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 ? |
8751 | 2^336350-344291 | 101252 | Enzo Doria | 07/2015 |
8752 | (17*10^101251-161)/9 | 101252 | Robert Price | 02/2021 |
8753 | (56*10^101222-47)/9 | 101223 | Robert Price | 02/2019 |
8754 | 2^336243-108777 | 101220 | Enzo Doria | 03/2014 |
8755 | (31*2^336189-1)/19 | 101204 | Lelio R Paula | 03/2014 |
8756 | 23535^19954+19954^23535 | 101202 | Hans Havermann | 11/2018 |
8757 | (78557*2^336178+1)/1268757 | 101199 | Lelio R Paula | 07/2014 |
8758 | 2^336150-20393 | 101192 | Enzo Doria | 06/2013 |
8759 | 23996^16473+16473^23996 | 101186 | Hans Havermann | 11/2018 |
8760 | 2^336112-1865249 | 101180 | Enzo Doria | 03/2016 |
8761 | 2^336103-109015 | 101178 | Enzo Doria | 04/2014 |
8762 | (2^336095+2078)/2 | 101175 | Stefano Morozzi | 09/2020 |
8763 | 2^336060+22201 | 101165 | Enzo Doria | 09/2023 |
8764 | 13^90809-96 | 101157 | Lelio R Paula | 10/2016 |
8765 | 26199^7258+7258^26199 | 101150 | Hans Havermann | 11/2018 |
8766 | 19^79098-30 | 101147 | Lelio R Paula | 10/2016 |
8767 | 19^79095+88 | 101144 | Lelio R Paula | 10/2016 |
8768 | (30826*34^66040-59)/11 | 101143 | Eric Chen | 03/2023 |
8769 | 12401^24702-24702^12401 | 101117 | Norbert Schneider | 07/2022 |
8770 | 15^85976+32 | 101116 | Lelio R Paula | 10/2016 |
8771 | 2^335872-2776215 | 101108 | Tony Prest / Never Odd Or Even | 01/2019 |
8772 | (3^211950+7)/28854513015920697088 | 101107 | Lelio R Paula | 11/2013 |
8773 | (4^167933+17)/3 | 101106 | Robert Price | 02/2017 |
8774 | 2^335858+153 | 101104 | Lelio R Paula | 10/2009 |
8775 | (2^335872-2776871)/5/5/479 | 101104 | Tony Prest / Never Odd Or Even | 01/2019 |
8776 | (2^335872-2758787)/23/31/20809 | 101101 | Tony Prest / Never Odd Or Even | 01/2019 |
8777 | (2^335839+2114)/2 | 101098 | Stefano Morozzi | 09/2020 |
8778 | (2^335872-2795083)/3/3/113/283/148367 | 101097 | Tony Prest / Never Odd Or Even | 01/2019 |
8780 | 6^129918-35 | 101096 | Lelio R Paula | 10/2016 |
8779 | (2*10^101096-107)/3 | 101096 | Robert Price | 11/2020 |
8781 | 3^211885-70 | 101095 | Lelio R Paula | 09/2016 |
8782 | (29*2^335835-1)/550599 | 101093 | Lelio R Paula | 03/2014 |
8783 | (2^335872-2794771)/3/5/5/61/232312440293 | 101093 | Tony Prest / Never Odd Or Even | 01/2019 |
8784 | 2^335818-20033 | 101092 | Enzo Doria | 06/2013 |
8785 | 2^335813-110083 | 101090 | Enzo Doria | 04/2014 |
8786 | 2^335804+2155 | 101088 | Lelio R Paula | 09/2016 |
8787 | (2^335872-2756679)/7/7/7/13/31/4987/437653/8734021 | 101087 | Tony Prest / Never Odd Or Even | 01/2019 |
8788 | 2^335784-16145 | 101082 | Enzo Doria | 04/2013 |
