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) !!!
Submit your PRPs
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
751 | 2^750725+60227 | 225991 | Enzo Doria | 02/2016 |
752 | 2^750525-113743 | 225931 | Enzo Doria | 09/2014 |
753 | (2^750151-1)/(429934042631*7590093831289*397764574647511*8361437834787151*17383638888678527263) | 225744 | Dario Alpern | 10/2014 |
754 | 2^749444-1861889 | 225606 | Enzo Doria | 12/2015 |
755 | tau(311^16452) | 225560 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
756 | 2^749096+1876861 | 225501 | Enzo Doria | 07/2016 |
757 | (519643#+43033)/(23*1871*2706283) | 225379 | Makoto Morimoto | 08/2019 |
758 | 2^748657+79607 | 225369 | Enzo Doria | 06/2016 |
759 | 2^746653-291583 | 224765 | Enzo Doria | 07/2015 |
760 | 2^746461-114043 | 224708 | Enzo Doria | 11/2014 |
761 | 2^745098-350471 | 224297 | Enzo Doria | 06/2015 |
762 | (154*10^224061-1)/9 | 224063 | Serge Batalov | 12/2014 |
763 | 2^744211-1845787 | 224030 | Enzo Doria | 07/2015 |
764 | 2^743592-134579 | 223844 | Hans Havermann | 10/2018 |
765 | 2^742452+3 | 223501 | Paul Bourdelais | 06/2019 |
766 | 234178^9+9^234178 | 223463 | Anatoly F. Selevich | 07/2011 |
767 | 2^741651-461197 | 223260 | Enzo Doria | 07/2015 |
768 | 2^741362-1845251 | 223173 | Enzo Doria | 07/2015 |
769 | 2^740455-1863877 | 222900 | Enzo Doria | 12/2015 |
770 | 2^739636-2792219 | 222653 | Enzo Doria | 02/2016 |
771 | 11^213654-180 | 222498 | Enzo Doria | 01/2013 |
772 | 2^738957-1859743 | 222449 | Enzo Doria | 11/2015 |
773 | (54^128189+1)/55 | 222073 | Paul Bourdelais | 11/2018 |
774 | (10^221654+179)/9 | 221654 | Serge Batalov | 12/2014 |
775 | 2^736281+79817 | 221643 | Enzo Doria | 06/2016 |
776 | 10^221631+247617 | 221632 | Patrick De Geest | 10/2013 |
777 | 10^221631-9 | 221631 | Alexander Gramolin | 02/2011 |
778 | 2^735970-1839821 | 221550 | Enzo Doria | 06/2015 |
779 | 2^735367+60293 | 221368 | Enzo Doria | 03/2016 |
780 | (29^151153+1)/30 | 221045 | Paul Bourdelais | 09/2017 |
781 | 2^733264-2784089 | 220735 | Enzo Doria | 10/2015 |
782 | 2^733114-24555 | 220690 | Enzo Doria | 09/2013 |
783 | 2^731891-636259 | 220322 | Enzo Doria | 12/2013 |
784 | 2^731124-25539 | 220091 | Enzo Doria | 09/2013 |
785 | 2^730972-462029 | 220045 | Enzo Doria | 07/2015 |
786 | 2^730833+1868627 | 220003 | Enzo Doria | 04/2016 |
787 | 2^730645-2787133 | 219947 | Enzo Doria | 11/2015 |
788 | 2^730547-25581 | 219917 | Enzo Doria | 09/2013 |
789 | 2^730460+82291 | 219891 | Enzo Doria | 09/2016 |
790 | (2^730339+1)/(3*27839062003*7943469324347) | 219831 | Gord Palameta | 12/2018 |
791 | L(1051849) | 219824 | Renaud Lifchitz | 03/2009 |
792 | 11^210961-240 | 219694 | Enzo Doria | 01/2013 |
793 | 2^729611-113437 | 219635 | Enzo Doria | 08/2014 |
794 | F(1049897) | 219416 | Henri Lifchitz | 10/2008 |
795 | 2^728604+1879321 | 219332 | Enzo Doria | 10/2016 |
796 | 33^144412+2 | 219292 | Robert Price | 04/2015 |
797 | 2^728277-183313 | 219234 | Hans Havermann | 10/2018 |
798 | 2^727998-1862711 | 219150 | Enzo Doria | 12/2015 |
