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 | (2^285882+2830)/2 | 86059 | Stefano Morozzi | 01/2020 |
7752 | 2^285836+65835 | 86046 | Lelio R Paula | 09/2013 |
7753 | 21587^9670+9670^21587 | 86034 | Hans Havermann | 08/2020 |
7754 | (2^285766+78557)/3 | 86024 | Lelio R Paula | 01/2013 |
7755 | (2^285732+2682)/2 | 86014 | Stefano Morozzi | 01/2020 |
7756 | (5*10^86011-107)/3 | 86012 | Robert Price | 01/2017 |
7757 | 6^110522+163 | 86003 | Lelio R Paula | 12/2015 |
7758 | 41^53323-40^53323 | 85999 | Robert Price | 07/2012 |
7759 | 2^285678-109617 | 85998 | Enzo Doria | 04/2014 |
7760 | 6^110506-16673 | 85991 | Enzo Doria | 02/2013 |
7761 | 2^285569-741 | 85965 | Lelio R Paula | 09/2013 |
7762 | (32*10^85964+31)/9 | 85965 | Robert Price | 05/2015 |
7763 | 133^40466+132 | 85944 | Norbert Schneider | 02/2019 |
7764 | 23378^4743+4743^23378 | 85939 | Hans Havermann | 08/2020 |
7765 | 6*10^85910-7 | 85911 | Dmitry Domanov | 11/2010 |
7766 | (412*10^85897-7)/9 | 85899 | Robert Price | 10/2017 |
7767 | 4*10^85881+63 | 85882 | Robert Price | 06/2016 |
7768 | (2^285257+1514)/2 | 85871 | Stefano Morozzi | 01/2020 |
7769 | (2^285248+3738)/2 | 85868 | Stefano Morozzi | 01/2020 |
7770 | (10^85864+53)/9 | 85864 | Robert Price | 01/2015 |
7772 | 2^285230+345 | 85863 | Lelio R Paula | 09/2013 |
7771 | 21774^8777+8777^21774 | 85863 | Hans Havermann | 08/2020 |
7773 | (5*10^85855-473)/9 | 85855 | Robert Price | 12/2017 |
7774 | 2^285199+75683 | 85854 | Enzo Doria | 04/2016 |
7775 | (76*10^85847-211)/9 | 85848 | Robert Price | 10/2017 |
7776 | p(17832)#/2+512 | 85845 | Robert Price | 05/2017 |
7777 | (16*10^85839+167)/3 | 85840 | Robert Price | 10/2017 |
7778 | 11^82426-1230 | 85838 | Enzo Doria | 11/2012 |
7779 | (14727^20593-1)/14726 | 85830 | Serge Batalov | 08/2020 |
7780 | 2^285113-109881 | 85828 | Enzo Doria | 04/2014 |
7781 | (2^285033+1670)/2 | 85804 | Stefano Morozzi | 01/2020 |
7782 | (2^285033+3854)/2 | 85804 | Stefano Morozzi | 01/2020 |
7783 | (2^285035+3194)/2 | 85804 | Stefano Morozzi | 01/2020 |
7784 | 5^122746-92 | 85796 | Lelio R Paula | 10/2015 |
7785 | 2^285001-113461 | 85794 | Enzo Doria | 08/2014 |
7787 | (11*10^85782-131)/3 | 85783 | Robert Price | 02/2017 |
7786 | (2^284963+2778)/2 | 85783 | Stefano Morozzi | 01/2020 |
7788 | (2^284960+3570)/2 | 85782 | Stefano Morozzi | 01/2020 |
7789 | (2^284937+1706)/2 | 85775 | Stefano Morozzi | 01/2020 |
7790 | 2^284913-110491 | 85768 | Enzo Doria | 05/2014 |
7791 | 6^110220+47 | 85768 | Lelio R Paula | 12/2015 |