8789 | 11^97051+62 | 101069 | Lelio R Paula | 10/2016 |
8790 | 3*10^101065+11 | 101066 | Robert Price | 07/2018 |
8791 | 27820^4293+4293^27820 | 101064 | Hans Havermann | 10/2018 |
8792 | (2^335714+3750)/2 | 101060 | Stefano Morozzi | 01/2021 |
8793 | (2^335690+1038)/2 | 101053 | Stefano Morozzi | 09/2020 |
8794 | (2^335691-3)/26515 | 101049 | Lelio R Paula | 12/2013 |
8795 | 2^335658-94395 | 101044 | Lelio R Paula | 09/2016 |
8796 | 24514^13215+13215^24514 | 101024 | Hans Havermann | 11/2018 |
8797 | 2^335586+169 | 101022 | Lelio R Paula | 10/2009 |
8798 | (56^57787-1)/55 | 101021 | Paul Bourdelais | 08/2017 |
8799 | (2^335562+1258)/2 | 101014 | Stefano Morozzi | 09/2020 |
8800 | 2^335541+4849845 | 101008 | Paul Bourdelais | 04/2022 |
8801 | (38*10^100999+169)/9 | 101000 | Robert Price | 11/2021 |
8802 | (19*10^100997+737)/9 | 100998 | Robert Price | 07/2018 |
8803 | 47599!2+1024 | 100991 | Robert Price | 02/2015 |
8804 | 2^335468-18629 | 100986 | Enzo Doria | 05/2013 |
8805 | 2^335454-20961 | 100982 | Enzo Doria | 06/2013 |
8806 | (2^335454+3258)/2 | 100982 | Stefano Morozzi | 02/2021 |
8807 | 10^100978+61 | 100979 | Lelio R Paula | 09/2016 |
8808 | 10^100978-156437 | 100978 | Patrick De Geest | 11/2016 |
8809 | (2^335422+3114)/2 | 100972 | Stefano Morozzi | 09/2020 |
8810 | (35*10^100970-89)/9 | 100971 | Robert Price | 02/2022 |
8811 | (15*2^335417+1)/481 | 100970 | Lelio R Paula | 12/2013 |
8812 | 2^335403-1859167 | 100967 | Enzo Doria | 03/2016 |
8813 | 23599^18976+18976^23599 | 100962 | Hans Havermann | 10/2018 |
8814 | 6*10^100958+43 | 100959 | Robert Price | 07/2020 |
8815 | 9*2^335364-5 | 100956 | Lelio R Paula | 08/2014 |
8816 | (7998^25867-1)/7997 | 100955 | Serge Batalov | 06/2022 |
8817 | 2^335349+72287 | 100951 | Enzo Doria | 04/2016 |
8818 | 2^335329+16095 | 100945 | Lelio R Paula | 09/2016 |
8819 | 232823#+233663 | 100937 | Makoto Morimoto | 10/2017 |
8820 | (232823#+10141)/(10141*1674461) | 100927 | Makoto Morimoto | 06/2019 |
8821 | 2^335237-20893 | 100917 | Enzo Doria | 06/2013 |
8822 | 2^335222-19337 | 100912 | Enzo Doria | 05/2013 |
8823 | 2^335222-109985 | 100912 | Enzo Doria | 04/2014 |
8824 | 2^335182-17237 | 100900 | Enzo Doria | 05/2013 |
8825 | (2^335107+1634)/2 | 100877 | Stefano Morozzi | 09/2020 |
8826 | 2^335043-16737 | 100858 | Enzo Doria | 05/2013 |
8827 | (21181*2^335049+1)/1343811 | 100858 | Lelio R Paula | 07/2014 |
8828 | 2^335038-20633 | 100857 | Enzo Doria | 06/2013 |
8829 | 2^335031+1873373 | 100855 | Enzo Doria | 06/2016 |
8830 | 23489^19650+19650^23489 | 100847 | Hans Havermann | 10/2018 |
8831 | 3^211354-8816 | 100842 | Enzo Doria | 05/2014 |
