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 ? |
6751 | 5*10^104326+77 | 104327 | Robert Price | 12/2018 |
6752 | (2^346564+2026)/2 | 104326 | Stefano Morozzi | 09/2020 |
6753 | (8925^26569-1)/(8925^163-1) | 104320 | Ryan Propper | 11/2023 |
6756 | 2^346527-111817 | 104316 | Enzo Doria | 05/2014 |
6755 | 6^134055+101 | 104316 | Lelio R Paula | 11/2016 |
6754 | fibonacci(499147)-4 | 104316 | Jon Grantham | 08/2023 |
6757 | 92105!4+8 | 104312 | Robert Price | 09/2019 |
6758 | (7^123407-5^123407)/2 | 104291 | Jon Grantham | 08/2023 |
6759 | 6^134013+163 | 104283 | Lelio R Paula | 11/2016 |
6760 | 10^(2*52140+1)-10^52140-1 | 104281 | Robert Price | 07/2023 |
6761 | (22*10^104272-61)/3 | 104273 | Robert Price | 03/2022 |
6762 | (2^346318+862)/2 | 104252 | Stefano Morozzi | 02/2021 |
6764 | 18^83049-89 | 104250 | Lelio R Paula | 01/2017 |
6763 | (2^346311+606)/2 | 104250 | Stefano Morozzi | 09/2020 |
6765 | (166*10^104230+11)/3 | 104232 | Robert Price | 03/2021 |
6766 | 2^346213+484049 | 104221 | Enzo Doria | 04/2017 |
6767 | 18^83021-91 | 104214 | Lelio R Paula | 01/2017 |
6768 | 10^104212+128779 | 104213 | Patrick De Geest | 05/2016 |
6769 | 10^104212+237447 | 104213 | Patrick De Geest | 05/2016 |
6770 | 10^104212+242139 | 104213 | Patrick De Geest | 05/2016 |
6771 | 10^104212+269199 | 104213 | Patrick De Geest | 05/2016 |
6772 | 24254^19803+19803^24254 | 104213 | Hans Havermann | 09/2020 |
6773 | (10^104212-71)/1 | 104212 | Serge Batalov | 12/2014 |
6774 | 23678^23821-23821^23678 | 104202 | Norbert Schneider | 01/2020 |
6775 | 2^346078+7695 | 104180 | Lelio R Paula | 12/2016 |
6776 | 2^346055-19537 | 104173 | Enzo Doria | 05/2013 |
6777 | (3*2^346057+1)/77 | 104173 | Lelio R Paula | 12/2013 |
6778 | 2^346037-20125 | 104168 | Enzo Doria | 06/2013 |
6779 | (34*10^104168-7)/9/13/551179 | 104162 | Makoto Morimoto | 05/2016 |
6780 | 6^133854-17993 | 104159 | Enzo Doria | 03/2013 |
6781 | 2^345990+1874239 | 104154 | Enzo Doria | 06/2016 |
6782 | 2^345988-20373 | 104153 | Enzo Doria | 06/2013 |
6783 | (2^345965+2214)/2 | 104146 | Stefano Morozzi | 09/2020 |
6784 | (26*10^104132-503)/9 | 104133 | Robert Price | 07/2018 |
6785 | 3*5^148968+8 | 104125 | Lelio R Paula | 11/2014 |
6786 | (7*10^104113+53)/3 | 104114 | Robert Price | 05/2018 |
6787 | p(21253)#/2+8 | 104113 | Robert Price | 01/2017 |
6788 | 33582^1259+1259^33582 | 104106 | Hans Havermann | 09/2020 |
6789 | 32958^1441+1441^32958 | 104104 | Hans Havermann | 09/2020 |
6790 | 29127^3748+3748^29127 | 104095 | Hans Havermann | 09/2020 |
6791 | 322^41507+41507^322 | 104094 | Norbert Schneider | 12/2014 |