799 | 2^727729-114073 | 219069 | Enzo Doria | 12/2014 |
800 | 2^727496-1851809 | 218999 | Enzo Doria | 08/2015 |
801 | 2^726591+2793143 | 218726 | Enzo Doria | 02/2016 |
802 | 2^725534-2791931 | 218408 | Enzo Doria | 02/2016 |
803 | Phi(220531,10)/2686456812680882641 | 217774 | Makoto Kamada | 10/2020 |
804 | 2^722184+1877821 | 217400 | Enzo Doria | 08/2016 |
805 | (109*10^217360-1)/9 | 217362 | Serge Batalov | 12/2014 |
806 | 281*499979#/46410-2702372 | 216849 | Robert Smith | 08/2016 |
807 | 281*499979#/46410+2400766 | 216849 | Robert Smith | 08/2016 |
808 | tau(311^15786) | 216429 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
809 | 2^718756-114107 | 216368 | Enzo Doria | 12/2014 |
810 | 2^717305+80327 | 215931 | Enzo Doria | 07/2016 |
811 | 2^717069-59773 | 215860 | Enzo Doria | 10/2015 |
812 | 30^146021-29^146021 | 215691 | Jean-Louis Charton | 09/2018 |
813 | 277^88301-276^88301 | 215674 | Jean-Louis Charton | 06/2012 |
814 | 227^91413-40 | 215372 | Enzo Doria | 04/2015 |
815 | 2^714717+782897 | 215152 | Enzo Doria | 11/2016 |
816 | tau(373^15186) | 214796 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
817 | 2^713241+1879157 | 214707 | Enzo Doria | 08/2016 |
818 | 11^206081-10^206081 | 214612 | Jean-Louis Charton | 10/2011 |
819 | 2^712801-69553 | 214575 | Enzo Doria | 01/2016 |
820 | 2^712603-25357 | 214515 | Enzo Doria | 09/2013 |
821 | 2^711721-1846723 | 214250 | Enzo Doria | 07/2015 |
822 | 2^710730-396765 | 213952 | Enzo Doria | 09/2013 |
823 | 2^708468-114329 | 213271 | Enzo Doria | 02/2015 |
824 | (163*10^212779-1)/9 | 212781 | Serge Batalov | 12/2014 |
825 | 2^706838-69041 | 212780 | Enzo Doria | 11/2015 |
826 | 2^704577-113871 | 212099 | Enzo Doria | 10/2014 |
827 | 2^704110-2792441 | 211959 | Enzo Doria | 02/2016 |
828 | 2^703943-1862947 | 211908 | Enzo Doria | 12/2015 |
829 | (60^119083+1)/61 | 211746 | Paul Bourdelais | 10/2018 |
830 | (10^211610-91)/9 | 211610 | Serge Batalov | 12/2014 |
831 | 227^89797+2 | 211565 | Lelio R Paula | 08/2012 |
832 | 2^702194-3 | 211382 | Paul Bourdelais | 03/2012 |
833 | 2^702064+1874911 | 211343 | Enzo Doria | 07/2016 |
834 | 2^701968-113873 | 211314 | Enzo Doria | 10/2014 |
835 | 2^701713-113779 | 211237 | Enzo Doria | 09/2014 |
836 | 2^701449-25201 | 211158 | Enzo Doria | 09/2013 |
837 | 2^701403-2786977 | 211144 | Enzo Doria | 11/2015 |
838 | 2^701189-25423 | 211079 | Enzo Doria | 09/2013 |
839 | 2^701162-459161 | 211071 | Enzo Doria | 05/2015 |
840 | (24^152783-1)/23 | 210872 | Paul Bourdelais | 04/2018 |
841 | 2^700400+9385 | 210842 | Lelio R Paula | 01/2014 |
842 | 2^700131+1874693 | 210761 | Enzo Doria | 06/2016 |
843 | (2^700095+225414)/2 | 210750 | Stefano Morozzi | 04/2020 |
844 | (2^700090+278254)/2 | 210748 | Stefano Morozzi | 04/2020 |
845 | (2^700075+303666)/2 | 210744 | Stefano Morozzi | 04/2020 |
846 | (2^700073+75882)/2 | 210743 | Stefano Morozzi | 04/2020 |
847 | (2^700074+328890)/2 | 210743 | Stefano Morozzi | 04/2020 |
848 | (2^700052+79162)/2 | 210737 | Stefano Morozzi | 04/2020 |