8832 | 2^334984-112757 | 100841 | Enzo Doria | 06/2014 |
8833 | (746*10^100827+1)/9 | 100829 | Robert Price | 03/2020 |
8834 | 29451^2648+2648^29451 | 100809 | Hans Havermann | 09/2018 |
8835 | 25583^8716+8716^25583 | 100806 | Hans Havermann | 08/2018 |
8837 | 2^334857+48527 | 100803 | Lelio R Paula | 08/2011 |
8836 | 12^93406+29 | 100803 | Lelio R Paula | 10/2016 |
8838 | 28780^3179+3179^28780 | 100796 | Hans Havermann | 09/2018 |
8839 | (5*2^334825+1)/47 | 100792 | Lelio R Paula | 12/2013 |
8840 | (328^40063-1)/327 | 100791 | Serge Batalov | 06/2022 |
8841 | 2^334809-13 | 100788 | Lelio R Paula | 04/2011 |
8842 | (3552^28387-1)/3551 | 100784 | Serge Batalov | 06/2022 |
8843 | (82*10^100780-1)/9 | 100781 | Robert Price | 12/2014 |
8844 | 31395^1622+1622^31395 | 100780 | Hans Havermann | 09/2018 |
8845 | (2^334779+47)/5 | 100778 | Lelio R Paula | 12/2013 |
8846 | 2^334751+2049 | 100771 | Lelio R Paula | 09/2016 |
8847 | 13^90443-98 | 100749 | Lelio R Paula | 10/2016 |
8848 | (2^334674+3190)/2 | 100747 | Stefano Morozzi | 09/2020 |
8849 | (2^334651+1050)/2 | 100740 | Stefano Morozzi | 09/2020 |
8850 | (7*10^100737+71)/3 | 100738 | Robert Price | 05/2018 |
8851 | (24737*2^334628+1)/3 | 100737 | Lelio R Paula | 07/2014 |
8852 | 15^85639-28 | 100720 | Lelio R Paula | 10/2016 |
8853 | 27068^5259+5259^27068 | 100718 | Hans Havermann | 08/2018 |
8854 | 2^334562+577294575 | 100711 | Lelio R Paula | 09/2016 |
8855 | 4*3^211077-5 | 100710 | Lelio R Paula | 08/2014 |
8856 | 2^334524+145 | 100702 | Lelio R Paula | 07/2009 |
8857 | (2^334451+714)/2 | 100680 | Stefano Morozzi | 09/2020 |
8858 | 27648^4375+4375^27648 | 100666 | Hans Havermann | 09/2018 |
8859 | (1017*2^334391-1)/3535 | 100662 | Lelio R Paula | 05/2015 |
8860 | (2^334309+3122)/2 | 100637 | Stefano Morozzi | 09/2020 |
8861 | 2^334294-113111 | 100633 | Enzo Doria | 07/2014 |
8862 | 25581^8582+8582^25581 | 100626 | Hans Havermann | 08/2018 |
8864 | 6^129308-17723 | 100622 | Enzo Doria | 03/2013 |
8863 | (19*10^100627+287)/9/292157 | 100622 | Alfred Reich | 11/2022 |
8865 | 6^129305+95 | 100619 | Lelio R Paula | 10/2016 |
8866 | 2^334244-1167 | 100618 | Lelio R Paula | 09/2016 |
8867 | (2^334233+3066)/2 | 100614 | Stefano Morozzi | 09/2020 |
8868 | (76*10^100611+77)/9 | 100612 | Robert Price | 10/2019 |
8869 | 7*10^100606-93 | 100607 | Robert Price | 08/2020 |
8870 | (26*10^100566+49)/3 | 100567 | Robert Price | 08/2019 |
8871 | (21*2^334050+1)/935 | 100558 | Lelio R Paula | 12/2013 |
8872 | 24885^10984+10984^24885 | 100555 | Hans Havermann | 07/2018 |
8873 | (1351^32119-1)/1350 | 100551 | Serge Batalov | 06/2022 |
