 | 82938 | Robert Price | 08/2013 |
7760 | 2^275512-109677 | 82938 | Enzo Doria | 04/2014 |
7761 | (2^275450+1698)/2 | 82919 | Stefano Morozzi | 01/2020 |
7762 | 2^275446-110045 | 82918 | Enzo Doria | 04/2014 |
7763 | (2^275473+47)/29526097447 | 82916 | Lelio R Paula | 12/2012 |
7764 | 3^173755-14 | 82903 | Robert Price | 08/2013 |
7765 | (182*10^82898+43)/9 | 82900 | Robert Price | 11/2017 |
7766 | 2^275363-112309 | 82893 | Enzo Doria | 06/2014 |
7767 | 61^46427-60 | 82888 | Norbert Schneider | 10/2014 |
7768 | 3*10^82880-7^82880 | 82881 | Mike Oakes | 02/2010 |
7769 | 2^275290+90195 | 82871 | Lelio R Paula | 07/2013 |
7770 | (35*10^82870-71)/9 | 82871 | Robert Price | 05/2015 |
7771 | 2^275270-1859621 | 82865 | Enzo Doria | 03/2016 |
7772 | 2^275259+81 | 82862 | Henri Lifchitz | 01/2007 |
7773 | 21436^7335+7335^21436 | 82859 | Hans Havermann | 05/2020 |
7774 | (2^275247+1590)/2 | 82858 | Stefano Morozzi | 01/2020 |
7775 | 22515^4784+4784^22515 | 82851 | Hans Havermann | 05/2020 |
7776 | 6*10^82842-31 | 82843 | Robert Price | 07/2020 |
7777 | (10013^20707-1)/10012 | 82836 | Serge Batalov | 08/2020 |
7778 | (113703^16384+1)/2 | 82834 | Richard Fischer | 04/2018 |
7779 | (2^275146+954)/2 | 82827 | Stefano Morozzi | 01/2020 |
7780 | 7^98003+98003^7+1 | 82823 | Robert Price | 05/2014 |
7781 | (2^275101+3566)/2 | 82814 | Stefano Morozzi | 01/2020 |
7782 | (2^275050+3150)/2 | 82798 | Stefano Morozzi | 01/2020 |
7783 | (2^275044+2482)/2 | 82797 | Stefano Morozzi | 01/2020 |
7784 | 10^82790+126043 | 82791 | Patrick De Geest | 07/2013 |
7785 | 10^82790-47 | 82790 | Robert Price | 05/2011 |
7786 | 20762^9711+9711^20762 | 82784 | Hans Havermann | 05/2020 |
7787 | 2^274985-112663 | 82779 | Enzo Doria | 06/2014 |
7788 | 2^274935+15015 | 82764 | Lelio R Paula | 07/2013 |
7789 | (17*2^274922+1)/3 | 82761 | Lelio R Paula | 12/2012 |
7790 | [(55^5)^9511-(22434^2)^9511]/19 | 82759 | Richard Fischer | 11/2019 |
7791 | (43*10^82752-61)/9 | 82753 | Robert Price | 06/2015 |
7792 | (3*2^274862+1)/1027 | 82740 | Lelio R Paula | 12/2012 |
7793 | (2^274851+3126)/2 | 82739 | Stefano Morozzi | 01/2020 |
7794 | (31*2^274848-1)/4065 | 82736 | Lelio R Paula | 02/2014 |
7795 | (4^137405+23)/3 | 82726 | Robert Price | 02/2017 |
7796 | (2^274798+414)/2 | 82723 | Stefano Morozzi | 01/2020 |
7797 | 2^274791-110491 | 82721 | Enzo Doria | 05/2014 |
7798 | (11*2^274776-1)/95 | 82715 | Lelio R Paula | 02/2014 |
7799 | (47*10^82706-119)/9 | 82707 | Robert Price | 09/2017 |
7800 | 2^274704-111395 | 82695 | Enzo Doria | 05/2014 |
7801 | 2^274686-3911 | 82689 | Yves Minet | 04/2017 |
7802 | (1629^25741-1)/1628 | 82675 | Serge Batalov | 08/2020 |
7803 | 21474^7075+7075^21474 | 82670 | Hans Havermann | 06/2020 |
7804 | (2^274663-1017)/48871318292279 | 82669 | Lelio R Paula | 12/2012 |
7805 | (3^173263-5)/14498 | 82664 | Lelio R Paula | 12/2012 |
7806 | 20488^10815+10815^20488 | 82650 | Hans Havermann | 07/2020 |
