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 | 18917^4392+4392^18917 | 68909 | Hans Havermann | 09/2019 |
7752 | 2^228902+73519 | 68907 | Enzo Doria | 04/2016 |
7753 | (47*2^228884+1)/63 | 68901 | Lelio R Paula | 04/2015 |
7754 | 16542^14591+14591^16542 | 68883 | Hans Havermann | 09/2019 |
7755 | 2^228821+76427 | 68882 | Enzo Doria | 05/2016 |
7756 | 3^144340+50 | 68868 | Lelio R Paula | 09/2013 |
7757 | 2^228758-112967 | 68864 | Enzo Doria | 06/2014 |
7758 | 2^228743-111379 | 68859 | Enzo Doria | 05/2014 |
7759 | 2^228741-110911 | 68858 | Enzo Doria | 05/2014 |
7761 | lucasU(1926,1,20964)-lucasU(1926,1,20963) | 68857 | Mike Oakes | 05/2009 |
7760 | 3*5^98511-8 | 68857 | Lelio R Paula | 09/2014 |
7762 | 2^228717+467 | 68851 | Lelio R Paula | 02/2011 |
7763 | Phi(153218,10)/2211855049/23592660859/497714117291 | 68849 | Maksym Voznyy | 10/2011 |
7764 | (509203*2^228665-1)/35 | 68840 | Lelio R Paula | 04/2015 |
7765 | 8^76213-7^76213 | 68828 | Ananda Tallur & Jean-Louis Charton | 11/2003 |
7766 | 11^66074-1050 | 68809 | Enzo Doria | 11/2012 |
7767 | (68*10^68798-11)/3 | 68800 | Robert Price | 10/2017 |
7768 | 2^228484+181 | 68781 | Lelio R Paula | 02/2012 |
7769 | 2^228479-58695 | 68780 | Lelio R Paula | 04/2012 |
7770 | 2^228465-15853 | 68775 | Enzo Doria | 04/2013 |
7771 | 11^66040-5400 | 68774 | Enzo Doria | 01/2013 |
7772 | (13*2^228441-1)/575 | 68766 | Lelio R Paula | 01/2014 |
7773 | (17*2^228393+1)/295 | 68752 | Lelio R Paula | 08/2012 |
7774 | 17859^7072+7072^17859 | 68749 | Hans Havermann | 09/2019 |
7775 | 2*10^68744+41 | 68745 | Robert Price | 01/2017 |
7776 | 11^66011-150 | 68744 | Enzo Doria | 08/2012 |
7777 | (13*2^228304-1)/81 | 68726 | Lelio R Paula | 01/2014 |
7778 | 2^228295-111079 | 68724 | Enzo Doria | 05/2014 |
7780 | (3^144039+7)/22 | 68723 | Lelio R Paula | 04/2012 |
7779 | (8*10^68722+49)/3 | 68723 | Robert Price | 03/2016 |
7781 | (35*10^68718+1)/9 | 68719 | Serge Batalov | 11/2010 |
7782 | 2^228271-228271 | 68717 | Henri Lifchitz | 11/2004 |
7783 | 2^228257+82875 | 68713 | Lelio R Paula | 02/2013 |
7784 | 3^144013+40 | 68712 | Lelio R Paula | 09/2013 |
7785 | 13^61671-86 | 68699 | Lelio R Paula | 09/2013 |
7786 | (46*10^68698+683)/9 | 68699 | Robert Price | 12/2017 |
7787 | 17^55828-72 | 68694 | Lelio R Paula | 09/2013 |
7788 | 18675^4762+4762^18675 | 68683 | Hans Havermann | 09/2019 |
7789 | 6^88259+61 | 68679 | Lelio R Paula | 09/2013 |
7790 | 18^54699-5 | 68663 | Lelio R Paula | 09/2013 |
7791 | 33^45216+2 | 68662 | Lelio R Paula | 05/2013 |