8874 | (78557*2^334018+1)/4326021 | 100548 | Lelio R Paula | 07/2014 |
8875 | (35*10^100542+1)/9 | 100543 | Serge Batalov | 11/2010 |
8876 | 23578^18367+18367^23578 | 100538 | Hans Havermann | 07/2018 |
8877 | 13^90251+24 | 100535 | Lelio R Paula | 10/2016 |
8878 | 2^333963+789 | 100533 | Lelio R Paula | 08/2011 |
8879 | 2^333919-1840087 | 100520 | Enzo Doria | 06/2015 |
8880 | (2^333916+1698)/2 | 100519 | Stefano Morozzi | 09/2020 |
8881 | 2^333911+72623 | 100518 | Enzo Doria | 04/2016 |
8882 | 24605^12168+12168^24605 | 100517 | Hans Havermann | 08/2018 |
8883 | 66*10^100502+7 | 100504 | Robert Price | 07/2020 |
8884 | 18^80064-67 | 100503 | Lelio R Paula | 10/2016 |
8885 | (13*10^100487+137)/3 | 100488 | Robert Price | 06/2021 |
8886 | 5^143742-48 | 100472 | Lelio R Paula | 10/2016 |
8887 | 24095^14784+14784^24095 | 100472 | Hans Havermann | 08/2018 |
8888 | 2^333757-112123 | 100471 | Enzo Doria | 06/2014 |
8889 | (7953^25759-1)/7952 | 100470 | Serge Batalov | 06/2022 |
8890 | 7*10^100457-57 | 100458 | Robert Price | 06/2020 |
8891 | 27358^4693+4693^27358 | 100444 | Hans Havermann | 07/2018 |
8892 | 8*10^100439-51 | 100440 | Robert Price | 08/2019 |
8893 | 2^333603-108861 | 100425 | Enzo Doria | 03/2014 |
8894 | 9*7^118818-2 | 100414 | Lelio R Paula | 11/2014 |
8895 | 2^333546-17097 | 100408 | Enzo Doria | 05/2013 |
8896 | (19*10^100403+287)/9/3 | 100403 | Alfred Reich | 11/2022 |
8897 | 17^81597+72 | 100401 | Lelio R Paula | 10/2016 |
8898 | (3613^28219-1)/3612 | 100396 | Serge Batalov | 06/2022 |
8899 | p(20559)#/2+8 | 100384 | Robert Price | 01/2017 |
8900 | 11^96390-24 | 100380 | Lelio R Paula | 10/2016 |
8901 | (2^333385+2730)/2 | 100359 | Stefano Morozzi | 09/2020 |
8902 | 4832563059*2^333333+30029 | 100353 | Donovan Johnson | 01/2008 |
8903 | 23565^18136+18136^23565 | 100353 | Hans Havermann | 07/2018 |
8904 | 2^333354+763 | 100350 | Lelio R Paula | 08/2011 |
8905 | (1608961*2^333333+1)/3 | 100349 | Alex_soldier | 09/2022 |
8906 | (424517*2^333333+1)/5 | 100349 | Alex_soldier | 09/2022 |
8907 | (887087*2^333333+1)/5 | 100349 | Alex_soldier | 09/2022 |
8908 | (1159703*2^333333+1)/7 | 100349 | Alex_soldier | 09/2022 |
8909 | (25*10^100343-67)/3 | 100344 | Robert Price | 06/2019 |
8910 | 2^333289-109483 | 100330 | Enzo Doria | 04/2014 |
8911 | 2^333240-2790209 | 100316 | Enzo Doria | 03/2016 |
8912 | 23601^17798+17798^23601 | 100314 | Hans Havermann | 05/2018 |
8914 | (2^333227-2^166614+1)/5 | 100311 | | 11/2006 |
8913 | 2^333225+719 | 100311 | Lelio R Paula | 08/2011 |
8915 | 2^333221-2785333 | 100310 | Enzo Doria | 10/2015 |
8916 | 76^53332-75 | 100308 | Norbert Schneider | 02/2015 |
