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 ? |
5751 | 655^37342+37342^655 | 105165 | Norbert Schneider | 08/2018 |
5752 | 13^94403+94 | 105160 | Lelio R Paula | 12/2016 |
5753 | 2^349301-112309 | 105151 | Enzo Doria | 06/2014 |
5754 | 30291^2960+2960^30291 | 105149 | Hans Havermann | 09/2020 |
5755 | 27350^6981+6981^27350 | 105132 | Hans Havermann | 09/2020 |
5757 | 9*2^349220+5 | 105127 | Lelio R Paula | 09/2014 |
5756 | (2^349225+3122)/2 | 105127 | Stefano Morozzi | 09/2020 |
5758 | 7*3^220317+8 | 105119 | Lelio R Paula | 09/2014 |
5759 | 7^124386+88 | 105119 | Lelio R Paula | 11/2016 |
5760 | (2^349197+570)/2 | 105119 | Stefano Morozzi | 09/2020 |
5761 | 2^349171-19657 | 105111 | Enzo Doria | 06/2013 |
5762 | (2^349115+590)/2 | 105094 | Stefano Morozzi | 09/2020 |
5763 | (2^(6*58185+1)-5)/3 | 105093 | Kevin Acres | 11/2013 |
5764 | (2^349111-5)/3 | 105093 | Lelio R Paula | 12/2013 |
5765 | 675^37142+37142^675 | 105087 | Norbert Schneider | 08/2018 |
5766 | 2^348981-2797573 | 105054 | Enzo Doria | 04/2016 |
5767 | 2^348869-6555 | 105021 | Lelio R Paula | 01/2017 |
5768 | 5^150217+78 | 104998 | Lelio R Paula | 11/2016 |
5769 | (88*10^104989-1)/3 | 104991 | Robert Price | 04/2020 |
5770 | (28*10^104972+149)/3 | 104973 | Robert Price | 12/2019 |
5771 | 24927^16262+16262^24927 | 104972 | Hans Havermann | 10/2020 |
5772 | 2^348703-113005 | 104971 | Enzo Doria | 06/2014 |
5773 | (191*10^104956-17)/3 | 104958 | Robert Price | 11/2021 |
5775 | 2^348612-809 | 104943 | Yves Minet | 05/2013 |
5774 | (73*10^104942-1)/9 | 104943 | Robert Price | 01/2015 |
5776 | 2^348583-350407 | 104934 | Enzo Doria | 06/2015 |
5777 | 2^348531-949 | 104919 | Yves Minet | 05/2012 |
5778 | 20^80636-59 | 104910 | Lelio R Paula | 12/2016 |
5779 | 8*10^104893-13 | 104894 | Robert Price | 08/2019 |
5781 | 2^348437+35 | 104890 | Lelio R Paula | 09/2011 |
5780 | 25208^14487+14487^25208 | 104890 | Hans Havermann | 09/2020 |
5782 | 2^348348+135 | 104864 | Lelio R Paula | 09/2011 |
5783 | (29*10^104858+349)/9 | 104859 | Robert Price | 06/2021 |
5784 | 6*10^104852-31 | 104853 | Robert Price | 07/2020 |
5786 | (2^348284+9)/25 | 104843 | Lelio R Paula | 12/2013 |
5785 | 75^55914-74 | 104843 | Norbert Schneider | 01/2015 |
5787 | 2^348268+255255 | 104840 | Lelio R Paula | 01/2017 |
5788 | 35540^891+891^35540 | 104839 | Hans Havermann | 09/2020 |
5789 | 5^149985-52 | 104836 | Lelio R Paula | 11/2016 |
5790 | 2^348224-20249 | 104826 | Enzo Doria | 06/2013 |
5791 | (2^348217+2522)/2 | 104824 | Stefano Morozzi | 09/2020 |
5792 | 2^348212-20105 | 104823 | Enzo Doria | 06/2013 |
