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 | 2^296377-16245 | 89219 | Enzo Doria | 04/2013 |
6752 | 2^296372-64875 | 89217 | Lelio R Paula | 11/2013 |
6753 | 5*6^114648-7 | 89215 | Lelio R Paula | 10/2014 |
6754 | (2^296361+1262)/2 | 89214 | Stefano Morozzi | 01/2020 |
6755 | 3*10^89206+13 | 89207 | Robert Price | 08/2017 |
6756 | 7*3^186960-4 | 89204 | Lelio R Paula | 08/2014 |
6757 | 2^296290-112133 | 89193 | Enzo Doria | 06/2014 |
6758 | 6^114620-17003 | 89192 | Enzo Doria | 02/2013 |
6759 | (4*10^89189+41)/9 | 89189 | Robert Price | 11/2014 |
6760 | 2^296261-161181 | 89184 | Enzo Doria | 04/2013 |
6761 | (19*2^296269-1)/530623 | 89182 | Lelio R Paula | 02/2014 |
6762 | 3^186903-10 | 89176 | Robert Price | 09/2013 |
6763 | 7*10^89169-3 | 89170 | Dmitry Domanov | 11/2010 |
6764 | (2^296221-1017)/35 | 89170 | Lelio R Paula | 01/2013 |
6765 | (43*10^89169+47)/9 | 89170 | Robert Price | 12/2017 |
6766 | (10^89160+29)/3 | 89160 | Serge Batalov | 12/2014 |
6767 | (28*10^89141-97)/3 | 89142 | Robert Price | 07/2016 |
6768 | 2^296116-734639 | 89140 | Paul Bourdelais | 04/2013 |
6769 | (2^296101+1250)/2 | 89135 | Stefano Morozzi | 01/2020 |
6770 | 22574^8883+8883^22574 | 89135 | Hans Havermann | 08/2020 |
6771 | 6^114544-133 | 89133 | Lelio R Paula | 12/2015 |
6772 | 2^296076-113297 | 89128 | Enzo Doria | 07/2014 |
6773 | (11*2^296060-1)/215 | 89122 | Lelio R Paula | 02/2014 |
6774 | 5*10^89118+81 | 89119 | Robert Price | 08/2017 |
6775 | 2^296037-1411 | 89117 | Yves Minet | 01/2013 |
6776 | 2^296020-2604809 | 89111 | Enzo Doria | 05/2015 |
6777 | 2^296016+81 | 89110 | Henri Lifchitz | 02/2007 |
6778 | 2^296010-11 | 89108 | Lelio R Paula | 10/2012 |
6779 | 2^295998-111911 | 89105 | Enzo Doria | 05/2014 |
6780 | (2^296014+47)/93991149 | 89102 | Lelio R Paula | 01/2013 |
6781 | 29529^1040+1040^29529 | 89090 | Hans Havermann | 08/2020 |
6782 | 2^295939-16621 | 89087 | Enzo Doria | 04/2013 |
6783 | 2^295934-109691 | 89086 | Enzo Doria | 04/2014 |
6784 | (2^295921+2246)/2 | 89081 | Stefano Morozzi | 01/2020 |
6785 | (185*10^89070-11)/3 | 89072 | Robert Price | 10/2020 |
6786 | (185*10^89063+7)/3 | 89065 | Robert Price | 02/2017 |
6787 | 2^295831+143 | 89055 | Lelio R Paula | 10/2012 |
6788 | 31^59713-30 | 89054 | Theodore Burton | 02/2007 |
6789 | (7*2^295834-1)/27663 | 89052 | Lelio R Paula | 02/2014 |
6790 | 79731!4-32 | 89049 | Robert Price | 11/2019 |
6792 | 123^42608+2 | 89047 | Lelio R Paula | 05/2013 |
6791 | 6^114433-114433^6 | 89047 | Norbert Schneider | 04/2017 |