8917 | (29*10^100302+583)/9 | 100303 | Robert Price | 06/2021 |
8918 | 6^128892-17573 | 100298 | Enzo Doria | 03/2013 |
8919 | (47*10^100283-119)/9 | 100284 | Robert Price | 05/2019 |
8920 | 2^333105-113215 | 100275 | Enzo Doria | 07/2014 |
8921 | (31*2^333102+1)/2375 | 100272 | Lelio R Paula | 11/2013 |
8922 | 4*10^100257-83 | 100258 | Robert Price | 10/2018 |
8923 | 17^81476-74 | 100253 | Lelio R Paula | 10/2016 |
8924 | 23392^19207+19207^23392 | 100199 | Hans Havermann | 06/2018 |
8925 | (2^332830+750)/2 | 100192 | Stefano Morozzi | 09/2020 |
8926 | (206*10^100189-17)/9 | 100191 | Robert Price | 01/2020 |
8927 | 17^81417-16 | 100180 | Theodore Burton | 11/2005 |
8928 | (2^332773+590)/2 | 100175 | Stefano Morozzi | 09/2020 |
8929 | 2*23^73562+215 | 100172 | Eric Chen | 08/2023 |
8930 | 2^332739-113215 | 100165 | Enzo Doria | 07/2014 |
8931 | 2^332730-1865621 | 100162 | Enzo Doria | 03/2016 |
8932 | 91^51127-2 | 100160 | Lelio R Paula | 09/2013 |
8933 | 15^85159-32 | 100155 | Lelio R Paula | 10/2016 |
8934 | 6^128661+97 | 100118 | Lelio R Paula | 10/2016 |
8935 | 19^78285+40 | 100108 | Lelio R Paula | 07/2016 |
8936 | 6^128640+5 | 100102 | Lelio R Paula | 11/2014 |
8937 | 10^100095+89973 | 100096 | Patrick De Geest | 10/2022 |
8938 | 10^100095-400001 | 100095 | Robert Price | 09/2021 |
8939 | 10^100095-203693 | 100095 | Patrick De Geest | 10/2022 |
8940 | 22779^22970-22970^22779 | 100093 | Norbert Schneider | 08/2020 |
8941 | (2^332468+3130)/2 | 100083 | Stefano Morozzi | 09/2020 |
8942 | (14*10^100075+61)/3 | 100076 | Robert Price | 12/2018 |
8943 | (2^332435+3098)/2 | 100073 | Stefano Morozzi | 09/2020 |
8944 | 28562^3189+3189^28562 | 100072 | Hans Havermann | 05/2018 |
8945 | 6^128565-127 | 100044 | Lelio R Paula | 08/2016 |
8946 | (2^332317+2106)/2 | 100038 | Stefano Morozzi | 09/2020 |
8948 | 2^332309-133 | 100035 | Lelio R Paula | 02/2011 |
8949 | 2^332307+731 | 100035 | Lelio R Paula | 07/2011 |
8947 | 937^33662-33662^937 | 100035 | Norbert Schneider | 05/2023 |
8950 | 2^332301+12927 | 100033 | Lelio R Paula | 04/2013 |
8951 | 23406^18781+18781^23406 | 100031 | Hans Havermann | 04/2018 |
8952 | 2^332277-4573 | 100026 | Yves Minet | 03/2019 |
8953 | 12^92684-13 | 100023 | Lelio R Paula | 07/2016 |
8954 | 70^54207-54207^70 | 100018 | Norbert Schneider | 04/2023 |
8955 | 78557*2^332193+55669 | 100005 | Jorge Menendez | 04/2019 |
8956 | 78557*2^332192+72689 | 100005 | Jorge Menendez | 04/2019 |
8957 | 10^100003+45799 | 100004 | Stefano Morozzi | 04/2019 |
8958 | 10^100003+155461 | 100004 | Stefano Morozzi | 04/2019 |