5793 | 24647^17898+17898^24647 | 104819 | Hans Havermann | 10/2020 |
5794 | 20^80562+19 | 104814 | Lelio R Paula | 12/2016 |
5795 | 2*10^104811-87 | 104812 | Robert Price | 04/2018 |
5796 | 2^348149-20053 | 104804 | Enzo Doria | 06/2013 |
5797 | 2^348095-171 | 104788 | Lelio R Paula | 09/2011 |
5798 | 25383^13412+13412^25383 | 104769 | Hans Havermann | 09/2020 |
5799 | 2^348003-20377 | 104760 | Enzo Doria | 06/2013 |
5800 | 2^347994-741 | 104757 | Lelio R Paula | 12/2016 |
5801 | (2^347993+2234)/2 | 104757 | Stefano Morozzi | 01/2021 |
5802 | (2^347990+790)/2 | 104756 | Stefano Morozzi | 09/2020 |
5803 | 25137^14702+14702^25137 | 104756 | Hans Havermann | 09/2020 |
5804 | (13*2^347972-1)/282309 | 104746 | Lelio R Paula | 03/2014 |
5805 | 12883^25484-25484^12883 | 104740 | Norbert Schneider | 08/2022 |
5806 | 2^347910-109937 | 104732 | Enzo Doria | 04/2014 |
5807 | (88*10^104709-637)/9 | 104710 | Robert Price | 11/2020 |
5808 | (13*10^104702+71)/3 | 104703 | Robert Price | 06/2021 |
5809 | 5^149794+28 | 104702 | Lelio R Paula | 11/2016 |
5810 | 105^51800-2 | 104698 | Lelio R Paula | 10/2017 |
5811 | (16^86939+3^86939)/19 | 104684 | Robert Price | 06/2013 |
5812 | (15^89009-1)/14 | 104682 | Paul Bourdelais | 03/2010 |
5813 | 2^347731-18609 | 104678 | Enzo Doria | 05/2013 |
5814 | 4*6^134519+7 | 104677 | Lelio R Paula | 11/2014 |
5815 | (34*10^104672-259)/9 | 104673 | Robert Price | 12/2021 |
5816 | 25321^13588+13588^25321 | 104656 | Hans Havermann | 09/2020 |
5817 | 2^347612-18545 | 104642 | Enzo Doria | 05/2013 |
5820 | 6^134465+95 | 104635 | Lelio R Paula | 11/2016 |
5818 | (2^347588+3718)/2 | 104635 | Stefano Morozzi | 09/2020 |
5819 | (166*10^104633-7)/3 | 104635 | Robert Price | 11/2020 |
5821 | 2^347517-17301 | 104614 | Enzo Doria | 05/2013 |
5822 | (2^347493+3186)/2 | 104606 | Stefano Morozzi | 09/2020 |
5823 | (2^347490+2574)/2 | 104605 | Stefano Morozzi | 09/2020 |
5824 | 3^219222-310 | 104596 | Lelio R Paula | 11/2016 |
5825 | 25993^10554+10554^25993 | 104581 | Hans Havermann | 09/2020 |
5826 | (15*2^347399-1)/13 | 104578 | Lelio R Paula | 03/2014 |
5827 | 2^347372-17853 | 104570 | Enzo Doria | 05/2013 |
5828 | 2^347314+99 | 104552 | Lelio R Paula | 09/2011 |
5829 | 2^347300-22233 | 104548 | Enzo Doria | 06/2013 |
5830 | 2^347287-19845 | 104544 | Enzo Doria | 06/2013 |
5831 | 2^347258-231191 | 104536 | Harrison Mesh | 06/2022 |
5832 | (31*2^347254-1)/891 | 104533 | Lelio R Paula | 03/2014 |
5833 | 5*F(250091)*(F(250091)-1)+1 | 104532 | Serge Batalov | 04/2015 |