6793 | 2^295800-459659 | 89045 | Enzo Doria | 05/2015 |
6795 | (52*10^89037-1)/3 | 89039 | Robert Price | 02/2017 |
6794 | (2^295782+1870)/2 | 89039 | Stefano Morozzi | 01/2020 |
6796 | (6^114398-11)/5 | 89019 | Paul Bourdelais | 11/2011 |
6797 | 11^85470-5850 | 89008 | Enzo Doria | 02/2013 |
6798 | (2^295650+522)/2 | 89000 | Stefano Morozzi | 01/2020 |
6799 | (27*2^295634+1)/4687 | 88993 | Lelio R Paula | 01/2013 |
6800 | (p(18424)#-4)/2 | 88985 | Robert Price | 09/2017 |
6801 | 2^295564-189 | 88974 | Lelio R Paula | 10/2012 |
6802 | 2^295557+73097 | 88972 | Enzo Doria | 04/2016 |
6803 | 11^85433-5760 | 88970 | Enzo Doria | 01/2013 |
6804 | 23074^7165+7165^23074 | 88956 | Hans Havermann | 08/2020 |
6805 | (2^295479+3050)/2 | 88948 | Stefano Morozzi | 01/2020 |
6806 | 6*10^88938-91 | 88939 | Robert Price | 11/2017 |
6807 | 15^75619-88 | 88935 | Lelio R Paula | 11/2015 |
6808 | (2^295433+2234)/2 | 88934 | Stefano Morozzi | 01/2020 |
6809 | (15589^21211-1)/15588 | 88930 | Serge Batalov | 08/2020 |
6810 | 23776^5499+5499^23776 | 88929 | Hans Havermann | 08/2020 |
6811 | 21112^16291+16291^21112 | 88923 | Hans Havermann | 08/2020 |
6812 | 2^295385-85 | 88920 | Lelio R Paula | 10/2012 |
6813 | 2^295385-16333 | 88920 | Enzo Doria | 04/2013 |
6814 | 79617!4+32 | 88909 | Robert Price | 09/2019 |
6815 | 22077^10640+10640^22077 | 88903 | Hans Havermann | 08/2020 |
6816 | 2^295309+59 | 88897 | Lelio R Paula | 10/2012 |
6817 | 2^295288-109799 | 88891 | Enzo Doria | 04/2014 |
6818 | 10^88888+170373 | 88889 | Stefano Morozzi | 07/2018 |
6819 | 10^88888+271399 | 88889 | Stefano Morozzi | 07/2018 |
6820 | 10^88888+375397 | 88889 | Stefano Morozzi | 07/2018 |
6821 | 10^88888+548823 | 88889 | Stefano Morozzi | 07/2018 |
6822 | 10^88888+820861 | 88889 | Stefano Morozzi | 07/2018 |
6823 | 10^88888+1206349 | 88889 | Stefano Morozzi | 07/2018 |
6824 | 10^88888+1292047 | 88889 | Stefano Morozzi | 07/2018 |
6825 | 10^88888+1456029 | 88889 | Stefano Morozzi | 07/2018 |
6826 | 10^88888-490287 | 88888 | Patrick De Geest | 09/2018 |
6827 | 2^295245+65 | 88878 | Lelio R Paula | 10/2012 |
6828 | Padovan(727734) | 88874 | Eric W. Weisstein | 04/2011 |
6829 | (2^295231+1554)/2 | 88874 | Stefano Morozzi | 01/2020 |
6830 | 79585!4-32 | 88870 | Robert Price | 11/2019 |
6831 | 18^70793-17 | 88865 | Lelio R Paula | 11/2015 |
6832 | 6^114196-17573 | 88862 | Enzo Doria | 02/2013 |
6833 | 302^35829+35829^302 | 88857 | Norbert Schneider | 12/2014 |
6834 | 3^186223+16 | 88851 | Robert Price | 10/2013 |