7793 | (2^228073-5)/3 | 68657 | Lelio R Paula | 04/2012 |
7792 | (2^(6*38012+1)-5)/3 | 68657 | Kevin Acres | 11/2013 |
7794 | 2^228055-111607 | 68652 | Enzo Doria | 05/2014 |
7795 | 2^228054-114761 | 68652 | Enzo Doria | 05/2015 |
7796 | 2^228041+1185 | 68648 | Lelio R Paula | 04/2012 |
7797 | 2^228040-1037 | 68647 | Yves Minet | 11/2012 |
7798 | 2*F(328450)-1 | 68642 | Henri Lifchitz | 12/2009 |
7799 | 5^98201-34 | 68640 | Lelio R Paula | 06/2013 |
7800 | 9^71910+3*8^71910-2*4^71910-3*2^71910 | 68620 | Mike Oakes | 12/2010 |
7801 | 18177^5948+5948^18177 | 68607 | Hans Havermann | 09/2019 |
7802 | 7*2^227780-3 | 68570 | Lelio R Paula | 04/2014 |
7803 | 15327^16382-16382^15327 | 68567 | Norbert Schneider | 11/2018 |
7804 | 2^227733-121 | 68555 | Lelio R Paula | 02/2012 |
7805 | 16460^14607+14607^16460 | 68549 | Hans Havermann | 09/2019 |
7806 | 2^227702-16967 | 68546 | Enzo Doria | 05/2013 |
7807 | 2^227689-111655 | 68542 | Enzo Doria | 05/2014 |
7808 | gcd(V(823915),5*F(164783)*(F(164783)-1)+1) | 68536 | Serge Batalov | 04/2015 |
7809 | 5^98051+2 | 68535 | Ray Chandler | 07/2011 |
7810 | 5*10^68524+51 | 68525 | Robert Price | 01/2017 |
7811 | (11*10^68520-149)/3 | 68521 | Robert Price | 01/2017 |
7812 | 16624^13235+13235^16624 | 68520 | Hans Havermann | 09/2019 |
7813 | 16359^15430+15430^16359 | 68518 | Hans Havermann | 08/2019 |
7814 | 19^53573-78 | 68507 | Lelio R Paula | 09/2013 |
7815 | (68*10^68504+13)/9 | 68505 | Ray Chandler | 10/2010 |
7816 | 9*5^98003+4 | 68503 | Lelio R Paula | 09/2014 |
7817 | 21121^1750+1750^21121 | 68497 | Hans Havermann | 10/2019 |
7818 | 2^227528+51 | 68493 | Lelio R Paula | 02/2012 |
7819 | (146*28^47316+1)/3 | 68476 | Karsten Bonath | 06/2018 |
7820 | 3^143497+8 | 68466 | Lelio R Paula | 04/2012 |
7821 | 2^227430-95 | 68464 | Lelio R Paula | 02/2012 |
7822 | (8*10^68455+13)/3 | 68456 | Robert Price | 11/2017 |
7823 | 3*5^97934-2 | 68454 | Lelio R Paula | 09/2014 |
7824 | 20^52613+77 | 68452 | Lelio R Paula | 09/2013 |
7825 | 2^227371+32769 | 68446 | Lelio R Paula | 04/2012 |
7826 | (3^143465+4)/21270420067 | 68440 | Lelio R Paula | 04/2012 |
7827 | CopelandErdos(68433) | 68433 | Eric W. Weisstein | 08/2013 |
7828 | 2^227323+75431 | 68432 | Enzo Doria | 04/2016 |
7829 | 2^227314-1965 | 68429 | Lelio R Paula | 04/2012 |
7830 | 2^227303-110617 | 68426 | Enzo Doria | 05/2014 |
7832 | 7^80963+66 | 68422 | Lelio R Paula | 07/2013 |
7831 | 2^227290+73849 | 68422 | Enzo Doria | 04/2016 |
7833 | (14^59693-1)/13 | 68415 | Paul Bourdelais | 06/2008 |