849 | (2^700052+304458)/2 | 210737 | Stefano Morozzi | 04/2020 |
850 | (2^700048+376578)/2 | 210736 | Stefano Morozzi | 04/2020 |
851 | 11^202357-1140 | 210734 | Enzo Doria | 12/2012 |
852 | (2^700021+25874)/2 | 210728 | Stefano Morozzi | 04/2020 |
853 | (2^700018+377518)/2 | 210727 | Stefano Morozzi | 04/2020 |
854 | 2^698929-2788423 | 210399 | Enzo Doria | 11/2015 |
855 | (5^300893+4^300893)/9 | 210315 | Jean-Louis Charton | 11/2011 |
856 | 2^698316+1878601 | 210215 | Enzo Doria | 08/2016 |
857 | 2^697036+1878751 | 209829 | Enzo Doria | 08/2016 |
858 | (2^696343-1)/11141489/36009913139329 | 209600 | Tony Prest / Never Odd Or Even | 02/2014 |
859 | 2^696254-115151 | 209594 | Enzo Doria | 06/2015 |
860 | (35^135623+1)/36 | 209410 | Paul Bourdelais | 07/2018 |
861 | 2^694633+1887257 | 209106 | Enzo Doria | 05/2017 |
862 | 2^693959+180923 | 208903 | Enzo Doria | 07/2016 |
863 | 2^693462-1847561 | 208753 | Enzo Doria | 07/2015 |
864 | 2^690890-59981 | 207979 | Enzo Doria | 11/2015 |
865 | (38^131591+1)/39 | 207884 | Paul Bourdelais | 07/2018 |
866 | 2^690163+1877063 | 207760 | Enzo Doria | 07/2016 |
867 | 2^689286-29381 | 207496 | Enzo Doria | 08/2015 |
868 | 2^689122-114521 | 207447 | Enzo Doria | 04/2015 |
869 | 2^688373+6077 | 207221 | Enzo Doria | 02/2016 |
870 | 2^688201+1875527 | 207170 | Enzo Doria | 07/2016 |
871 | (42^127609+1)/43 | 207140 | Paul Bourdelais | 07/2018 |
872 | 2^687619+81563 | 206994 | Enzo Doria | 08/2016 |
873 | 2^686572-1845299 | 206679 | Enzo Doria | 07/2015 |
874 | (21^156217-1)/20 | 206552 | Paul Bourdelais | 04/2010 |
875 | (2^685739+1)/(3*87774593) | 206420 | Gord Palameta | 08/2018 |
876 | 2^685578+3 | 206380 | Paul Bourdelais | 04/2012 |
877 | 2^684836-49349 | 206157 | Enzo Doria | 10/2015 |
878 | (73^110603-1)/72 | 206088 | Paul Bourdelais | 06/2018 |
879 | 2^684163-2790967 | 205954 | Enzo Doria | 12/2015 |
880 | 326^81943-325^81943 | 205941 | Jean-Louis Charton | 07/2012 |
881 | (2^684127-1)/23765203727 | 205933 | Jens K. Andersen | 04/2009 |
882 | 4^341233-3^341233 | 205443 | Jean-Louis Charton | 01/2006 |
883 | 2^681545+467 | 205166 | Lelio R Paula | 11/2013 |
884 | 2^681306+81049 | 205094 | Enzo Doria | 07/2016 |
885 | 2^680697+1877057 | 204911 | Enzo Doria | 07/2016 |
886 | 2^680415+71783 | 204826 | Enzo Doria | 04/2016 |
887 | 2^679860-459995 | 204659 | Enzo Doria | 06/2015 |
888 | 2^679593+1879547 | 204578 | Enzo Doria | 09/2016 |
889 | 2^678651+1875023 | 204295 | Enzo Doria | 07/2016 |
890 | 2^678561+678561 | 204268 | Henri Lifchitz | 05/2006 |
891 | 2^678101-114043 | 204129 | Enzo Doria | 11/2014 |
892 | 2^678028+1875631 | 204107 | Enzo Doria | 06/2016 |
893 | 2^676992-59783 | 203795 | Enzo Doria | 10/2015 |
894 | 2^676972+1866871 | 203789 | Enzo Doria | 03/2016 |
895 | 2^676761-9 | 203726 | Robert Price | 01/2017 |
896 | 2^676563-2792407 | 203666 | Enzo Doria | 02/2016 |
897 | 2^676306-114713 | 203589 | Enzo Doria | 04/2015 |
898 | (2^675977-1)/(1686378749257*7171117283326998925471) | 203456 | Dario Alpern | 07/2015 |
899 | 2^674627-25161 | 203083 | Enzo Doria | 09/2013 |