6792 | 6^133768-17423 | 104092 | Enzo Doria | 03/2013 |
6793 | 2^345786-19557 | 104092 | Enzo Doria | 05/2013 |
6794 | 171^46606-2 | 104072 | Lelio R Paula | 10/2017 |
6795 | (29*10^104071+91)/3 | 104072 | Robert Price | 04/2020 |
6796 | norm(Mod(x,x^3-x-1)^852167-1) | 104070 | Paul Vanderveen | 09/2022 |
6797 | 2^345704-149 | 104068 | Lelio R Paula | 09/2011 |
6798 | 25821^10712+10712^25821 | 104056 | Hans Havermann | 09/2020 |
6799 | (2^345662+2038)/2 | 104055 | Stefano Morozzi | 09/2020 |
6800 | 6^133704-17273 | 104042 | Enzo Doria | 02/2013 |
6801 | (3*2^345590-1)/23 | 104033 | Lelio R Paula | 03/2014 |
6802 | (19*2^345590+1)/33859 | 104030 | Lelio R Paula | 12/2013 |
6803 | 2^345547+11 | 104021 | Lelio R Paula | 09/2011 |
6804 | (19*10^104014-49)/3 | 104015 | Robert Price | 05/2019 |
6805 | (2^345519+1794)/2 | 104012 | Stefano Morozzi | 01/2021 |
6806 | 2^345504+723 | 104008 | Lelio R Paula | 09/2011 |
6807 | 2^345498-20037 | 104006 | Enzo Doria | 06/2013 |
6809 | 18^82854-37 | 104005 | Lelio R Paula | 01/2017 |
6808 | (2^345497+2070)/2 | 104005 | Stefano Morozzi | 09/2020 |
6810 | 2^345486-109065 | 104002 | Enzo Doria | 04/2014 |
6811 | 2^345487-1851507 | 104002 | Enzo Doria | 08/2015 |
6812 | 2^345476+777 | 103999 | Lelio R Paula | 09/2011 |
6813 | (2^345477+1554)/2 | 103999 | Stefano Morozzi | 09/2020 |
6814 | 4*10^103981-83 | 103982 | Robert Price | 10/2018 |
6815 | (2^345327+3206)/2 | 103954 | Stefano Morozzi | 09/2020 |
6816 | (5*2^345321-1)/3 | 103953 | Lelio R Paula | 03/2014 |
6817 | (2^345267-9)/539049127 | 103927 | Lelio R Paula | 12/2013 |
6818 | (19*10^103907+77)/3 | 103908 | Robert Price | 05/2019 |
6819 | (78557*2^345163+1)/78589 | 103905 | Lelio R Paula | 07/2014 |
6820 | 2^345106-112181 | 103888 | Enzo Doria | 06/2014 |
6821 | (2^345105+1326)/2 | 103887 | Stefano Morozzi | 02/2021 |
6822 | 2^345079+99 | 103880 | Lelio R Paula | 09/2011 |
6823 | (2^345034+1198)/2 | 103866 | Stefano Morozzi | 09/2020 |
6824 | 11^99736-1500 | 103865 | Enzo Doria | 12/2012 |
6825 | (569^37699-1)/568 | 103863 | Serge Batalov | 06/2022 |
6826 | (2^345010+2082)/2 | 103859 | Stefano Morozzi | 09/2020 |
6827 | 23263^23784-23784^23263 | 103857 | Norbert Schneider | 01/2020 |
6828 | 3*10^103844+31 | 103845 | Robert Price | 06/2021 |
6829 | 444*33^68368+19 | 103821 | Eric Chen | 05/2023 |
6830 | 23927^21814+21814^23927 | 103813 | Hans Havermann | 09/2020 |
6831 | 2^344842-20157 | 103808 | Enzo Doria | 06/2013 |
6832 | (47*2^344822+1)/1071 | 103801 | Lelio R Paula | 05/2015 |
6833 | 2^344802-20765 | 103796 | Enzo Doria | 06/2013 |