6835 | 3*2^295139+7 | 88847 | Lelio R Paula | 08/2014 |
6836 | 3^186212-113008 | 88846 | Enzo Doria | 07/2014 |
6837 | 2*10^88844+23 | 88845 | Robert Price | 12/2020 |
6838 | 23415^6224+6224^23415 | 88839 | Hans Havermann | 08/2020 |
6839 | 87^45803-86 | 88836 | Norbert Schneider | 11/2015 |
6840 | 2^295094-110273 | 88833 | Enzo Doria | 05/2014 |
6841 | 2^295080+1869151 | 88828 | Enzo Doria | 05/2016 |
6842 | (2^295061+2174)/2 | 88822 | Stefano Morozzi | 01/2020 |
6843 | 2^295027-113407 | 88812 | Enzo Doria | 07/2014 |
6844 | (2*10^88808-71)/3 | 88808 | Robert Price | 01/2017 |
6845 | (2^295009+2102)/2 | 88807 | Stefano Morozzi | 01/2020 |
6846 | 13^79722-80 | 88806 | Lelio R Paula | 11/2015 |
6847 | 21112^16077+16077^21112 | 88802 | Hans Havermann | 08/2020 |
6848 | 3^186103+104 | 88794 | Lelio R Paula | 12/2015 |
6849 | (3*2^294971+1)/25565 | 88792 | Lelio R Paula | 01/2013 |
6850 | (4025^24631-1)/4024 | 88786 | Serge Batalov | 08/2020 |
6851 | 2^294937-115003 | 88785 | Enzo Doria | 06/2015 |
6852 | (2^294923+3134)/2 | 88781 | Stefano Morozzi | 01/2020 |
6853 | (15326^21211-1)/15325 | 88773 | Serge Batalov | 08/2020 |
6854 | 2551^26056+26056^2551 | 88766 | Norbert Schneider | 01/2020 |
6855 | 2^294868-109709 | 88765 | Enzo Doria | 04/2014 |
6856 | 2^294864-2985 | 88763 | Yves Minet | 09/2014 |
6857 | (17*10^88761-41)/3 | 88762 | Robert Price | 09/2017 |
6858 | (2^294834+2854)/2 | 88754 | Stefano Morozzi | 01/2020 |
6859 | (2^294823+2466)/2 | 88751 | Stefano Morozzi | 01/2020 |
6860 | (2^294764+642)/2 | 88733 | Stefano Morozzi | 01/2020 |
6861 | 2^294754+15015 | 88730 | Lelio R Paula | 10/2013 |
6862 | (3^185951-7)/20 | 88720 | Lelio R Paula | 01/2013 |
6863 | (2^294688+3186)/2 | 88710 | Stefano Morozzi | 01/2020 |
6864 | (2^294673+3906)/2 | 88706 | Stefano Morozzi | 01/2020 |
6865 | (2^294661+5)/443693383 | 88694 | Lelio R Paula | 01/2013 |
6866 | (2^294631+2514)/2 | 88693 | Stefano Morozzi | 01/2020 |
6867 | (10^88680+11)/3 | 88680 | Robert Price | 11/2014 |
6868 | 2*7^104928-9 | 88675 | Lelio R Paula | 10/2014 |
6869 | 11^85150+12 | 88675 | Lelio R Paula | 11/2015 |
6870 | 8*6^113951+5 | 88673 | Lelio R Paula | 10/2014 |
6871 | (2^294549+3110)/2 | 88668 | Stefano Morozzi | 01/2020 |
6873 | 6^113940-18563 | 88663 | Enzo Doria | 03/2013 |
6872 | 38^56123-56123^38 | 88663 | Norbert Schneider | 02/2017 |
6874 | 2^294492-65 | 88651 | Lelio R Paula | 10/2012 |
6875 | (2^294481+2822)/2 | 88648 | Stefano Morozzi | 01/2020 |
6876 | 23874^5165+5165^23874 | 88646 | Hans Havermann | 08/2020 |