7792 | (1966^26041-1)/1965 | 85765 | Serge Batalov | 08/2020 |
7793 | 20698^13909+13909^20698 | 85758 | Hans Havermann | 07/2020 |
7794 | 3*6^110202-5 | 85755 | Lelio R Paula | 10/2014 |
7795 | 9*10^85746+37 | 85747 | Robert Price | 12/2020 |
7796 | (157^39047-1)/156 | 85742 | Alexander Zhirkov | 06/2016 |
7797 | (3^179705+7)/250 | 85739 | Lelio R Paula | 12/2012 |
7798 | (9127^21649-1)/9126 | 85734 | Serge Batalov | 08/2020 |
7799 | 2^284734-6555 | 85714 | Lelio R Paula | 09/2013 |
7800 | (10^85712+53)/9 | 85712 | Robert Price | 01/2015 |
7801 | 3*10^85703+11 | 85704 | Robert Price | 11/2017 |
7802 | (6883*2^284668-1)/3 | 85697 | Lelio R Paula | 04/2015 |
7803 | 20728^13621+13621^20728 | 85694 | Hans Havermann | 07/2020 |
7804 | 22126^7459+7459^22126 | 85687 | Hans Havermann | 07/2020 |
7805 | (9311^21589-1)/9310 | 85683 | Serge Batalov | 08/2020 |
7806 | 20^65855+83 | 85680 | Lelio R Paula | 01/2016 |
7807 | 6^110105+53 | 85679 | Lelio R Paula | 12/2015 |
7808 | 3*10^85672-73 | 85673 | Robert Price | 02/2021 |
7809 | (10^85666-97)/3 | 85666 | Robert Kinser | 10/2017 |
7811 | 2^284567-7605 | 85664 | Lelio R Paula | 09/2013 |
7810 | 9*5^122555-2 | 85664 | Lelio R Paula | 10/2014 |
7812 | 5^122554-14 | 85662 | Lelio R Paula | 10/2015 |
7813 | 9*5^122543+4 | 85655 | Lelio R Paula | 10/2014 |
7814 | 3^179489-2^(179489-1) | 85639 | Robert Price | 01/2016 |
7815 | (2^284487+1970)/2 | 85639 | Stefano Morozzi | 01/2020 |
7816 | (2^284477+1614)/2 | 85636 | Stefano Morozzi | 01/2020 |
7817 | 23693^4114+4114^23693 | 85633 | Hans Havermann | 07/2020 |
7818 | 19690^19941+19941^19690 | 85632 | Norbert Schneider | 03/2020 |
7819 | 2^284429+177 | 85622 | Lelio R Paula | 08/2012 |
7820 | 2*3^179452+5 | 85621 | Lelio R Paula | 07/2014 |
7821 | 21036^11711+11711^21036 | 85587 | Hans Havermann | 07/2020 |
7823 | (19^66919+3^66919)/22 | 85572 | Robert Price | 07/2013 |
7822 | (2^284265+602)/2 | 85572 | Stefano Morozzi | 01/2020 |
7826 | 66^47023-65 | 85561 | Norbert Schneider | 11/2014 |
7824 | (2^284226+3802)/2 | 85561 | Stefano Morozzi | 01/2020 |
7825 | (2^284228+2170)/2 | 85561 | Stefano Morozzi | 01/2020 |
7827 | 6^109943+13 | 85553 | Lelio R Paula | 12/2015 |
7828 | 2^284183-4747 | 85548 | Yves Minet | 05/2019 |
7829 | (5637^22807-1)/5636 | 85547 | Serge Batalov | 08/2020 |
7830 | 20782^13071+13071^20782 | 85546 | Hans Havermann | 08/2020 |
7831 | (2^284167-509203)/34564775 | 85536 | Lelio R Paula | 01/2013 |
7832 | (2^284136+2262)/2 | 85534 | Stefano Morozzi | 01/2020 |
7833 | 2^284133-3 | 85533 | M. Frind & P. Underwood | 07/2002 |