6834 | 2^344794-3821 | 103794 | Yves Minet | 05/2011 |
6835 | 2^344782-109103 | 103790 | Enzo Doria | 04/2014 |
6836 | 3^217521-112292 | 103784 | Enzo Doria | 06/2014 |
6837 | 2^344759-3505 | 103783 | Lelio R Paula | 11/2016 |
6838 | 2^344735-147 | 103776 | Lelio R Paula | 09/2011 |
6839 | L(496533)-p(496533) | 103770 | Robert Price | 02/2018 |
6840 | 2*10^103749-69 | 103750 | Robert Price | 03/2018 |
6841 | (55459*2^344617+1)/3 | 103745 | Lelio R Paula | 07/2014 |
6842 | 2^344607-108891 | 103738 | Enzo Doria | 03/2014 |
6844 | (2^344608+3798)/2 | 103738 | Stefano Morozzi | 02/2021 |
6843 | (1097^34123-1)/1096 | 103738 | Serge Batalov | 06/2022 |
6845 | 2^344560+22411 | 103723 | Enzo Doria | 09/2023 |
6846 | (148*10^103718-1)/3 | 103720 | Robert Price | 03/2020 |
6847 | 2^344541-19561 | 103718 | Enzo Doria | 05/2013 |
6848 | (1017*2^344529-1)/569 | 103714 | Lelio R Paula | 05/2015 |
6849 | (8896^26263-1)/8895 | 103714 | Serge Batalov | 06/2022 |
6850 | (2^344503+1758)/2 | 103706 | Stefano Morozzi | 09/2020 |
6851 | (4*10^103703-1)/3 | 103704 | Serge Batalov | 08/2009 |
6852 | 2^344444-455 | 103688 | Yves Minet | 03/2012 |
6853 | (4*10^103678-43)/3 | 103679 | Robert Price | 01/2022 |
6854 | (13*53^238539*617^9)/10^307661 | 103673 | Hans Havermann | 06/2017 |
6855 | (2^344367+2738)/2 | 103665 | Stefano Morozzi | 01/2021 |
6857 | 2^344317-109933 | 103650 | Enzo Doria | 04/2014 |
6856 | 26055^9508+9508^26055 | 103650 | Hans Havermann | 09/2020 |
6858 | 2^344309-1852033 | 103648 | Enzo Doria | 08/2015 |
6859 | 25729^10672+10672^25729 | 103643 | Hans Havermann | 09/2020 |
6860 | (2^344289+1166)/2 | 103642 | Stefano Morozzi | 09/2020 |
6861 | 3*10^103637+61 | 103638 | Robert Price | 09/2020 |
6862 | (2^344246+1614)/2 | 103629 | Stefano Morozzi | 09/2020 |
6863 | 2^344213-21253 | 103619 | Enzo Doria | 06/2013 |
6864 | 2^344196-465 | 103614 | Lelio R Paula | 11/2016 |
6865 | 2^344187-20001 | 103611 | Enzo Doria | 06/2013 |
6866 | (117191#+(355573#)/(117191#))/(200233679) | 103602 | Makoto Morimoto | 10/2017 |
6867 | (237^43627-1)/236 | 103601 | Serge Batalov | 06/2022 |
6868 | 20^79621-19^79621 | 103590 | Jean-Louis Charton | 03/2011 |
6869 | Phi(155391,10)/1243129 | 103586 | Maksym Voznyy | 09/2015 |
6870 | (9^108553-2^108553)/7 | 103586 | Jon Grantham | 07/2023 |
6871 | 2^344085-20365 | 103580 | Enzo Doria | 06/2013 |
6872 | 5^148169+54 | 103566 | Lelio R Paula | 11/2016 |
6873 | 2^344017-108795 | 103560 | Enzo Doria | 03/2014 |
6874 | (2^343974+2670)/2 | 103547 | Stefano Morozzi | 09/2020 |
6875 | 2^343971-5077 | 103546 | Lelio R Paula | 09/2011 |