6877 | (2^294469+1466)/2 | 88644 | Stefano Morozzi | 01/2020 |
6878 | 7*10^88634-27 | 88635 | Dylan Delgado | 08/2019 |
6879 | 20^68120-53 | 88627 | Lelio R Paula | 01/2016 |
6880 | 122^42477-121 | 88623 | Norbert Schneider | 10/2018 |
6881 | 2^294391-113257 | 88621 | Enzo Doria | 07/2014 |
6882 | 5^126784-15476 | 88619 | Enzo Doria | 02/2013 |
6883 | (2^294335+554)/2 | 88604 | Stefano Morozzi | 01/2020 |
6884 | 2^294327+9 | 88602 | Lelio R Paula | 10/2012 |
6885 | 2^294320+18915 | 88600 | Lelio R Paula | 10/2013 |
6886 | (98*10^88595-89)/99 | 88595 | Ray Chandler | 07/2011 |
6887 | 20723^18834+18834^20723 | 88590 | Hans Havermann | 08/2020 |
6888 | (2^294268+1402)/2 | 88584 | Stefano Morozzi | 01/2020 |
6889 | (17*2^294262-1)/13 | 88582 | Lelio R Paula | 02/2014 |
6890 | 6*10^88571+77 | 88572 | Robert Price | 06/2020 |
6891 | (2^294145+1346)/2 | 88547 | Stefano Morozzi | 01/2020 |
6892 | (2*10^88545+7)/9 | 88545 | Erik Branger | 03/2011 |
6893 | 2^294137-133 | 88545 | Lelio R Paula | 10/2012 |
6894 | (2^294109+3494)/2 | 88536 | Stefano Morozzi | 01/2020 |
6895 | 23515^5818+5818^23515 | 88529 | Hans Havermann | 08/2020 |
6896 | (2^294092-1017)/1171 | 88528 | Lelio R Paula | 01/2013 |
6898 | 4^147036+147036^4+1 | 88525 | Robert Price | 04/2014 |
6897 | 22541^8458+8458^22541 | 88525 | Hans Havermann | 08/2020 |
6899 | (14^77213-5^77213)/9 | 88496 | Robert Price | 04/2013 |
6900 | 2^293965+2793377 | 88493 | Enzo Doria | 03/2016 |
6901 | (2^293956+3306)/2 | 88490 | Stefano Morozzi | 01/2020 |
6902 | 11^84971-150 | 88489 | Enzo Doria | 08/2012 |
6903 | (2^293944+1066)/2 | 88486 | Stefano Morozzi | 01/2020 |
6904 | 9*10^88485+91 | 88486 | Robert Price | 10/2020 |
6905 | 48^52626-47 | 88478 | Norbert Schneider | 08/2018 |
6906 | (2^293956-1017)/39851376696677 | 88476 | Lelio R Paula | 01/2013 |
6907 | 11^84958-2190 | 88475 | Enzo Doria | 12/2012 |
6908 | (2^293902+3070)/2 | 88474 | Stefano Morozzi | 01/2020 |
6909 | 2^293880-111347 | 88467 | Enzo Doria | 05/2014 |
6910 | (106*10^88459-1)/3 | 88461 | Serge Batalov | 12/2014 |
6911 | 10^88457+278733 | 88458 | Patrick De Geest | 07/2017 |
6912 | 10^88457-401 | 88457 | Robert Price | 06/2017 |
6913 | (2^293834+3018)/2 | 88453 | Stefano Morozzi | 01/2020 |
6914 | (2^293747+1626)/2 | 88427 | Stefano Morozzi | 01/2020 |
6915 | 22080^10109+10109^22080 | 88424 | Hans Havermann | 08/2020 |
6916 | 11^84908-1680 | 88423 | Enzo Doria | 12/2012 |
6917 | (2^293703+3486)/2 | 88414 | Stefano Morozzi | 01/2020 |
6918 | 2^293679-112075 | 88407 | Enzo Doria | 06/2014 |