900 | 2^674603-69337 | 203076 | Enzo Doria | 12/2015 |
901 | 2^674459+74213 | 203033 | Enzo Doria | 04/2016 |
902 | 2^674116-458879 | 202930 | Enzo Doria | 04/2015 |
903 | 2^674076-114755 | 202918 | Enzo Doria | 04/2015 |
904 | 5^289937-4^289937 | 202658 | Jean-Louis Charton | 03/2013 |
905 | (467617#+14534)/(2^2*13*43) | 202603 | Makoto Morimoto | 06/2019 |
906 | 2^672556-1835519 | 202460 | Enzo Doria | 04/2015 |
907 | 2^671446-351401 | 202126 | Enzo Doria | 07/2015 |
908 | 2^671373+77747 | 202104 | Enzo Doria | 05/2016 |
909 | 2^670726-114845 | 201909 | Enzo Doria | 05/2015 |
910 | 2^670539-25045 | 201853 | Enzo Doria | 09/2013 |
911 | 43284^43519-43519^43284 | 201769 | Norbert Schneider | 08/2018 |
912 | 2^670213+282317 | 201755 | Enzo Doria | 09/2016 |
913 | 2^669854-113501 | 201647 | Enzo Doria | 08/2014 |
914 | 2^669718+81619 | 201606 | Enzo Doria | 08/2016 |
915 | (145*10^201504-1)/9 | 201506 | Serge Batalov | 12/2014 |
916 | 2^668938-1839821 | 201371 | Enzo Doria | 06/2015 |
917 | 2^668492+1879303 | 201237 | Enzo Doria | 08/2016 |
918 | 6^258413-5^258413 | 201085 | Jean-Louis Charton | 03/2013 |
919 | 10194^50161-50161^10194 | 201063 | Norbert Schneider | 09/2020 |
920 | 2^667894+1875589 | 201057 | Enzo Doria | 06/2016 |
921 | 2^667569-1836559 | 200959 | Enzo Doria | 05/2015 |
922 | (172*10^200746-1)/9 | 200748 | Serge Batalov | 12/2014 |
923 | 2^666430+782569 | 200616 | Enzo Doria | 10/2016 |
924 | (86^103471+1)/87 | 200163 | Paul Bourdelais | 05/2018 |
925 | 103*(10^100000+20110)^2-4707*(10^100000+20110)+50383 | 200003 | Robert Price | 05/2017 |
926 | 103*(10^100000+33798)^2-4707*(10^100000+33798)+50383 | 200003 | Robert Price | 01/2018 |
927 | numbpart(32235776596) | 200002 | Serge Batalov | 04/2017 |
928 | (10^200001+149)/3 | 200001 | Serge Batalov | 12/2014 |
929 | 10^200000+571791 | 200001 | Patrick De Geest | 08/2019 |
930 | (3^419180-621444)/3 | 200000 | Stefano Morozzi | 06/2019 |
931 | (3^419180-1941384)/3 | 200000 | Stefano Morozzi | 08/2019 |
932 | 10^200000-45557 | 200000 | Patrick De Geest | 08/2019 |
933 | (10^200000-1)/9-10^58838 | 200000 | Akihiko Yoshida | 04/2020 |
934 | (299*10^199801-17)/3 | 199803 | Robert Price | 04/2020 |
935 | 6*10^199788+77 | 199789 | Robert Price | 06/2020 |
936 | (14*10^199447+73)/3 | 199448 | Robert Price | 12/2018 |
937 | 2^662103-29287 | 199313 | Enzo Doria | 08/2015 |
938 | (2*10^199283+457)/9 | 199283 | Robert Price | 01/2018 |
939 | (10^199248-19)/9 | 199248 | Robert Price | 01/2018 |
940 | (194*10^199146-17)/3 | 199148 | Robert Price | 09/2020 |
941 | (151*10^199003-1)/3 | 199005 | Robert Price | 03/2020 |
942 | 9*10^198946-73 | 198947 | Robert Price | 01/2021 |
943 | 45*10^198931-7 | 198933 | Robert Price | 07/2020 |
944 | 2^660482-2891 | 198825 | Yves Minet | 05/2016 |
945 | 2^659583+1879223 | 198555 | Enzo Doria | 08/2016 |
946 | (53*10^198529+1)/9 | 198530 | Serge Batalov | 07/2010 |
947 | (49*10^198403+311)/9 | 198404 | Robert Price | 06/2019 |
948 | 2^658828-29819 | 198327 | Enzo Doria | 08/2015 |