7834 | 2^227256-2786249 | 68411 | Enzo Doria | 03/2016 |
7836 | 8*10^68400-7 | 68401 | Dmitry Domanov | 11/2010 |
7835 | (15*2^227228-1)/583 | 68401 | Lelio R Paula | 01/2014 |
7837 | 2^227210+1869169 | 68398 | Enzo Doria | 05/2016 |
7838 | 2^227159-1838635 | 68382 | Enzo Doria | 05/2015 |
7839 | 44*10^68368+3 | 68370 | Robert Price | 08/2017 |
7840 | 2^227115-7807 | 68369 | Lelio R Paula | 10/2009 |
7841 | 18469^5032+5032^18469 | 68368 | Hans Havermann | 08/2019 |
7842 | 2^227078-47355 | 68358 | Lelio R Paula | 04/2012 |
7843 | 16927^10920+10920^16927 | 68355 | Hans Havermann | 10/2019 |
7844 | (2^227063-1017)/349 | 68351 | Lelio R Paula | 04/2012 |
7845 | 2^227057+148091 | 68351 | Lelio R Paula | 04/2012 |
7846 | 2^227046+187 | 68348 | Lelio R Paula | 02/2012 |
7847 | 2^227009-4693 | 68337 | Yves Minet | 05/2019 |
7848 | 9*10^68331+59 | 68332 | Robert Price | 10/2017 |
7849 | 11^65608-720 | 68324 | Enzo Doria | 11/2012 |
7850 | (32*10^68320+319)/9 | 68321 | Robert Price | 10/2017 |
7851 | 33408!2-2^33408-1 | 68315 | Joe McLean | 05/2017 |
7852 | 2^226894-123 | 68302 | Lelio R Paula | 02/2012 |
7853 | 10^68297+141559 | 68298 | Patrick De Geest | 05/2013 |
7854 | 10^68297-93 | 68297 | Robert Price | 05/2011 |
7855 | 680^24107-679^24107 | 68284 | Jean-Louis Charton | 07/2010 |
7856 | 12^63267+59 | 68277 | Lelio R Paula | 07/2013 |
7857 | 2^226785-1159 | 68270 | Yves Minet | 11/2012 |
7858 | 7^80769+90 | 68258 | Lelio R Paula | 07/2013 |
7859 | (88*10^68256-1)/3 | 68258 | Robert Price | 10/2017 |
7860 | 2^226741+66587 | 68256 | Yves Minet | 10/2010 |
7861 | 2^226720-112403 | 68250 | Enzo Doria | 06/2014 |
7862 | (78557*2^226694+1)/3 | 68247 | Lelio R Paula | 06/2014 |
7863 | (83*10^68231+7)/9 | 68232 | Robert Price | 10/2015 |
7864 | 65^37636+16 | 68231 | Lelio R Paula | 01/2016 |
7865 | 5^97615-4^(97615+1) | 68230 | Robert Price | 04/2016 |
7866 | 12^63200-43 | 68205 | Lelio R Paula | 07/2013 |
7867 | -330*Bern(21940)/223790620733 | 68201 | Serge Batalov | 04/2015 |
7868 | 3^142925+8680 | 68193 | Lelio R Paula | 06/2013 |
7869 | 3*2^226512-7 | 68188 | Lelio R Paula | 04/2014 |
7870 | 14^59484+57 | 68177 | Lelio R Paula | 08/2013 |
7871 | 11^65438-2190 | 68147 | Enzo Doria | 12/2012 |
7872 | 91^34784-2 | 68144 | Lelio R Paula | 09/2013 |
7873 | 16589^12810+12810^16589 | 68141 | Hans Havermann | 10/2019 |
7874 | (2^226336-3)*6-7 | 68135 | Gilbert Mozzo | 06/2016 |
7875 | (22*10^68133-37)/3 | 68134 | Robert Price | 05/2017 |
7876 | 5^97476+36 | 68133 | Lelio R Paula | 06/2013 |