7834 | 2^284113+315 | 85527 | Lelio R Paula | 09/2013 |
7835 | (2^284098+3082)/2 | 85522 | Stefano Morozzi | 01/2020 |
7836 | (2^284070+1482)/2 | 85514 | Stefano Morozzi | 01/2020 |
7837 | (2^284068+2866)/2 | 85513 | Stefano Morozzi | 01/2020 |
7838 | (8721^21701-1)/8720 | 85511 | Serge Batalov | 08/2020 |
7839 | 22085^7438+7438^22085 | 85502 | Hans Havermann | 07/2020 |
7840 | (28*10^85500+149)/3 | 85501 | Robert Price | 10/2017 |
7841 | (16*10^85495-19)/3 | 85496 | Robert Price | 03/2016 |
7842 | 2^284004+177 | 85494 | Lelio R Paula | 08/2012 |
7843 | (16*10^85473-91)/3 | 85474 | Robert Price | 06/2016 |
7844 | 66*10^85468+7 | 85470 | Robert Price | 07/2020 |
7845 | 3^179120+22 | 85462 | Robert Price | 12/2013 |
7846 | 2^283872-2792249 | 85454 | Enzo Doria | 03/2016 |
7847 | (4703^23269-1)/4702 | 85449 | Serge Batalov | 08/2020 |
7848 | (29*10^85438+259)/9 | 85439 | Robert Price | 04/2017 |
7849 | 2^283817+82875 | 85438 | Lelio R Paula | 09/2013 |
7850 | (2^283815-5)/201 | 85435 | Lelio R Paula | 01/2013 |
7851 | 2^283796-125 | 85432 | Lelio R Paula | 08/2012 |
7852 | (10174^21319-1)/10173 | 85432 | Serge Batalov | 08/2020 |
7853 | (29*2^283784-1)/13 | 85428 | Lelio R Paula | 02/2014 |
7854 | 15^72592-28 | 85375 | Lelio R Paula | 11/2015 |
7856 | 2^283604-58695 | 85374 | Lelio R Paula | 09/2013 |
7855 | 21863^8034+8034^21863 | 85374 | Hans Havermann | 08/2020 |
7858 | (44*10^85370-503)/9 | 85371 | Robert Price | 10/2017 |
7857 | (2^283597+1854)/2 | 85371 | Stefano Morozzi | 01/2020 |
7859 | 496^31671+31671^496 | 85369 | Norbert Schneider | 07/2017 |
7860 | (2^283578+478)/2 | 85366 | Stefano Morozzi | 01/2020 |
7861 | (15*2^283583-1)/103051 | 85364 | Lelio R Paula | 02/2014 |
7862 | 15^72572+92 | 85352 | Lelio R Paula | 11/2015 |
7863 | 2^283473-1615 | 85334 | Lelio R Paula | 09/2013 |
7864 | (2^283469+2934)/2 | 85333 | Stefano Morozzi | 01/2020 |
7865 | 2^283466-15 | 85332 | Lelio R Paula | 08/2012 |
7866 | (2^283437+3446)/2 | 85323 | Stefano Morozzi | 01/2020 |
7867 | 19966^18761+18761^19966 | 85320 | Hans Havermann | 07/2020 |
7868 | (2^283411+3894)/2 | 85315 | Stefano Morozzi | 01/2020 |
7869 | 11^81916-7020 | 85307 | Enzo Doria | 02/2013 |
7870 | 6^109626-7 | 85306 | Lelio R Paula | 11/2014 |
7871 | 2^283374-110327 | 85305 | Enzo Doria | 05/2014 |
7872 | 33*10^85297+7 | 85299 | Robert Price | 07/2016 |
7873 | (6010^22573-1)/6009 | 85297 | Serge Batalov | 08/2020 |
7874 | 1137^27908+27908^1137 | 85281 | Norbert Schneider | 04/2020 |
7875 | 18^67933+41 | 85275 | Lelio R Paula | 11/2015 |