7807 | (2^274554+894)/2 | 82649 | Stefano Morozzi | 01/2020 |
7809 | 2^274511-64875 | 82637 | Lelio R Paula | 08/2013 |
7808 | (2^274512+1222)/2 | 82637 | Stefano Morozzi | 01/2020 |
7810 | 12^76572-83 | 82636 | Lelio R Paula | 11/2015 |
7811 | 3^173191-2^173191 | 82634 | Mike Oakes | 02/2004 |
7812 | (2^274500+2970)/2 | 82633 | Stefano Morozzi | 01/2020 |
7813 | 20507^10698+10698^20507 | 82629 | Hans Havermann | 07/2020 |
7814 | (2^274429+3726)/2 | 82612 | Stefano Morozzi | 01/2020 |
7815 | 14^72072-15 | 82604 | Lelio R Paula | 11/2015 |
7816 | (2^274374+2950)/2 | 82595 | Stefano Morozzi | 01/2020 |
7817 | (55459*2^274339+1)/3 | 82589 | Lelio R Paula | 06/2014 |
7818 | 2^274351+123 | 82588 | Lelio R Paula | 06/2012 |
7819 | (1017*2^274339-1)/3515 | 82584 | Lelio R Paula | 04/2015 |
7820 | 21208^7833+7833^21208 | 82583 | Hans Havermann | 07/2020 |
7821 | (2^274248+750)/2 | 82557 | Stefano Morozzi | 01/2020 |
7822 | (2^274240+822)/2 | 82555 | Stefano Morozzi | 01/2020 |
7823 | 6^106086+113 | 82551 | Lelio R Paula | 12/2015 |
7824 | (17*10^82540-377)/9 | 82541 | Robert Price | 11/2017 |
7825 | (3470^23311-1)/3469 | 82526 | Serge Batalov | 08/2020 |
7826 | 2^274093-58695 | 82511 | Lelio R Paula | 07/2013 |
7827 | 7*10^82509-47 | 82510 | Robert Price | 02/2017 |
7828 | 2^274081-6555 | 82507 | Lelio R Paula | 07/2013 |
7830 | 2^274074-274074+1 | 82505 | Robert Price | 10/2014 |
7831 | 14^71985+47 | 82505 | Lelio R Paula | 11/2015 |
7829 | 5*10^82504+21 | 82505 | Robert Price | 01/2017 |
7832 | (2^274062+498)/2 | 82501 | Stefano Morozzi | 01/2020 |
7833 | 6^106014-139 | 82495 | Lelio R Paula | 12/2015 |
7834 | (2^274046+47)/753 | 82494 | Lelio R Paula | 12/2012 |
7835 | 2^274008-1838099 | 82485 | Enzo Doria | 05/2015 |
7836 | 2^273981-110029 | 82477 | Enzo Doria | 04/2014 |
7837 | 19^64490-30 | 82467 | Lelio R Paula | 12/2015 |
7838 | (2^273864+3150)/2 | 82441 | Stefano Morozzi | 01/2020 |
7839 | 21328^7333+7333^21328 | 82439 | Hans Havermann | 07/2020 |
7840 | 20829^9070+9070^20829 | 82434 | Hans Havermann | 07/2020 |
7843 | 2^273785+11715 | 82418 | Lelio R Paula | 07/2013 |
7842 | 8*10^82417+51 | 82418 | Robert Price | 05/2017 |
7841 | (2^273786+2242)/2 | 82418 | Stefano Morozzi | 01/2020 |
7844 | 11^79133-10 | 82409 | Theodore Burton | 11/2005 |
7845 | (2^273752+1882)/2 | 82408 | Stefano Morozzi | 01/2020 |
7846 | 91^42063+90 | 82404 | Norbert Schneider | 01/2016 |
7848 | 2^273662-63 | 82381 | Lelio R Paula | 06/2012 |
7847 | 2^273661+255 | 82381 | Lelio R Paula | 07/2013 |
7849 | (62*10^82377-17)/9 | 82378 | Robert Price | 09/2015 |
7850 | 2^273608+524287 | 82365 | Paul Underwood | 08/2018 |
7851 | 8*10^82360-51 | 82361 | Robert Price | 10/2017 |
7852 | (79*10^82347-1)/3 | 82349 | Robert Price | 11/2017 |
7853 | (11*10^82323+91)/3 | 82323 | Robert Price | 02/2017 |
7854 | (189977#+19081)/(19081*17260483) | 82317 | Makoto Morimoto | 10/2017 |
7855 | (189977#+196924)/(2*7*13*541*28874441) | 82316 | Makoto Morimoto | 10/2017 |