8959 | 10^100003+211389 | 100004 | Stefano Morozzi | 04/2019 |
8960 | 10^100003+381027 | 100004 | Stefano Morozzi | 04/2019 |
8961 | 10^100003+540703 | 100004 | Stefano Morozzi | 04/2019 |
8962 | 10^100003+823977 | 100004 | Stefano Morozzi | 04/2019 |
8974 | 10^100002+236023 | 100003 | Stefano Morozzi | 05/2018 |
8975 | 10^100002+433741 | 100003 | Stefano Morozzi | 05/2018 |
8963 | 2^332202+104659 | 100003 | Stefano Morozzi | 04/2019 |
8964 | 2^332202+857539 | 100003 | Stefano Morozzi | 04/2019 |
8965 | 2^332202+181773 | 100003 | Stefano Morozzi | 04/2019 |
8966 | 2^332202+687543 | 100003 | Stefano Morozzi | 04/2019 |
8967 | 2^332202+941595 | 100003 | Stefano Morozzi | 04/2019 |
8968 | 2^332202+445813 | 100003 | Stefano Morozzi | 04/2019 |
8969 | 2^332202+1599727 | 100003 | Stefano Morozzi | 04/2019 |
8970 | 2^332202+1389567 | 100003 | Stefano Morozzi | 04/2019 |
8971 | 2^332202+1206315 | 100003 | Stefano Morozzi | 04/2019 |
8972 | 2^332202+1958647 | 100003 | Stefano Morozzi | 04/2019 |
8973 | 10^100003-562857 | 100003 | Patrick De Geest | 07/2019 |
8976 | (10^100002-1)/9-10^33441 | 100002 | Akihiko Yoshida | 03/2018 |
8977 | (10^100002-1)/9-10^44692 | 100002 | Akihiko Yoshida | 03/2018 |
8978 | 10^100002-249519 | 100002 | Patrick De Geest | 07/2018 |
8979 | 10^100001+314199 | 100002 | Patrick De Geest | 07/2018 |
8999 | 10^100000+3938473 | 100001 | Robert Price | 05/2015 |
9000 | 10^100000+3979761 | 100001 | Robert Price | 05/2015 |
8996 | 10^100000+4858267 | 100001 | Robert Price | 12/2015 |
8997 | 10^100000+5113299 | 100001 | Robert Price | 02/2016 |
8998 | 10^100000+5169411 | 100001 | Robert Price | 02/2016 |
8988 | 10^100000+5354043 | 100001 | Robert Price | 05/2016 |
8989 | 10^100000+5717979 | 100001 | Robert Price | 05/2016 |
8990 | 10^100000+6080631 | 100001 | Robert Price | 07/2016 |
8991 | 10^100000+6308469 | 100001 | Robert Price | 07/2016 |
8992 | 10^100000+6512023 | 100001 | Robert Price | 08/2016 |
8993 | 10^100000+6596557 | 100001 | Robert Price | 08/2016 |
8994 | 10^100000+7115409 | 100001 | Robert Price | 10/2016 |
8995 | 10^100000+7235803 | 100001 | Robert Price | 10/2016 |
8980 | 10^100000+11187931 | 100001 | Robert Price | 03/2018 |
8981 | 10^100000+11264721 | 100001 | Robert Price | 04/2018 |
8982 | 10^100000+11477661 | 100001 | Robert Price | 05/2018 |
8983 | 10^100000+11644653 | 100001 | Robert Price | 07/2018 |
8984 | 10^100001-431439 | 100001 | Patrick De Geest | 07/2018 |
8985 | 10^100000+11769399 | 100001 | Robert Price | 08/2018 |
8986 | 10^100000+11848159 | 100001 | Robert Price | 08/2018 |
8987 | 10^100000+11863467 | 100001 | Robert Price | 08/2018 |