7877 | 3*5^97474-2 | 68132 | Lelio R Paula | 09/2014 |
7878 | 12^63130+49 | 68129 | Lelio R Paula | 07/2013 |
7879 | 2^226281+5385 | 68118 | Lelio R Paula | 04/2012 |
7880 | 2^226257-3049 | 68111 | Yves Minet | 10/2014 |
7881 | (28*10^68107-43)/3 | 68108 | Robert Price | 04/2016 |
7882 | 2^226243-2587 | 68106 | Yves Minet | 07/2014 |
7883 | 20^52346-11 | 68104 | Lelio R Paula | 09/2013 |
7884 | 2^226205-99 | 68095 | Lelio R Paula | 02/2012 |
7885 | 2^226187-49 | 68090 | Lelio R Paula | 02/2012 |
7886 | 3^142708+76 | 68090 | Lelio R Paula | 06/2013 |
7887 | 7*10^68088-89 | 68089 | Robert Price | 01/2017 |
7888 | 18562^4653+4653^18562 | 68081 | Hans Havermann | 09/2019 |
7889 | 2^226154+71839 | 68080 | Enzo Doria | 04/2016 |
7890 | (10^68078+59)/3 | 68078 | Serge Batalov | 12/2014 |
7892 | 14^59387+69 | 68066 | Lelio R Paula | 08/2013 |
7891 | 2^226108-113429 | 68066 | Enzo Doria | 07/2014 |
7893 | (62*10^68059-197)/9 | 68060 | Robert Price | 10/2017 |
7894 | (24^49307-1)/23 | 68053 | Robert Price | 04/2012 |
7895 | 15^57859-28 | 68048 | Lelio R Paula | 08/2013 |
7896 | (509203*2^226003-1)/13 | 68039 | Lelio R Paula | 04/2015 |
7897 | 2^226015-111361 | 68038 | Enzo Doria | 05/2014 |
7898 | (10^(2*34016+1)+72*10^34016-1)/9 | 68033 | Robert Price | 09/2017 |
7899 | 4*10^68020+91 | 68021 | Robert Price | 11/2017 |
7900 | 16181^15948+15948^16181 | 68004 | Hans Havermann | 09/2019 |
7901 | 156*(10^(4*17000)-1)/101-1 | 68001 | Robert Price | 05/2014 |
7902 | 2^225878-1017 | 67997 | Lelio R Paula | 04/2012 |
7903 | 3*8^75289+5^75289-3*4^75289+3*2^75289 | 67994 | Mike Oakes | 11/2010 |
7904 | 3*2^225865+5 | 67993 | Lelio R Paula | 04/2014 |
7905 | 11^65284-5820 | 67987 | Enzo Doria | 01/2013 |
7906 | 19^53166+70 | 67987 | Lelio R Paula | 08/2013 |
7907 | 2^225837+75407 | 67984 | Enzo Doria | 04/2016 |
7908 | 3^142464-112654 | 67973 | Enzo Doria | 06/2014 |
7909 | 16851^10802+10802^16851 | 67969 | Hans Havermann | 09/2019 |
7910 | 2^225777+95 | 67966 | Lelio R Paula | 02/2012 |
7911 | (2^225641-1017)/138815 | 67920 | Lelio R Paula | 04/2012 |
7912 | 2^225609-18445 | 67916 | Enzo Doria | 05/2013 |
7913 | 2^225542-177 | 67895 | Lelio R Paula | 02/2012 |
7914 | 2^225536-29 | 67894 | Henri Lifchitz | 05/2005 |
7915 | 11^65190-3390 | 67889 | Enzo Doria | 12/2012 |
7916 | 62271!4-1024 | 67878 | Robert Price | 11/2019 |
7917 | 16971^9986+9986^16971 | 67874 | Hans Havermann | 09/2019 |
7918 | 3^142249-2^(142249-1) | 67871 | Robert Price | 01/2016 |