7856 | 15^69984+32 | 82308 | Lelio R Paula | 11/2015 |
7857 | 9*6^105746+7 | 82288 | Lelio R Paula | 10/2014 |
7858 | 89^42211-88 | 82286 | Norbert Schneider | 01/2016 |
7859 | gcd(V(1476435),5*F(295287)*(F(295287)+1)+1) | 82281 | Serge Batalov | 04/2015 |
7860 | 20596^9883+9883^20596 | 82279 | Hans Havermann | 06/2020 |
7861 | (19*10^82270-43)/3 | 82271 | Robert Price | 10/2017 |
7862 | 2^273282-110241 | 82267 | Enzo Doria | 05/2014 |
7863 | (2^273277+3746)/2 | 82265 | Stefano Morozzi | 01/2020 |
7864 | (13^73849+4^73849)/17 | 82263 | Robert Price | 05/2013 |
7865 | 10^82254+9^(82254+1) | 82255 | Robert Price | 08/2016 |
7866 | 10^82254+457773 | 82255 | Patrick De Geest | 12/2016 |
7867 | 10^82254-99609 | 82254 | Patrick De Geest | 12/2016 |
7868 | (754*10^82250-7)/9 | 82252 | Robert Price | 11/2017 |
7869 | 18^65524+13 | 82251 | Lelio R Paula | 11/2015 |
7870 | 2^273206-107 | 82244 | Lelio R Paula | 06/2012 |
7871 | 2^273193-459145 | 82240 | Enzo Doria | 05/2015 |
7873 | 2^273126+36937 | 82220 | Yves Minet | 06/2010 |
7872 | 39505!2+128 | 82220 | Robert Price | 03/2015 |
7874 | (2^273090+2214)/2 | 82208 | Stefano Morozzi | 01/2020 |
7875 | 2^273053-112111 | 82198 | Enzo Doria | 06/2014 |
7876 | (2^273052+3138)/2 | 82197 | Stefano Morozzi | 01/2020 |
7877 | 2^273016+1185 | 82187 | Lelio R Paula | 07/2013 |
7878 | (2^273008+3)/7 | 82183 | Lelio R Paula | 12/2012 |
7879 | (2^272992+2230)/2 | 82179 | Stefano Morozzi | 01/2020 |
7881 | 5^117554-15386 | 82167 | Enzo Doria | 02/2013 |
7880 | 21158^7647+7647^21158 | 82167 | Hans Havermann | 06/2020 |
7882 | 20194^11715+11715^20194 | 82165 | Hans Havermann | 06/2020 |
7883 | (38*10^82161+403)/9 | 82162 | Robert Price | 11/2017 |
7884 | 2^272907-111439 | 82154 | Enzo Doria | 05/2014 |
7885 | (67607*2^272901+1)/50790115 | 82149 | Lelio R Paula | 06/2014 |
7886 | 2^272875-69 | 82144 | Lelio R Paula | 06/2012 |
7887 | (3997^22807-1)/3996 | 82142 | Serge Batalov | 08/2020 |
7888 | 7^97197+4 | 82141 | Robert Price | 01/2014 |
7889 | (4*6^105554+1)/5 | 82137 | Serge Batalov | 10/2014 |
7890 | 2^272826-4433 | 82129 | Yves Minet | 03/2019 |
7891 | 6^105531-109 | 82120 | Lelio R Paula | 12/2015 |
7892 | 2^272766-65 | 82111 | Lelio R Paula | 06/2012 |
7893 | 19^64205-78 | 82103 | Lelio R Paula | 12/2015 |
7894 | 2^272702+1869979 | 82092 | Enzo Doria | 05/2016 |
7895 | (29*2^272687-1)/9 | 82088 | Lelio R Paula | 02/2014 |
7896 | (5*10^82085-119)/3 | 82086 | Robert Price | 04/2016 |
7897 | (2^272665+1334)/2 | 82081 | Stefano Morozzi | 01/2020 |
7898 | 2^272660-110049 | 82079 | Enzo Doria | 04/2014 |
7899 | 6^105474+67 | 82075 | Lelio R Paula | 12/2015 |
7900 | 2^272637+82875 | 82072 | Lelio R Paula | 07/2013 |
7901 | (6330^21589-1)/6329 | 82065 | Serge Batalov | 08/2020 |
7902 | 2^272587-31 | 82057 | Lelio R Paula | 06/2012 |
7903 | (1969^24907-1)/1968 | 82047 | Serge Batalov | 08/2020 |
7904 | (41*10^82044-59)/9 | 82045 | Robert Price | 10/2017 |