6876 | (7*10^103531+71)/3 | 103532 | Robert Price | 05/2018 |
6877 | 2^343899+141 | 103524 | Lelio R Paula | 08/2011 |
6878 | (2^343933+1)/(3*98294339989393) | 103520 | Gord Palameta | 08/2018 |
6879 | 6^133029-67 | 103517 | Lelio R Paula | 11/2016 |
6880 | (17*2^343877+1)/13625 | 103515 | Lelio R Paula | 12/2013 |
6881 | 66*10^103507+7 | 103509 | Robert Price | 07/2020 |
6882 | 26297^8614+8614^26297 | 103485 | Hans Havermann | 09/2020 |
6883 | (2^343764+2658)/2 | 103483 | Stefano Morozzi | 09/2020 |
6884 | 26499^8036+8036^26499 | 103480 | Hans Havermann | 09/2020 |
6885 | (136*10^103474-1)/9 | 103476 | Serge Batalov | 12/2014 |
6886 | 5^148014-24 | 103458 | Lelio R Paula | 11/2016 |
6887 | 28316^4503+4503^28316 | 103453 | Hans Havermann | 09/2020 |
6888 | (22*10^103450+53)/3 | 103451 | Robert Price | 07/2019 |
6889 | 2^343604+9931 | 103436 | Yves Minet | 01/2011 |
6890 | 2^343601-21189 | 103435 | Enzo Doria | 06/2013 |
6891 | (10^103423+173)/3 | 103423 | Robert Price | 05/2021 |
6892 | 2^343508-115229 | 103407 | Enzo Doria | 06/2015 |
6893 | (8766^26227-1)/8765 | 103404 | Serge Batalov | 06/2022 |
6894 | (509203*2^343491-1)/16147 | 103403 | Lelio R Paula | 05/2015 |
6895 | 3^216685-113056 | 103386 | Enzo Doria | 07/2014 |
6896 | 24829^14580+14580^24829 | 103382 | Hans Havermann | 08/2020 |
6897 | 91355!4+1024 | 103381 | Robert Price | 09/2019 |
6898 | 44^62903-43^62903 | 103379 | Jean-Louis Charton | 08/2012 |
6899 | 11^99262-20 | 103371 | Lelio R Paula | 12/2016 |
6900 | (509203*2^343357-1)/5 | 103366 | Lelio R Paula | 05/2015 |
6901 | (2^343363+3570)/2 | 103363 | Stefano Morozzi | 09/2020 |
6902 | 2^343356-2791889 | 103361 | Enzo Doria | 03/2016 |
6903 | (25*10^103360-241)/9 | 103361 | Robert Price | 05/2018 |
6904 | Phi(104989,10)/2939693 | 103350 | Maksym Voznyy | 09/2015 |
6905 | (371*33^68040-19)/32 | 103321 | Eric Chen | 05/2023 |
6906 | 24826^14511+14511^24826 | 103319 | Hans Havermann | 09/2020 |
6907 | 2^343182-171 | 103309 | Lelio R Paula | 08/2011 |
6908 | (23619^23619+23620^23620)/7187 | 103294 | David Broadhurst | 09/2009 |
6909 | (16487^24649-1)/(16487^157-1) | 103287 | Ryan Propper | 11/2023 |
6910 | (10680^25639-1)/10679 | 103285 | Serge Batalov | 06/2022 |
6911 | (7427^26683-1)/7426 | 103282 | Serge Batalov | 06/2022 |
6912 | 2^343079-1835611 | 103278 | Enzo Doria | 05/2015 |
6913 | 2^343063-1147 | 103273 | Yves Minet | 06/2013 |
6914 | 24741^14930+14930^24741 | 103271 | Hans Havermann | 09/2020 |
6915 | (16*10^103255-73)/3 | 103256 | Robert Price | 07/2020 |
6916 | (219^44119-1)/218 | 103256 | Serge Batalov | 06/2022 |
