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 ? |
751 | 2^983620+60451 | 296100 | Phil Moore | 03/2008 |
752 | 2^982721-2683 | 295829 | Yves Minet | 02/2016 |
753 | (23*10^295581+7)/3 | 295582 | Robert Price | 05/2023 |
754 | 2^981298-2788421 | 295401 | Enzo Doria | 12/2015 |
755 | 2^981163-2790967 | 295360 | Enzo Doria | 12/2015 |
756 | 2^981016-15 | 295316 | Stefano Morozzi | 07/2022 |
757 | 8*10^295160-13 | 295161 | Robert Price | 08/2023 |
758 | 2^979594+1865839 | 294888 | Enzo Doria | 02/2016 |
759 | 2*10^294527-91 | 294528 | Robert Price | 02/2023 |
760 | 10^294475+49 | 294476 | Robert Price | 02/2023 |
761 | (11*10^294308+91)/3 | 294309 | Robert Price | 03/2023 |
762 | 2^976760+1885021 | 294035 | Enzo Doria | 02/2017 |
763 | 3*2^976518+5 | 293962 | Jon Grantham | 07/2023 |
764 | 2^974441+1878407 | 293336 | Enzo Doria | 08/2016 |
765 | 2^974180+1876831 | 293258 | Enzo Doria | 07/2016 |
766 | 2^974170+453319 | 293255 | Enzo Doria | 01/2017 |
767 | 2^972744-114419 | 292826 | Enzo Doria | 03/2015 |
768 | 11^281143+6 | 292781 | Paul Bourdelais | 03/2021 |
769 | (4*10^292619-91)/3 | 292620 | Robert Price | 12/2022 |
770 | CopelandErdos(292447) | 292447 | Mark Rodenkirch | 12/2015 |
771 | 3*10^292344-97 | 292345 | Robert Price | 01/2023 |
772 | 2^970911+1873373 | 292274 | Enzo Doria | 07/2016 |
773 | 2^970670+383059 | 292201 | Enzo Doria | 12/2016 |
774 | 2^970141-100363 | 292042 | Enzo Doria | 02/2014 |
775 | 2^969945+1874717 | 291983 | Enzo Doria | 06/2016 |
776 | 2^969384+1880701 | 291814 | Enzo Doria | 10/2016 |
777 | (13*10^291721-43)/3 | 291722 | Robert Price | 08/2023 |
778 | (3^611333-2^611333)/(99035947*87109884549431) | 291659 | Mike Oakes | 02/2017 |
779 | 3*2^963892+5 | 290161 | Jon Grantham | 07/2023 |
780 | (151*10^289877-1)/3 | 289879 | Robert Price | 07/2023 |
781 | 2^962194+60139 | 289650 | Enzo Doria | 02/2016 |
782 | 2^959307-1861507 | 288781 | Enzo Doria | 11/2015 |
783 | (17*10^288692+31)/3 | 288693 | Robert Price | 04/2023 |
784 | 14^251797-3 | 288592 | Paul Bourdelais | 12/2020 |
785 | (2*10^288371-23)/3 | 288371 | Robert Price | 10/2022 |
786 | (31*10^288238-1)/3 | 288240 | Robert Price | 02/2023 |
787 | (7^341063-2^341063)/5 | 288231 | Jon Grantham | 08/2023 |
788 | 2^957372+283531 | 288198 | Enzo Doria | 02/2017 |
789 | 2^957192-1864049 | 288144 | Enzo Doria | 01/2016 |
790 | 2^956446+1875199 | 287919 | Enzo Doria | 06/2016 |
791 | 7*10^287854+39 | 287855 | Robert Price | 01/2023 |
792 | (17^233689+1)/18 | 287542 | Paul Bourdelais | 09/2013 |
793 | 2^955178+1883587 | 287538 | Enzo Doria | 12/2016 |
794 | 2^954127-954127^2 | 287221 | Henri Lifchitz | 01/2021 |
795 | 2^954001+1875257 | 287183 | Enzo Doria | 07/2016 |
796 | 2*10^287175-49 | 287176 | Robert Price | 05/2023 |
797 | (22*10^287085-19)/3 | 287086 | Robert Price | 07/2023 |
798 | 2^953488+22411 | 287029 | Enzo Doria | 09/2023 |
799 | (23891^65536+1)/2 | 286932 | Serge Batalov | 03/2018 |
800 | 5*3^601336-2^601336 | 286911 | Jon Grantham | 08/2023 |
801 | 2^952268+1878601 | 286662 | Enzo Doria | 09/2016 |
802 | 2^950801+81557 | 286220 | Enzo Doria | 08/2016 |
803 | 22845^65536+22844^65536 | 285658 | Kellen Shenton | 03/2021 |
804 | (11*2^948635-13)/15 | 285568 | Renaud Lifchitz | 11/2011 |
805 | (54^164839+1)/55 | 285565 | Paul Bourdelais | 07/2019 |
806 | 2^948510+782569 | 285530 | Enzo Doria | 10/2016 |
807 | 2^948374-114137 | 285490 | Enzo Doria | 01/2015 |
808 | 2^947739+1872923 | 285298 | Enzo Doria | 06/2016 |
809 | 7*10^284751+9 | 284752 | Robert Price | 04/2023 |
810 | 2^944561-21 | 284342 | Stefano Morozzi | 07/2022 |
811 | (4*10^284330-31)/9 | 284330 | Serge Batalov | 11/2022 |
812 | (32*10^283826-23)/9 | 283827 | Serge Batalov | 01/2023 |