5834 | 7*2^347239-3 | 104531 | Lelio R Paula | 09/2014 |
5835 | (3^219098-8)/514147439969 | 104525 | Lelio R Paula | 12/2013 |
5836 | 5*3^219047+8 | 104513 | Lelio R Paula | 09/2014 |
5837 | 2^347148-109073 | 104502 | Enzo Doria | 04/2014 |
5838 | 2^347129-20145 | 104497 | Enzo Doria | 06/2013 |
5839 | 4*10^104461-57 | 104462 | Robert Price | 10/2018 |
5840 | (2^346989+590)/2 | 104454 | Stefano Morozzi | 09/2020 |
5841 | 2^346923-19585 | 104435 | Enzo Doria | 05/2013 |
5842 | (11*2^346864-1)/175 | 104416 | Lelio R Paula | 03/2014 |
5843 | 2^346854+346854^2+1 | 104414 | Robert Price | 06/2014 |
5844 | 2^346834+11715 | 104408 | Lelio R Paula | 04/2015 |
5845 | 24276^19961+19961^24276 | 104392 | Hans Havermann | 09/2020 |
5846 | 79^55009+78 | 104387 | Norbert Schneider | 02/2015 |
5847 | 3^218744+58 | 104368 | Lelio R Paula | 11/2016 |
5848 | 9*10^104361+89 | 104362 | Robert Price | 11/2019 |
5849 | 25048^14665+14665^25048 | 104358 | Hans Havermann | 09/2020 |
5850 | 2^346649+65 | 104352 | Lelio R Paula | 09/2011 |
5851 | 2^346638-17265 | 104349 | Enzo Doria | 05/2013 |
5852 | (2^346606+2082)/2 | 104339 | Stefano Morozzi | 09/2020 |
5853 | (2^346607+2070)/2 | 104339 | Stefano Morozzi | 09/2020 |
5854 | (2^346582+3202)/2 | 104332 | Stefano Morozzi | 09/2020 |
5855 | 26092^9967+9967^26092 | 104331 | Hans Havermann | 09/2020 |
5856 | 5*10^104326+77 | 104327 | Robert Price | 12/2018 |
5857 | (2^346564+2026)/2 | 104326 | Stefano Morozzi | 09/2020 |
5859 | 2^346527-111817 | 104316 | Enzo Doria | 05/2014 |
5858 | 6^134055+101 | 104316 | Lelio R Paula | 11/2016 |
5860 | 92105!4+8 | 104312 | Robert Price | 09/2019 |
5861 | 6^134013+163 | 104283 | Lelio R Paula | 11/2016 |
5862 | (22*10^104272-61)/3 | 104273 | Robert Price | 03/2022 |
5863 | (2^346318+862)/2 | 104252 | Stefano Morozzi | 02/2021 |
5865 | 18^83049-89 | 104250 | Lelio R Paula | 01/2017 |
5864 | (2^346311+606)/2 | 104250 | Stefano Morozzi | 09/2020 |
5866 | (166*10^104230+11)/3 | 104232 | Robert Price | 03/2021 |
5867 | 2^346213+484049 | 104221 | Enzo Doria | 04/2017 |
5868 | 18^83021-91 | 104214 | Lelio R Paula | 01/2017 |
5869 | 10^104212+128779 | 104213 | Patrick De Geest | 05/2016 |
5870 | 10^104212+237447 | 104213 | Patrick De Geest | 05/2016 |
5871 | 10^104212+242139 | 104213 | Patrick De Geest | 05/2016 |
5872 | 10^104212+269199 | 104213 | Patrick De Geest | 05/2016 |
5873 | 24254^19803+19803^24254 | 104213 | Hans Havermann | 09/2020 |
5874 | (10^104212-71)/1 | 104212 | Serge Batalov | 12/2014 |
5875 | 23678^23821-23821^23678 | 104202 | Norbert Schneider | 01/2020 |