6917 | 9*6^132689+7 | 103254 | Lelio R Paula | 11/2014 |
6918 | 2^342979-18085 | 103247 | Enzo Doria | 05/2013 |
6919 | (16*10^103244-31)/3 | 103245 | Robert Price | 03/2019 |
6920 | 2^342966-16995 | 103244 | Lelio R Paula | 11/2016 |
6921 | 23797^21774+21774^23797 | 103230 | Hans Havermann | 09/2020 |
6922 | 25388^11637+11637^25388 | 103224 | Hans Havermann | 09/2020 |
6923 | (2^342876+2086)/2 | 103216 | Stefano Morozzi | 09/2020 |
6924 | (16*10^103214-31)/3 | 103215 | Robert Price | 03/2019 |
6926 | (37*10^103211+611)/9 | 103212 | Robert Price | 11/2018 |
6925 | (238*10^103210-1)/3 | 103212 | Robert Price | 04/2020 |
6927 | 5^147650+16 | 103203 | Lelio R Paula | 11/2016 |
6928 | gcd(Phi(852060,10),(10^42603+1)^3*(10^42603+1-10^21302)+(10^85206+10^42603)*(10^21302+1)+10^127809) | 103201 | Serge Batalov | 04/2015 |
6929 | 2^342817+2707067 | 103199 | Enzo Doria | 03/2016 |
6930 | 43^63177-42 | 103198 | Theodore Burton | 02/2007 |
6931 | 2^342805-21025 | 103195 | Enzo Doria | 06/2013 |
6932 | 24129^18890+18890^24129 | 103182 | Hans Havermann | 09/2020 |
6933 | (622*10^103171-1)/9 | 103173 | Robert Price | 11/2021 |
6934 | 2^342714+12285 | 103168 | Lelio R Paula | 11/2016 |
6935 | (2*10^103161-17)/3 | 103161 | Serge Batalov | 11/2022 |
6936 | 18^82178-4481 | 103156 | Enzo Doria | 03/2013 |
6937 | 2^342665-17349 | 103153 | Enzo Doria | 05/2013 |
6938 | 30408^2467+2467^30408 | 103150 | Hans Havermann | 09/2020 |
6939 | (5^147571-3^147571)/2 | 103148 | Kellen Shenton | 05/2022 |
6940 | (2^342615+3090)/2 | 103138 | Stefano Morozzi | 09/2020 |
6941 | 2^342612+175 | 103137 | Lelio R Paula | 08/2011 |
6942 | 5*2^342585-3 | 103130 | Lelio R Paula | 09/2014 |
6943 | (16*10^103127+77)/3 | 103128 | Robert Price | 03/2019 |
6944 | (2^342527+3774)/2 | 103111 | Stefano Morozzi | 02/2021 |
6945 | 2^342503-19209 | 103104 | Enzo Doria | 05/2013 |
6946 | (2^342484+2586)/2 | 103098 | Stefano Morozzi | 09/2020 |
6947 | 2^342476+51 | 103096 | Lelio R Paula | 08/2011 |
6948 | 2^342460-18885 | 103091 | Enzo Doria | 05/2013 |
6949 | (2^342442+862)/2 | 103086 | Stefano Morozzi | 02/2021 |
6950 | (194*10^103074-17)/3 | 103076 | Robert Price | 09/2020 |
6951 | 7^121968+82 | 103075 | Lelio R Paula | 10/2016 |
6952 | 2^342392-108863 | 103071 | Enzo Doria | 03/2014 |
6953 | (2^342389+806)/2 | 103070 | Stefano Morozzi | 01/2021 |
6954 | 3^216014+28 | 103065 | Lelio R Paula | 11/2016 |
6955 | 2^342353-108879 | 103059 | Enzo Doria | 03/2014 |
6956 | 10^103055+276081 | 103056 | Patrick De Geest | 12/2016 |
6957 | 10^103055-53 | 103055 | Lelio R Paula | 11/2016 |