949 | 2^658776-2791229 | 198312 | Enzo Doria | 01/2016 |
950 | (35*10^198294-377)/9 | 198295 | Robert Price | 07/2018 |
951 | (4*10^198268+137)/3 | 198269 | Robert Price | 03/2018 |
952 | 2^658544-1854029 | 198242 | Enzo Doria | 09/2015 |
953 | (157^90239-1)/156 | 198154 | Alexander Zhirkov | 06/2016 |
954 | (4*10^197778+173)/3 | 197779 | Robert Price | 02/2018 |
955 | 78*10^197702-7 | 197704 | Robert Price | 10/2020 |
956 | (29*10^197629-77)/3 | 197630 | Robert Price | 02/2020 |
957 | 2^656462+1874419 | 197615 | Enzo Doria | 06/2016 |
958 | (89^101363+1)/90 | 197595 | Paul Bourdelais | 11/2017 |
959 | 2^655893-344563 | 197444 | Enzo Doria | 07/2015 |
960 | 2^655875+71333 | 197439 | Enzo Doria | 04/2016 |
961 | 2^655845-9 | 197430 | Robert Price | 01/2017 |
962 | (299*10^197426+1)/3 | 197428 | Robert Price | 04/2020 |
963 | 2^655682-1849121 | 197380 | Enzo Doria | 08/2015 |
964 | 4*10^197268-89 | 197269 | Robert Price | 11/2018 |
965 | 2^655045-49603 | 197189 | Enzo Doria | 10/2015 |
966 | 2^655000-2784089 | 197175 | Enzo Doria | 10/2015 |
967 | (18^157051-1)/17 | 197141 | Paul Bourdelais | 03/2010 |
968 | 2^654400-39719 | 196995 | Enzo Doria | 09/2015 |
969 | (14*10^196965-221)/9 | 196966 | Robert Price | 04/2018 |
970 | 2^654216-459995 | 196939 | Enzo Doria | 06/2015 |
971 | (17*10^196921+31)/3 | 196922 | Robert Price | 01/2019 |
972 | (14*10^196885-89)/3 | 196886 | Robert Price | 05/2020 |
973 | 3*10^196881-59 | 196882 | Robert Price | 09/2018 |
974 | 2^653851+82295 | 196829 | Enzo Doria | 09/2016 |
975 | (38*10^196792+691)/9 | 196793 | Robert Price | 10/2018 |
976 | 2^653636-29429 | 196765 | Enzo Doria | 08/2015 |
977 | 2^653601-69673 | 196754 | Enzo Doria | 02/2016 |
978 | (44*10^196684+1)/9 | 196685 | Serge Batalov | 07/2010 |
979 | 2^652786-1852571 | 196509 | Enzo Doria | 08/2015 |
980 | 2^652752-1325 | 196498 | Yves Minet | 07/2015 |
981 | 2^652470-1859141 | 196414 | Enzo Doria | 10/2015 |
982 | 2^652307+1877483 | 196364 | Enzo Doria | 08/2016 |
983 | 2^652293-115273 | 196360 | Enzo Doria | 06/2015 |
984 | (47*10^196281+403)/9 | 196282 | Robert Price | 03/2019 |
985 | 6*10^196070+73 | 196071 | Robert Price | 04/2019 |
986 | 2^651218-1868201 | 196037 | Enzo Doria | 04/2016 |
987 | 2^651047-49687 | 195985 | Enzo Doria | 10/2015 |
988 | 2^650285-2785333 | 195756 | Enzo Doria | 10/2015 |
989 | (69^106397-1)/68 | 195647 | Paul Bourdelais | 10/2017 |
990 | 2^649811+1877573 | 195613 | Enzo Doria | 08/2016 |
991 | 2^649662-170531 | 195568 | Hans Havermann | 10/2018 |
992 | 2^649586+182839 | 195545 | Enzo Doria | 11/2016 |
993 | 2^649537-1867363 | 195531 | Enzo Doria | 04/2016 |
994 | 2^649536-1861499 | 195530 | Enzo Doria | 12/2015 |
995 | (103*10^195478-1)/3 | 195480 | Serge Batalov | 12/2014 |
996 | 2^649102-59351 | 195400 | Enzo Doria | 10/2015 |
997 | (2*10^195399+529)/9 | 195399 | Robert Price | 01/2018 |
998 | (202*10^195383-13)/9 | 195385 | Robert Price | 01/2020 |
999 | 2^648988+1178971 | 195365 | Enzo Doria | 06/2016 |
1000 | 2^648975-114319 | 195361 | Enzo Doria | 02/2015 |