7905 | 8*5^117368+9 | 82038 | Lelio R Paula | 10/2014 |
7906 | (101525^16384+1)/131074 | 82023 | Lelio R Paula | 11/2011 |
7907 | 4*5^117343-7 | 82020 | Lelio R Paula | 10/2014 |
7908 | (2^272381+506)/2 | 81995 | Stefano Morozzi | 01/2020 |
7909 | (265*10^81990+17)/3 | 81992 | Robert Price | 05/2016 |
7910 | 3^171843+10 | 81990 | Robert Price | 10/2013 |
7911 | (2^272361+3762)/2 | 81989 | Stefano Morozzi | 01/2020 |
7912 | (7*10^81985+107)/3 | 81986 | Robert Price | 02/2017 |
7913 | (3*2^272356+1)/177163 | 81983 | Lelio R Paula | 12/2012 |
7914 | 2^272296-111149 | 81970 | Enzo Doria | 05/2014 |
7915 | (2^272293+2810)/2 | 81969 | Stefano Morozzi | 01/2020 |
7916 | (2^272278+2118)/2 | 81964 | Stefano Morozzi | 01/2020 |
7917 | (9*2^272243-1)/7 | 81954 | Lelio R Paula | 02/2014 |
7918 | 2^272177-111223 | 81934 | Enzo Doria | 05/2014 |
7919 | (2^272127+3878)/2 | 81919 | Stefano Morozzi | 01/2020 |
7921 | (151009^15817+1)/151010 | 81912 | Faysal YAtim | 06/2017 |
7920 | (2^272104+1950)/2 | 81912 | Stefano Morozzi | 01/2020 |
7922 | 23160^3437+3437^23160 | 81898 | Hans Havermann | 05/2020 |
7923 | (16*10^81896-19)/3 | 81897 | Robert Price | 03/2016 |
7924 | (16*10^81872-73)/3 | 81873 | Robert Price | 07/2020 |
7925 | (2^271922+78557)/63 | 81855 | Lelio R Paula | 12/2012 |
7926 | (10^81848+77)/3 | 81848 | Robert Price | 01/2017 |
7927 | (24737*2^271883+1)/3757 | 81846 | Lelio R Paula | 06/2014 |
7928 | (2^271885+1670)/2 | 81846 | Stefano Morozzi | 01/2020 |
7929 | 18813^19144-19144^18813 | 81831 | Norbert Schneider | 02/2020 |
7930 | (14*10^81818-89)/3 | 81819 | Robert Price | 05/2020 |
7931 | 19492^15731+15731^19492 | 81804 | Hans Havermann | 07/2020 |
7932 | 20420^10137+10137^20420 | 81801 | Hans Havermann | 06/2020 |
7933 | (2^271734+3942)/2 | 81800 | Stefano Morozzi | 01/2020 |
7934 | (15429^19531-1)/15428 | 81799 | Serge Batalov | 08/2020 |
7935 | 10^81778+39 | 81779 | Robert Price | 03/2011 |
7936 | 10^81778-171999 | 81778 | Patrick De Geest | 07/2013 |
7937 | (2^271652+2886)/2 | 81776 | Stefano Morozzi | 01/2020 |
7938 | (14*10^81767-11)/3 | 81768 | Erik Branger | 09/2014 |
7939 | (2^271546+610)/2 | 81744 | Stefano Morozzi | 01/2020 |
7940 | 2^271541+1868417 | 81742 | Enzo Doria | 04/2016 |
7941 | (2^271549-1)/238749682487 | 81734 | Jens K. Andersen | 02/2009 |
7942 | 19339^16836+16836^19339 | 81732 | Hans Havermann | 06/2020 |
7943 | 2^271484-167 | 81725 | Lelio R Paula | 06/2012 |
7945 | (8*10^81709-611)/9 | 81709 | Robert Price | 11/2017 |
7944 | (2^271431+1790)/2 | 81709 | Stefano Morozzi | 01/2020 |
7946 | 19648^14399+14399^19648 | 81703 | Hans Havermann | 05/2020 |
7947 | 2^271382+12909 | 81695 | Lelio R Paula | 07/2013 |
7948 | 49^48334+48 | 81694 | Norbert Schneider | 07/2014 |
7949 | 7^96668+48 | 81694 | Lelio R Paula | 10/2015 |
7950 | 11^78435-1410 | 81682 | Enzo Doria | 12/2012 |
7951 | 19575^14882+14882^19575 | 81680 | Hans Havermann | 04/2020 |