6958 | (58*10^103046+419)/9 | 103047 | Robert Price | 05/2019 |
6959 | 7^121933+4 | 103046 | Lelio R Paula | 11/2014 |
6960 | (71*10^103028-143)/9 | 103029 | Robert Price | 10/2021 |
6961 | (2^342235+3618)/2 | 103023 | Stefano Morozzi | 09/2020 |
6962 | 19^80560-68 | 103017 | Lelio R Paula | 10/2016 |
6963 | 2^342209+76607 | 103016 | Enzo Doria | 05/2016 |
6965 | 328^40945+40945^328 | 103013 | Norbert Schneider | 12/2014 |
6964 | 12^95454-91 | 103013 | Lelio R Paula | 10/2016 |
6966 | 25833^9716+9716^25833 | 103009 | Hans Havermann | 01/2019 |
6967 | (893*10^103004+7)/9 | 103006 | Robert Price | 06/2021 |
6968 | (1475^32503-1)/1474 | 102993 | Serge Batalov | 06/2022 |
6969 | 65^56809-64^56809 | 102990 | Robert Price | 08/2013 |
6970 | 5^147338-6 | 102985 | Lelio R Paula | 11/2014 |
6971 | 2^342084-112739 | 102978 | Enzo Doria | 06/2014 |
6972 | 9*6^132330+7 | 102974 | Lelio R Paula | 11/2014 |
6973 | 2^342034-16737 | 102963 | Enzo Doria | 05/2013 |
6974 | (9787*2^342011-1)/25 | 102959 | Lelio R Paula | 05/2015 |
6975 | (3^215802-5)/124343756 | 102956 | Lelio R Paula | 12/2013 |
6976 | 2^341991-17421 | 102950 | Enzo Doria | 05/2013 |
6977 | 2^341969-112225 | 102943 | Enzo Doria | 06/2014 |
6978 | (9787*2^341864-1)/3 | 102915 | Lelio R Paula | 05/2015 |
6979 | 2^341859+1869413 | 102910 | Enzo Doria | 05/2016 |
6980 | 10^102909+61 | 102910 | Lelio R Paula | 11/2016 |
6981 | 10^102909-67049 | 102909 | Patrick De Geest | 12/2016 |
6982 | 23814^20935+20935^23814 | 102898 | Hans Havermann | 02/2019 |
6983 | 28345^4264+4264^28345 | 102888 | Hans Havermann | 02/2019 |
6984 | 25665^10204+10204^25665 | 102886 | Hans Havermann | 02/2019 |
6985 | 26288^8181+8181^26288 | 102860 | Hans Havermann | 02/2019 |
6986 | 2^341659+1836773 | 102850 | Enzo Doria | 03/2016 |
6987 | 2^341608-110639 | 102835 | Enzo Doria | 05/2014 |
6988 | 10^102824+49 | 102825 | Lelio R Paula | 11/2016 |
6989 | 10^102824-168011 | 102824 | Patrick De Geest | 12/2016 |
6990 | 11^98708-8 | 102794 | Lelio R Paula | 10/2016 |
6991 | 2^341466+4429 | 102792 | Lelio R Paula | 09/2011 |
6992 | 2^341460-107 | 102790 | Lelio R Paula | 08/2011 |
6993 | (2^341388+47)/21014943 | 102761 | Lelio R Paula | 12/2013 |
6994 | 2^341355+21 | 102759 | Lelio R Paula | 08/2011 |
6995 | (6496^26947-1)/6495 | 102736 | Serge Batalov | 06/2022 |
6996 | 2^341246+40755 | 102726 | Lelio R Paula | 11/2016 |
6997 | (2^341256-5)/991699 | 102723 | Lelio R Paula | 12/2013 |
6998 | 7^121547+78 | 102720 | Lelio R Paula | 10/2016 |
6999 | 2^341223-461197 | 102719 | Enzo Doria | 06/2015 |
7000 | 27838^4893+4893^27838 | 102711 | Hans Havermann | 03/2019 |