7919 | 11^65168-4830 | 67866 | Enzo Doria | 01/2013 |
7920 | 10^67863+58627 | 67864 | Patrick De Geest | 01/2016 |
7921 | 16258^14933+14933^16258 | 67864 | Hans Havermann | 09/2019 |
7922 | 10^67863-11 | 67863 | Robert Price | 05/2011 |
7923 | (1!)^2+(2!)^2+(3!)^2+...+(9566!)^2 | 67856 | T. D. Noe | 02/2006 |
7924 | (19*2^225357-1)/11 | 67840 | Lelio R Paula | 01/2014 |
7925 | 9*6^87164-7 | 67828 | Lelio R Paula | 09/2014 |
7926 | 131^32034+130 | 67825 | Norbert Schneider | 03/2015 |
7927 | (29*2^225314+1)/10881 | 67824 | Lelio R Paula | 08/2012 |
7928 | 17761^6586+6586^17761 | 67823 | Hans Havermann | 09/2019 |
7929 | (134*10^67816+7)/3 | 67818 | Robert Price | 02/2017 |
7930 | 2^225261+855 | 67811 | Lelio R Paula | 04/2012 |
7931 | (29*2^225252-1)/2933 | 67806 | Lelio R Paula | 01/2014 |
7932 | (4*10^67796+23)/9 | 67796 | Robert Price | 11/2014 |
7933 | 15477^16180-16180^15477 | 67790 | Norbert Schneider | 09/2018 |
7934 | (23*2^225177+1)/31 | 67785 | Lelio R Paula | 08/2012 |
7935 | (4*10^67784+149)/9 | 67784 | Robert Price | 03/2017 |
7936 | 8177^17322+17322^8177 | 67774 | Norbert Schneider | 03/2019 |
7937 | 2^225120+151 | 67768 | Lelio R Paula | 02/2012 |
7938 | (44^41233+1)/45 | 67763 | Robert Price | 11/2013 |
7939 | 89933!6-16 | 67747 | Robert Price | 08/2018 |
7940 | 8*10^67742+9 | 67743 | Dmitry Domanov | 11/2010 |
7941 | 2^224977+1089 | 67725 | Lelio R Paula | 04/2012 |
7942 | (2^224950+5)/19509 | 67713 | Lelio R Paula | 04/2012 |
7943 | 11^65011-5970 | 67702 | Enzo Doria | 01/2013 |
7944 | 88*10^67691+7 | 67693 | Robert Price | 11/2016 |
7945 | 11^65000-7140 | 67691 | Enzo Doria | 01/2013 |
7946 | 2^224841+59 | 67684 | Lelio R Paula | 02/2012 |
7947 | 33129!2+2^33129 | 67684 | Robert Price | 09/2015 |
7948 | 2^224827+8415 | 67680 | Lelio R Paula | 02/2013 |
7949 | (9913*2^224821-1)/19825 | 67678 | Lelio R Paula | 04/2015 |
7950 | 2^224815-49789 | 67677 | Henri Lifchitz | 04/2005 |
7951 | 7257^17528+17528^7257 | 67672 | Norbert Schneider | 08/2019 |
7952 | (43*10^67664+11)/9 | 67665 | Robert Price | 06/2015 |
7953 | (292*10^67659-1)/3 | 67661 | Robert Price | 01/2017 |
7954 | (17*2^224766+1)/29007 | 67659 | Lelio R Paula | 06/2012 |
7955 | 2^224744+577294575 | 67655 | Lelio R Paula | 02/2016 |
7956 | ((1+sqrt(7)+sqrt(6))^45863-1)*((1+sqrt(7)-sqrt(6))^45863-1)*((1-sqrt(7)+sqrt(6))^45863-1)*((1-sqrt(7 | 67653 | Mike Oakes | 05/2009 |
7957 | (2^224721+78557)/1739 | 67645 | Lelio R Paula | 05/2012 |
7958 | 2^224700+1915 | 67642 | Lelio R Paula | 04/2012 |