5876 | 2^346078+7695 | 104180 | Lelio R Paula | 12/2016 |
5877 | 2^346055-19537 | 104173 | Enzo Doria | 05/2013 |
5878 | (3*2^346057+1)/77 | 104173 | Lelio R Paula | 12/2013 |
5879 | 2^346037-20125 | 104168 | Enzo Doria | 06/2013 |
5880 | (34*10^104168-7)/9/13/551179 | 104162 | Makoto Morimoto | 05/2016 |
5881 | 6^133854-17993 | 104159 | Enzo Doria | 03/2013 |
5882 | 2^345990+1874239 | 104154 | Enzo Doria | 06/2016 |
5883 | 2^345988-20373 | 104153 | Enzo Doria | 06/2013 |
5884 | (2^345965+2214)/2 | 104146 | Stefano Morozzi | 09/2020 |
5885 | (26*10^104132-503)/9 | 104133 | Robert Price | 07/2018 |
5886 | 3*5^148968+8 | 104125 | Lelio R Paula | 11/2014 |
5887 | (7*10^104113+53)/3 | 104114 | Robert Price | 05/2018 |
5888 | p(21253)#/2+8 | 104113 | Robert Price | 01/2017 |
5889 | 33582^1259+1259^33582 | 104106 | Hans Havermann | 09/2020 |
5890 | 32958^1441+1441^32958 | 104104 | Hans Havermann | 09/2020 |
5891 | 29127^3748+3748^29127 | 104095 | Hans Havermann | 09/2020 |
5892 | 322^41507+41507^322 | 104094 | Norbert Schneider | 12/2014 |
5893 | 6^133768-17423 | 104092 | Enzo Doria | 03/2013 |
5894 | 2^345786-19557 | 104092 | Enzo Doria | 05/2013 |
5895 | 171^46606-2 | 104072 | Lelio R Paula | 10/2017 |
5896 | (29*10^104071+91)/3 | 104072 | Robert Price | 04/2020 |
5897 | norm(Mod(x,x^3-x-1)^852167-1) | 104070 | Paul Vanderveen | 09/2022 |
5898 | 2^345704-149 | 104068 | Lelio R Paula | 09/2011 |
5899 | 25821^10712+10712^25821 | 104056 | Hans Havermann | 09/2020 |
5900 | (2^345662+2038)/2 | 104055 | Stefano Morozzi | 09/2020 |
5901 | 6^133704-17273 | 104042 | Enzo Doria | 02/2013 |
5902 | (3*2^345590-1)/23 | 104033 | Lelio R Paula | 03/2014 |
5903 | (19*2^345590+1)/33859 | 104030 | Lelio R Paula | 12/2013 |
5904 | 2^345547+11 | 104021 | Lelio R Paula | 09/2011 |
5905 | (19*10^104014-49)/3 | 104015 | Robert Price | 05/2019 |
5906 | (2^345519+1794)/2 | 104012 | Stefano Morozzi | 01/2021 |
5907 | 2^345504+723 | 104008 | Lelio R Paula | 09/2011 |
5908 | 2^345498-20037 | 104006 | Enzo Doria | 06/2013 |
5910 | 18^82854-37 | 104005 | Lelio R Paula | 01/2017 |
5909 | (2^345497+2070)/2 | 104005 | Stefano Morozzi | 09/2020 |
5911 | 2^345486-109065 | 104002 | Enzo Doria | 04/2014 |
5912 | 2^345487-1851507 | 104002 | Enzo Doria | 08/2015 |
5913 | 2^345476+777 | 103999 | Lelio R Paula | 09/2011 |
5914 | (2^345477+1554)/2 | 103999 | Stefano Morozzi | 09/2020 |
5915 | 4*10^103981-83 | 103982 | Robert Price | 10/2018 |
5916 | (2^345327+3206)/2 | 103954 | Stefano Morozzi | 09/2020 |
5917 | (5*2^345321-1)/3 | 103953 | Lelio R Paula | 03/2014 |