7952 | 5*10^81653+9 | 81654 | Dmitry Domanov | 11/2010 |
7953 | (2^271224+1230)/2 | 81647 | Stefano Morozzi | 01/2020 |
7954 | (2^271211-1)/613961495159 | 81631 | Jens K. Andersen | 02/2009 |
7955 | (2^271168+1806)/2 | 81630 | Stefano Morozzi | 01/2020 |
7956 | 3^171068+52 | 81621 | Lelio R Paula | 12/2015 |
7957 | 2^271115-110407 | 81614 | Enzo Doria | 05/2014 |
7958 | 2^271111-110635 | 81613 | Enzo Doria | 05/2014 |
7959 | 2^271098-111 | 81609 | Lelio R Paula | 06/2012 |
7960 | (77*10^81606+13)/9 | 81607 | Robert Price | 10/2015 |
7961 | 2^271064-2790209 | 81599 | Enzo Doria | 03/2016 |
7962 | (13*2^271053-1)/3095 | 81593 | Lelio R Paula | 02/2014 |
7963 | 2^271040-57 | 81592 | Lelio R Paula | 06/2012 |
7964 | (14*10^81585+73)/3 | 81586 | Robert Price | 04/2016 |
7965 | 127^38779+126 | 81584 | Norbert Schneider | 02/2015 |
7966 | 18^64966-43 | 81551 | Lelio R Paula | 11/2015 |
7967 | 19406^15909+15909^19406 | 81538 | Hans Havermann | 06/2020 |
7968 | (2^270843+1074)/2 | 81532 | Stefano Morozzi | 01/2020 |
7969 | (2^270840+47)/591 | 81529 | Lelio R Paula | 12/2012 |
7970 | 20^62654+87 | 81515 | Lelio R Paula | 01/2016 |
7971 | 20120^11241+11241^20120 | 81503 | Hans Havermann | 05/2020 |
7972 | (19*10^81488+191)/3 | 81489 | Robert Price | 01/2017 |
7973 | 8*10^81487+21 | 81488 | Robert Price | 02/2017 |
7974 | (73*10^81485+107)/9 | 81486 | Robert Price | 08/2016 |
7975 | 536^29847+29847^536 | 81458 | Norbert Schneider | 03/2017 |
7976 | 8*7^96378+9 | 81450 | Lelio R Paula | 10/2014 |
7977 | (2^270560+3862)/2 | 81447 | Stefano Morozzi | 01/2020 |
7978 | (5728^21673-1)/5727 | 81444 | Serge Batalov | 08/2020 |
7979 | 2^270546-110345 | 81443 | Enzo Doria | 05/2014 |
7980 | 13^73105-60 | 81435 | Lelio R Paula | 11/2015 |
7981 | (2^270502+1518)/2 | 81429 | Stefano Morozzi | 01/2020 |
7982 | (2^270479+2114)/2 | 81422 | Stefano Morozzi | 01/2020 |
7983 | (2^270384+1218)/2 | 81394 | Stefano Morozzi | 01/2020 |
7984 | 44^49524+43 | 81391 | Norbert Schneider | 06/2014 |
7985 | (3*2^270368-1)/13 | 81389 | Lelio R Paula | 02/2014 |
7986 | 19548^14563+14563^19548 | 81384 | Hans Havermann | 05/2020 |
7987 | (2^270337+1130)/2 | 81380 | Stefano Morozzi | 01/2020 |
7988 | 27405^932+932^27405 | 81377 | Hans Havermann | 05/2020 |
7989 | 19396^15683+15683^19396 | 81375 | Hans Havermann | 05/2020 |
7991 | (10^81363+719)/9 | 81363 | Serge Batalov | 12/2014 |
7990 | (734^28393-1)/733 | 81363 | Serge Batalov | 08/2020 |
7992 | (10^81353+449)/9 | 81353 | Serge Batalov | 12/2014 |
7993 | 14^70978-89 | 81350 | Lelio R Paula | 11/2015 |
7994 | 11^78108-2040 | 81342 | Enzo Doria | 12/2012 |
7995 | 7*10^81327-57 | 81328 | Robert Price | 06/2020 |
7996 | (7*10^81328-79)/9 | 81328 | Robert Price | 01/2021 |
7997 | 19102^18065+18065^19102 | 81315 | Hans Havermann | 07/2020 |
7998 | 3^170400-58 | 81302 | Lelio R Paula | 12/2015 |
7999 | (17*10^81300-131)/3 | 81301 | Robert Price | 12/2017 |
8000 | 10^81297+36991 | 81298 | Patrick De Geest | 06/2013 |