7876 | 2^283271-109341 | 85274 | Enzo Doria | 04/2014 |
7877 | (2^283266+3678)/2 | 85272 | Stefano Morozzi | 01/2020 |
7878 | 19688^19855-19855^19688 | 85262 | Norbert Schneider | 03/2020 |
7879 | (2^283216+1926)/2 | 85257 | Stefano Morozzi | 01/2020 |
7880 | (25*10^85249-73)/3 | 85250 | Robert Price | 09/2016 |
7881 | 2*5^121959+3 | 85246 | Lelio R Paula | 10/2014 |
7882 | 19834^19834+8947 | 85235 | Henri Lifchitz | 11/2004 |
7883 | 2^283140+1915 | 85234 | Lelio R Paula | 09/2013 |
7884 | 3*10^85227-97 | 85228 | Robert Price | 08/2017 |
7885 | (16*10^85221-31)/3 | 85222 | Robert Price | 03/2016 |
7886 | (2^283096+3186)/2 | 85221 | Stefano Morozzi | 01/2020 |
7887 | 2^283083+1869383 | 85217 | Enzo Doria | 05/2016 |
7888 | (24737*2^283051+1)/17 | 85211 | Lelio R Paula | 06/2014 |
7889 | (2^283032+78557)/3 | 85201 | Lelio R Paula | 01/2013 |
7890 | 2^283026-109893 | 85200 | Enzo Doria | 04/2014 |
7891 | (2^283012+78557)/129 | 85193 | Lelio R Paula | 01/2013 |
7892 | (10^(2*42589+1)+72*10^42589-1)/9 | 85179 | Robert Price | 09/2017 |
7893 | 2^282860-109727 | 85150 | Enzo Doria | 04/2014 |
7894 | (2^282847+13)/3 | 85145 | Stefano Morozzi | 03/2020 |
7895 | 2^282819+1869413 | 85138 | Enzo Doria | 05/2016 |
7896 | 3^178428+2 | 85132 | Ray Chandler | 07/2011 |
7897 | 19985^18162+18162^19985 | 85120 | Hans Havermann | 07/2020 |
7898 | (22699*2^282742+1)/73 | 85117 | Lelio R Paula | 06/2014 |
7899 | (16^70687+5^70687)/21 | 85115 | Robert Price | 07/2013 |
7900 | (2^282732+738)/2 | 85111 | Stefano Morozzi | 01/2020 |
7901 | (2^282720+1590)/2 | 85107 | Stefano Morozzi | 01/2020 |
7902 | (3^178352-2)/7 | 85095 | Lelio R Paula | 12/2012 |
7903 | 22150^6939+6939^22150 | 85085 | Hans Havermann | 07/2020 |
7904 | 11^81699-2790 | 85081 | Enzo Doria | 12/2012 |
7905 | 2^282612-95 | 85075 | Lelio R Paula | 08/2012 |
7906 | 2^282609-4069 | 85074 | Yves Minet | 02/2018 |
7907 | 75^45366-2 | 85065 | Lelio R Paula | 08/2013 |
7908 | (2^282550+1918)/2 | 85056 | Stefano Morozzi | 01/2020 |
7909 | (2^282515+3786)/2 | 85046 | Stefano Morozzi | 01/2020 |
7910 | 2^282511+69 | 85045 | Lelio R Paula | 08/2012 |
7911 | 69^46249+2 | 85045 | Lelio R Paula | 06/2013 |
7912 | (2^282488+2382)/2 | 85038 | Stefano Morozzi | 01/2020 |
7913 | (2^282489+1862)/2 | 85038 | Stefano Morozzi | 01/2020 |
7914 | 3^178226+10 | 85036 | Robert Price | 10/2013 |
7915 | (2^282475+1050)/2 | 85034 | Stefano Morozzi | 01/2020 |
7916 | (182*10^85017+43)/9 | 85019 | Robert Price | 11/2017 |
7917 | (14*10^85010+73)/3 | 85011 | Robert Price | 04/2016 |