5918 | (2^345267-9)/539049127 | 103927 | Lelio R Paula | 12/2013 |
5919 | (19*10^103907+77)/3 | 103908 | Robert Price | 05/2019 |
5920 | (78557*2^345163+1)/78589 | 103905 | Lelio R Paula | 07/2014 |
5921 | 2^345106-112181 | 103888 | Enzo Doria | 06/2014 |
5922 | (2^345105+1326)/2 | 103887 | Stefano Morozzi | 02/2021 |
5923 | 2^345079+99 | 103880 | Lelio R Paula | 09/2011 |
5924 | (2^345034+1198)/2 | 103866 | Stefano Morozzi | 09/2020 |
5925 | 11^99736-1500 | 103865 | Enzo Doria | 12/2012 |
5926 | (569^37699-1)/568 | 103863 | Serge Batalov | 06/2022 |
5927 | (2^345010+2082)/2 | 103859 | Stefano Morozzi | 09/2020 |
5928 | 23263^23784-23784^23263 | 103857 | Norbert Schneider | 01/2020 |
5929 | 3*10^103844+31 | 103845 | Robert Price | 06/2021 |
5930 | 23927^21814+21814^23927 | 103813 | Hans Havermann | 09/2020 |
5931 | 2^344842-20157 | 103808 | Enzo Doria | 06/2013 |
5932 | (47*2^344822+1)/1071 | 103801 | Lelio R Paula | 05/2015 |
5933 | 2^344802-20765 | 103796 | Enzo Doria | 06/2013 |
5934 | 2^344794-3821 | 103794 | Yves Minet | 05/2011 |
5935 | 2^344782-109103 | 103790 | Enzo Doria | 04/2014 |
5936 | 3^217521-112292 | 103784 | Enzo Doria | 06/2014 |
5937 | 2^344759-3505 | 103783 | Lelio R Paula | 11/2016 |
5938 | 2^344735-147 | 103776 | Lelio R Paula | 09/2011 |
5939 | L(496533)-p(496533) | 103770 | Robert Price | 02/2018 |
5940 | 2*10^103749-69 | 103750 | Robert Price | 03/2018 |
5941 | (55459*2^344617+1)/3 | 103745 | Lelio R Paula | 07/2014 |
5942 | 2^344607-108891 | 103738 | Enzo Doria | 03/2014 |
5944 | (2^344608+3798)/2 | 103738 | Stefano Morozzi | 02/2021 |
5943 | (1097^34123-1)/1096 | 103738 | Serge Batalov | 06/2022 |
5945 | (148*10^103718-1)/3 | 103720 | Robert Price | 03/2020 |
5946 | 2^344541-19561 | 103718 | Enzo Doria | 05/2013 |
5947 | (1017*2^344529-1)/569 | 103714 | Lelio R Paula | 05/2015 |
5948 | (8896^26263-1)/8895 | 103714 | Serge Batalov | 06/2022 |
5949 | (2^344503+1758)/2 | 103706 | Stefano Morozzi | 09/2020 |
5950 | (4*10^103703-1)/3 | 103704 | Serge Batalov | 08/2009 |
5951 | 2^344444-455 | 103688 | Yves Minet | 03/2012 |
5952 | (4*10^103678-43)/3 | 103679 | Robert Price | 01/2022 |
5953 | (13*53^238539*617^9)/10^307661 | 103673 | Hans Havermann | 06/2017 |
5954 | (2^344367+2738)/2 | 103665 | Stefano Morozzi | 01/2021 |
5956 | 2^344317-109933 | 103650 | Enzo Doria | 04/2014 |
5955 | 26055^9508+9508^26055 | 103650 | Hans Havermann | 09/2020 |
5957 | 2^344309-1852033 | 103648 | Enzo Doria | 08/2015 |
5958 | 25729^10672+10672^25729 | 103643 | Hans Havermann | 09/2020 |
5959 | (2^344289+1166)/2 | 103642 | Stefano Morozzi | 09/2020 |