6919 | 6^113608-65 | 88405 | Lelio R Paula | 12/2015 |
6920 | 2^293624+82875 | 88390 | Lelio R Paula | 10/2013 |
6921 | (2^293618+918)/2 | 88388 | Stefano Morozzi | 01/2020 |
6922 | 2^293612-112337 | 88387 | Enzo Doria | 06/2014 |
6923 | 758^30693+30693^758 | 88386 | Norbert Schneider | 05/2017 |
6924 | gcd(V(1062695),5*F(212539)*(F(212539)-1)+1) | 88383 | Serge Batalov | 04/2015 |
6926 | 5*10^88377+77 | 88378 | Robert Price | 06/2016 |
6925 | (2^293584+1558)/2 | 88378 | Stefano Morozzi | 01/2020 |
6927 | (5*2^293526+1)/303 | 88359 | Lelio R Paula | 01/2013 |
6928 | 20918^16743+16743^20918 | 88355 | Hans Havermann | 08/2020 |
6929 | (2^293499+2390)/2 | 88352 | Stefano Morozzi | 01/2020 |
6930 | 20727^18292+18292^20727 | 88344 | Hans Havermann | 08/2020 |
6931 | 2^293461-2785843 | 88341 | Enzo Doria | 03/2016 |
6933 | 2^293416-1037 | 88328 | Yves Minet | 11/2012 |
6932 | 6*10^88327+43 | 88328 | Robert Price | 07/2020 |
6934 | 25109^3292+3292^25109 | 88320 | Hans Havermann | 08/2020 |
6935 | (2^293384-7)/3 | 88317 | Lelio R Paula | 01/2013 |
6936 | (3^185092-5)/4 | 88311 | Lelio R Paula | 01/2013 |
6937 | (5*10^88303-173)/3 | 88304 | Robert Price | 11/2017 |
6938 | 14^77039-57 | 88297 | Lelio R Paula | 11/2015 |
6939 | 3*10^88293-59 | 88294 | Robert Price | 10/2017 |
6940 | 5^126303+42 | 88283 | Lelio R Paula | 10/2015 |
6942 | 19^69025-18 | 88266 | Theodore Burton | 01/2006 |
6941 | 140^41128+139 | 88266 | Norbert Schneider | 04/2019 |
6943 | 5*10^88261-3 | 88262 | Dmitry Domanov | 11/2010 |
6944 | (2^293195+2238)/2 | 88261 | Stefano Morozzi | 01/2020 |
6945 | 8^97731-7^(97731-1) | 88260 | Robert Price | 04/2016 |
6946 | 5*10^88254+59 | 88255 | Robert Price | 09/2016 |
6948 | 4*3^184968-7 | 88253 | Lelio R Paula | 08/2014 |
6947 | 34*10^88251-9 | 88253 | Robert Price | 06/2020 |
6949 | 7^104426+88 | 88251 | Lelio R Paula | 10/2015 |
6950 | 2^293142+99 | 88245 | Lelio R Paula | 10/2012 |
6951 | 11^84736-2130 | 88244 | Enzo Doria | 12/2012 |
6952 | 23884^4945+4945^23884 | 88232 | Hans Havermann | 08/2020 |
6953 | 22080^9907+9907^22080 | 88231 | Hans Havermann | 08/2020 |
6954 | (10^110315+1)/(10^22063+1)/5824812682167648578460281641 | 88225 | Makoto Kamada | 04/2019 |
6955 | 2^293068-89 | 88223 | Lelio R Paula | 10/2012 |
6956 | 2126^26511+26511^2126 | 88218 | Norbert Schneider | 02/2020 |
6957 | 2^293014-109655 | 88207 | Enzo Doria | 04/2014 |
6958 | 8*7^104371-3 | 88205 | Lelio R Paula | 10/2014 |
6959 | 2^292995-2145 | 88201 | Lelio R Paula | 10/2013 |