7918 | 9*2^282338-7 | 84994 | Lelio R Paula | 08/2014 |
7919 | (9913*2^282330-1)/1578003 | 84988 | Lelio R Paula | 04/2015 |
7920 | 20025^17504+17504^20025 | 84969 | Hans Havermann | 07/2020 |
7921 | 2^282203+5 | 84952 | Lelio R Paula | 08/2012 |
7922 | 2^282184+13 | 84946 | Lelio R Paula | 08/2012 |
7923 | 11^81568-2400 | 84945 | Enzo Doria | 12/2012 |
7924 | 11^81562-2340 | 84939 | Enzo Doria | 12/2012 |
7925 | (41*10^84927+31)/9 | 84928 | Robert Price | 05/2015 |
7926 | (3923^23629-1)/3922 | 84911 | Serge Batalov | 08/2020 |
7927 | (2^282053+1406)/2 | 84907 | Stefano Morozzi | 01/2020 |
7928 | (2*10^84904-17)/3 | 84904 | Robert Price | 08/2014 |
7929 | (4^141017+23)/3 | 84901 | Robert Price | 02/2017 |
7930 | 6^109100-18653 | 84897 | Enzo Doria | 03/2013 |
7931 | 2^281973+1025 | 84883 | Lelio R Paula | 09/2013 |
7932 | 2^281951-109651 | 84876 | Enzo Doria | 04/2014 |
7933 | 21618^8431+8431^21618 | 84870 | Hans Havermann | 07/2020 |
7934 | 11^81491-5430 | 84865 | Enzo Doria | 01/2013 |
7935 | 10^84863+141057 | 84864 | Patrick De Geest | 02/2021 |
7936 | 10^84863-40001 | 84863 | Robert Price | 01/2021 |
7937 | ((1+sqrt(3)+sqrt(2))^79357-1)*((1+sqrt(3)-sqrt(2))^79357-1)*((1-sqrt(3)+sqrt(2))^79357-1)*((1-sqrt(3 | 84849 | Mike Oakes | 04/2009 |
7938 | (2^281827+650)/2 | 84839 | Stefano Morozzi | 01/2020 |
7939 | 2^281825-1861483 | 84838 | Enzo Doria | 03/2016 |
7940 | 2^281809-97305 | 84833 | Lelio R Paula | 09/2013 |
7942 | 2^281780-114893 | 84825 | Enzo Doria | 05/2015 |
7941 | (7654^21841-1)/7653 | 84825 | Serge Batalov | 08/2020 |
7943 | 3^177719+64 | 84794 | Lelio R Paula | 12/2015 |
7945 | 2^281675-199 | 84793 | Lelio R Paula | 08/2012 |
7944 | 2^281673-111223 | 84793 | Enzo Doria | 05/2014 |
7946 | (4*10^84779+149)/9 | 84779 | Robert Price | 03/2017 |
7948 | 129^40167-128 | 84777 | Norbert Schneider | 01/2019 |
7947 | (2^281622+3970)/2 | 84777 | Stefano Morozzi | 01/2020 |
7949 | (72*10^84772-27)/9 | 84773 | Ray Chandler | 01/2011 |
7950 | 11^81398-630 | 84768 | Enzo Doria | 11/2012 |
7951 | (25*2^281577-1)/23 | 84764 | Lelio R Paula | 02/2014 |
7952 | 20320^14819+14819^20320 | 84752 | Hans Havermann | 07/2020 |
7953 | 2^281500-455 | 84740 | Yves Minet | 03/2012 |
7954 | 10^84725+1145141919 | 84726 | Marisa Kirisame | 03/2015 |
7955 | 10^84725+142041 | 84726 | Patrick De Geest | 10/2015 |
7956 | 10^84725-138677 | 84725 | Patrick De Geest | 02/2016 |
7957 | 76^45044+75 | 84720 | Norbert Schneider | 02/2015 |
7958 | 54^48897-53 | 84709 | Norbert Schneider | 08/2014 |