5960 | 3*10^103637+61 | 103638 | Robert Price | 09/2020 |
5961 | (2^344246+1614)/2 | 103629 | Stefano Morozzi | 09/2020 |
5962 | 2^344213-21253 | 103619 | Enzo Doria | 06/2013 |
5963 | 2^344196-465 | 103614 | Lelio R Paula | 11/2016 |
5964 | 2^344187-20001 | 103611 | Enzo Doria | 06/2013 |
5965 | (117191#+(355573#)/(117191#))/(200233679) | 103602 | Makoto Morimoto | 10/2017 |
5966 | (237^43627-1)/236 | 103601 | Serge Batalov | 06/2022 |
5967 | 20^79621-19^79621 | 103590 | Jean-Louis Charton | 03/2011 |
5968 | Phi(155391,10)/1243129 | 103586 | Maksym Voznyy | 09/2015 |
5969 | 2^344085-20365 | 103580 | Enzo Doria | 06/2013 |
5970 | 5^148169+54 | 103566 | Lelio R Paula | 11/2016 |
5971 | 2^344017-108795 | 103560 | Enzo Doria | 03/2014 |
5972 | (2^343974+2670)/2 | 103547 | Stefano Morozzi | 09/2020 |
5973 | 2^343971-5077 | 103546 | Lelio R Paula | 09/2011 |
5974 | (7*10^103531+71)/3 | 103532 | Robert Price | 05/2018 |
5975 | 2^343899+141 | 103524 | Lelio R Paula | 08/2011 |
5976 | (2^343933+1)/(3*98294339989393) | 103520 | Gord Palameta | 08/2018 |
5977 | 6^133029-67 | 103517 | Lelio R Paula | 11/2016 |
5978 | (17*2^343877+1)/13625 | 103515 | Lelio R Paula | 12/2013 |
5979 | 66*10^103507+7 | 103509 | Robert Price | 07/2020 |
5980 | 26297^8614+8614^26297 | 103485 | Hans Havermann | 09/2020 |
5981 | (2^343764+2658)/2 | 103483 | Stefano Morozzi | 09/2020 |
5982 | 26499^8036+8036^26499 | 103480 | Hans Havermann | 09/2020 |
5983 | (136*10^103474-1)/9 | 103476 | Serge Batalov | 12/2014 |
5984 | 5^148014-24 | 103458 | Lelio R Paula | 11/2016 |
5985 | 28316^4503+4503^28316 | 103453 | Hans Havermann | 09/2020 |
5986 | (22*10^103450+53)/3 | 103451 | Robert Price | 07/2019 |
5987 | 2^343604+9931 | 103436 | Yves Minet | 01/2011 |
5988 | 2^343601-21189 | 103435 | Enzo Doria | 06/2013 |
5989 | (10^103423+173)/3 | 103423 | Robert Price | 05/2021 |
5990 | 2^343508-115229 | 103407 | Enzo Doria | 06/2015 |
5991 | (8766^26227-1)/8765 | 103404 | Serge Batalov | 06/2022 |
5992 | (509203*2^343491-1)/16147 | 103403 | Lelio R Paula | 05/2015 |
5993 | 3^216685-113056 | 103386 | Enzo Doria | 07/2014 |
5994 | 24829^14580+14580^24829 | 103382 | Hans Havermann | 08/2020 |
5995 | 91355!4+1024 | 103381 | Robert Price | 09/2019 |
5996 | 44^62903-43^62903 | 103379 | Jean-Louis Charton | 08/2012 |
5997 | 11^99262-20 | 103371 | Lelio R Paula | 12/2016 |
5998 | (509203*2^343357-1)/5 | 103366 | Lelio R Paula | 05/2015 |
5999 | (2^343363+3570)/2 | 103363 | Stefano Morozzi | 09/2020 |
6000 | 2^343356-2791889 | 103361 | Enzo Doria | 03/2016 |