7959 | (29*2^224694-1)/198905 | 67636 | Lelio R Paula | 01/2014 |
7960 | 2^224658-93 | 67629 | Lelio R Paula | 02/2012 |
7961 | 2^224624-13077 | 67619 | Enzo Doria | 10/2012 |
7962 | 18^53860-3319 | 67609 | Enzo Doria | 03/2013 |
7963 | (21181*2^224570+1)/5 | 67606 | Lelio R Paula | 06/2014 |
7964 | 89762!6-3 | 67606 | Robert Price | 08/2018 |
7965 | 2^224578-111041 | 67605 | Enzo Doria | 05/2014 |
7966 | 2^224569-113233 | 67603 | Enzo Doria | 07/2014 |
7967 | 2^224562-113171 | 67600 | Enzo Doria | 07/2014 |
7968 | 2^224556-113129 | 67599 | Enzo Doria | 07/2014 |
7969 | 6*7^79980-5 | 67592 | Lelio R Paula | 09/2014 |
7970 | (6883*2^224523-1)/16031 | 67588 | Lelio R Paula | 04/2015 |
7971 | 10^67579+164779 | 67580 | Patrick De Geest | 07/2017 |
7973 | (52*10^67578-61)/9 | 67579 | Robert Price | 09/2015 |
7972 | 10^67579-401 | 67579 | Robert Price | 06/2017 |
7974 | (17^54919-1)/16 | 67574 | Paul Bourdelais | 03/2010 |
7975 | 22115^1134+1134^22115 | 67553 | Hans Havermann | 08/2019 |
7976 | ((10^33771-1)*10000/9+6389)*10^33771+(10^33771-1)/9 | 67546 | Hans Havermann | 05/2019 |
7977 | (2^224363-9)/1951 | 67537 | Lelio R Paula | 04/2012 |
7978 | (709*10^67517-7)/9 | 67519 | Robert Price | 09/2017 |
7979 | 47719!3-59049 | 67516 | Robert Price | 12/2015 |
7980 | 16356^13411+13411^16356 | 67509 | Hans Havermann | 09/2019 |
7981 | (22^50287+1)/23 | 67506 | Robert Price | 06/2013 |
7982 | 16671^11198+11198^16671 | 67504 | Hans Havermann | 09/2019 |
7983 | 3*10^67499+67 | 67500 | Robert Price | 11/2017 |
7984 | 11^64811-5820 | 67494 | Enzo Doria | 01/2013 |
7985 | 131^31870+31870^131 | 67478 | Serge Batalov | 05/2014 |
7986 | 9*10^67468+7 | 67469 | Dmitry Domanov | 11/2010 |
7987 | (31*2^224115-1)/47723 | 67463 | Lelio R Paula | 01/2014 |
7988 | 128^32001+32001^128 | 67433 | Serge Batalov | 05/2014 |
7989 | (89*10^67431-17)/9 | 67432 | Robert Price | 11/2015 |
7990 | 2^223912-111059 | 67405 | Enzo Doria | 05/2014 |
7992 | 2^223913+73697 | 67405 | Enzo Doria | 04/2016 |
7991 | (2*10^67405-23)/3 | 67405 | Robert Price | 09/2017 |
7994 | (10^67404+179)/9 | 67404 | Serge Batalov | 12/2014 |
7993 | 18738^3955+3955^18738 | 67404 | Hans Havermann | 09/2016 |
7995 | 9*7^79728-2 | 67379 | Lelio R Paula | 09/2014 |
7996 | (28*10^67378-43)/3 | 67379 | Robert Price | 04/2016 |
7997 | (3^141234-7)/430241282 | 67378 | Lelio R Paula | 04/2012 |
7998 | 211^28988-2 | 67377 | Lelio R Paula | 09/2013 |
7999 | 4*10^67375+91 | 67376 | Robert Price | 11/2017 |
8000 | 2^223793-18069 | 67369 | Enzo Doria | 05/2013 |