6960 | 28534^1233+1233^28534 | 88198 | Hans Havermann | 08/2020 |
6961 | 5*10^88182+77 | 88183 | Robert Price | 06/2016 |
6963 | 2^292929+43877 | 88181 | Donovan Johnson | 04/2005 |
6962 | 6^113321+131 | 88181 | Lelio R Paula | 12/2015 |
6964 | (2^292919+3366)/2 | 88178 | Stefano Morozzi | 01/2020 |
6965 | 2^292904-583949 | 88173 | Paul Bourdelais | 04/2013 |
6966 | 2^292894-111851 | 88170 | Enzo Doria | 05/2014 |
6967 | 2^292872-2797349 | 88164 | Enzo Doria | 03/2016 |
6968 | (2^292864+1042)/2 | 88161 | Stefano Morozzi | 01/2020 |
6969 | 2^292860-109685 | 88160 | Enzo Doria | 04/2014 |
6970 | 9*6^113288-5 | 88157 | Lelio R Paula | 10/2014 |
6971 | (2^292735+2166)/2 | 88122 | Stefano Morozzi | 01/2020 |
6972 | (2^292735+2346)/2 | 88122 | Stefano Morozzi | 01/2020 |
6973 | (2^292705+1934)/2 | 88113 | Stefano Morozzi | 01/2020 |
6974 | (2*10^88112-23)/3 | 88112 | Robert Price | 09/2017 |
6975 | 29622^943+943^29622 | 88111 | Hans Havermann | 08/2020 |
6976 | (73*10^88105+17)/9 | 88106 | Robert Price | 10/2015 |
6977 | (2^292673+3026)/2 | 88104 | Stefano Morozzi | 01/2020 |
6978 | 11^84588+78 | 88090 | Lelio R Paula | 11/2015 |
6979 | (265*10^88077+11)/3 | 88079 | Robert Price | 03/2017 |
6980 | 91^44959-90^44959 | 88077 | Robert Price | 07/2015 |
6981 | 6^113177-5^113177 | 88069 | Jean-Louis Charton | 01/2009 |
6982 | 1316^28233+28233^1316 | 88066 | Norbert Schneider | 07/2020 |
6983 | (2^292543+2234)/2 | 88064 | Stefano Morozzi | 01/2020 |
6984 | (10^88020+101)/3 | 88020 | Robert Price | 04/2016 |
6985 | 52^51285-51 | 88006 | Norbert Schneider | 09/2018 |
6986 | 21002^15501+15501^21002 | 88006 | Hans Havermann | 08/2020 |
6987 | 20466^19951+19951^20466 | 88004 | Hans Havermann | 08/2020 |
6988 | (2^292324+2346)/2 | 87998 | Stefano Morozzi | 01/2020 |
6989 | 21634^11683+11683^21634 | 87998 | Hans Havermann | 08/2020 |
6990 | (6340^23143-1)/6339 | 87988 | Serge Batalov | 08/2020 |
6991 | 20674^18013+18013^20674 | 87980 | Hans Havermann | 08/2020 |
6992 | (10^(2*43982+1)-6*10^43982-1)/3 | 87965 | Robert Price | 02/2016 |
6993 | (424*10^87930-1)/9 | 87932 | Robert Price | 11/2017 |
6994 | (2^292100+3070)/2 | 87931 | Stefano Morozzi | 01/2020 |
6995 | p(18225)#/2+128 | 87929 | Robert Price | 04/2017 |
6996 | (10607^21841-1)/10606 | 87919 | Serge Batalov | 08/2020 |
6997 | (2^292047+1550)/2 | 87915 | Stefano Morozzi | 01/2020 |
6998 | 4*6^112977+7 | 87914 | Lelio R Paula | 10/2014 |
6999 | (2^292016+1062)/2 | 87906 | Stefano Morozzi | 01/2020 |
7000 | (11359^21673-1)/11358 | 87888 | Serge Batalov | 08/2020 |