7959 | (9913*2^281377-1)/65 | 84706 | Lelio R Paula | 04/2015 |
7960 | Phi(171124,10)/29946701/7761329021/58560344041 | 84700 | Maksym Voznyy | 11/2015 |
7961 | 2^281337-111223 | 84691 | Enzo Doria | 05/2014 |
7962 | 2^281308+54615 | 84683 | Lelio R Paula | 09/2013 |
7963 | 7*5^121144-6 | 84677 | Lelio R Paula | 10/2014 |
7964 | (2^281270+3858)/2 | 84671 | Stefano Morozzi | 01/2020 |
7965 | (2^281252+47)/3 | 84665 | Lelio R Paula | 01/2013 |
7966 | 11^81297-3150 | 84663 | Enzo Doria | 12/2012 |
7967 | 2^281208-303 | 84653 | Yves Minet | 02/2012 |
7968 | 197^36892-2 | 84648 | Lelio R Paula | 10/2017 |
7969 | (21*2^281187+1)/2483 | 84644 | Lelio R Paula | 01/2013 |
7970 | 11^81247-84 | 84611 | Lelio R Paula | 10/2015 |
7971 | (10876^20959-1)/10875 | 84597 | Serge Batalov | 08/2020 |
7972 | 2067^25516+25516^2067 | 84595 | Norbert Schneider | 11/2019 |
7974 | 17^68747-64 | 84590 | Lelio R Paula | 11/2015 |
7973 | (2^281002+3718)/2 | 84590 | Stefano Morozzi | 01/2020 |
7975 | 5^121018-15596 | 84588 | Enzo Doria | 02/2013 |
7976 | 6^108702+7 | 84587 | Lelio R Paula | 11/2014 |
7977 | 10^84582+100677 | 84583 | Patrick De Geest | 05/2016 |
7978 | 10^84582-8001 | 84582 | Robert Price | 03/2016 |
7979 | 1828^25929+25929^1828 | 84580 | Norbert Schneider | 12/2019 |
7980 | (2^280968+1446)/2 | 84580 | Stefano Morozzi | 01/2020 |
7981 | 3*5^121004-8 | 84579 | Lelio R Paula | 10/2014 |
7982 | 5^120996-84 | 84573 | Lelio R Paula | 10/2015 |
7983 | (2^280896+1522)/2 | 84558 | Stefano Morozzi | 01/2020 |
7984 | (10223*2^280881+1)/68419 | 84553 | Lelio R Paula | 06/2014 |
7985 | 2^280852+97681 | 84545 | Yves Minet | 08/2004 |
7986 | 2^280848-1203 | 84544 | Lelio R Paula | 09/2013 |
7987 | 5^120946-24 | 84538 | Lelio R Paula | 10/2015 |
7988 | 2^280810-2111 | 84533 | Yves Minet | 08/2013 |
7989 | 11^81161-5160 | 84521 | Enzo Doria | 01/2013 |
7990 | 2^280737-93 | 84511 | Lelio R Paula | 08/2012 |
7991 | (2^280737+494)/2 | 84510 | Stefano Morozzi | 01/2020 |
7992 | (2^280717+1166)/2 | 84504 | Stefano Morozzi | 01/2020 |
7993 | 2^280698+315 | 84499 | Lelio R Paula | 09/2013 |
7994 | 3^177099-340 | 84498 | Lelio R Paula | 12/2015 |
7995 | (8201^21589-1)/8200 | 84493 | Serge Batalov | 08/2020 |
7996 | (5*10^84469-107)/3 | 84470 | Robert Price | 01/2017 |
7997 | (3^177002+7)/6696022272161744 | 84436 | Lelio R Paula | 12/2012 |
7998 | (2^280500-1017)/823 | 84436 | Lelio R Paula | 01/2013 |
7999 | 24677^2640+2640^24677 | 84435 | Hans Havermann | 07/2020 |
8000 | 14^73660-5 | 84424 | Lelio R Paula | 11/2015 |