813 | 2^942533+82067 | 283731 | Enzo Doria | 09/2016 |
814 | 2^941469+39 | 283411 | Jon Grantham | 08/2023 |
815 | (10^283342-31)/3 | 283342 | Robert Price | 03/2023 |
816 | 2^940950+1874989 | 283255 | Enzo Doria | 06/2016 |
817 | 2^940426-2786141 | 283097 | Enzo Doria | 11/2015 |
818 | (2*10^282948-11)/9 | 282948 | Serge Batalov | 11/2022 |
819 | 2^939881-114115 | 282933 | Enzo Doria | 12/2014 |
820 | (14*10^282673-143)/3 | 282674 | Robert Price | 05/2023 |
821 | 10^282493+632551 | 282494 | Patrick De Geest | 02/2017 |
823 | 10^282493-9^282493 | 282493 | Jean-Louis Charton | 03/2007 |
822 | 10^282493-132327 | 282493 | Patrick De Geest | 02/2017 |
824 | 3^591827-2^591827 | 282374 | Mike Oakes | 08/2009 |
825 | Tau(509^18946) | 282048 | Nik Lygeros / Olivier Rozier | 05/2015 |
826 | 2^936716-1835519 | 281980 | Enzo Doria | 04/2015 |
827 | 2^936588+79411 | 281942 | Enzo Doria | 06/2016 |
828 | 2^936047+1877423 | 281779 | Enzo Doria | 08/2016 |
829 | 2^935713+1869377 | 281678 | Enzo Doria | 05/2016 |
830 | 2^933082+1877059 | 280886 | Enzo Doria | 08/2016 |
831 | (5*10^280710+1)/3 | 280711 | Serge Batalov | 10/2014 |
832 | 2^931976+25 | 280553 | Stefano Morozzi | 05/2021 |
833 | (42^172663+1)/43 | 280274 | Paul Bourdelais | 07/2019 |
834 | (10^279978+11)/3 | 279978 | Serge Batalov | 07/2015 |
835 | 2^929524-1864829 | 279815 | Enzo Doria | 01/2016 |
836 | 2^929329+22697 | 279756 | Enzo Doria | 09/2023 |
837 | 2^929152+60181 | 279703 | Enzo Doria | 02/2016 |
838 | 2^928621-2787883 | 279543 | Enzo Doria | 11/2015 |
839 | (2^927763+lucasV(-1,2,927763)+1)/2 | 279285 | Mike Oakes | 04/2017 |
840 | 2^927429+1882277 | 279184 | Enzo Doria | 12/2016 |
841 | 2^926766-11 | 278985 | Stefano Morozzi | 07/2022 |
842 | (71^150697-1)/70 | 278978 | Paul Bourdelais | 07/2019 |
843 | (33^183661-1)/32 | 278891 | Paul Bourdelais | 12/2018 |
844 | (166*10^278633-7)/3 | 278635 | Robert Price | 02/2023 |
845 | (2^925391+lucasV(-1,2,925391)+1)/2 | 278571 | Mike Oakes | 03/2017 |
846 | 2^925330-1301 | 278553 | Yves Minet | 06/2015 |
847 | (299*10^278546+1)/3 | 278548 | Robert Price | 01/2023 |
848 | (21^210599+1)/22 | 278457 | Paul Bourdelais | 08/2013 |
849 | (266*10^278325+1)/3 | 278327 | Robert Price | 01/2023 |
850 | 2^923261-69175 | 277930 | Enzo Doria | 11/2015 |
851 | 757^96487-756^96487 | 277796 | Jean-Louis Charton | 12/2012 |
852 | 2^922301+57 | 277641 | Stefano Morozzi | 05/2021 |
853 | 7*10^277478-39 | 277479 | Robert Price | 07/2023 |
854 | ((((((((((((((291487327^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2 | 277369 | Serge Batalov | 07/2016 |
855 | (5^396413-1)/4 | 277081 | Paul Bourdelais | 06/2010 |
856 | 2^920318+2792299 | 277044 | Enzo Doria | 02/2016 |
857 | (16*10^276537-31)/3 | 276538 | Robert Price | 03/2023 |
858 | 2^918596-2786939 | 276525 | Enzo Doria | 11/2015 |
859 | 2^916639+60563 | 275936 | Enzo Doria | 04/2016 |
860 | (2*10^275922+1)/3 | 275922 | Serge Batalov | 11/2022 |
861 | 2^916385-2786383 | 275860 | Enzo Doria | 11/2015 |
862 | 2^915928+82081 | 275722 | Enzo Doria | 09/2016 |
863 | (4*10^275671+11)/3 | 275672 | Tyler Busby | 02/2023 |
864 | 2^915576-114137 | 275616 | Enzo Doria | 01/2015 |
865 | 2^914828+1876741 | 275391 | Enzo Doria | 07/2016 |
866 | 2^914757-2203 | 275370 | Yves Minet | 12/2015 |
867 | 7*10^275157-27 | 275158 | Robert Price | 11/2022 |
868 | 2^912744+20011 | 274764 | Enzo Doria | 06/2023 |
869 | 2^911900+1871245 | 274510 | Enzo Doria | 05/2016 |
870 | (16*10^274251-91)/3 | 274252 | Robert Price | 11/2022 |
871 | 2^910999+9 | 274239 | Stefano Morozzi | 05/2021 |
872 | (2^910869-7)/5 | 274199 | Stefano Morozzi | 08/2022 |
873 | 2^910490+283429 | 274085 | Enzo Doria | 01/2017 |
874 | 7*10^273983+39 | 273984 | Robert Price | 01/2023 |
875 | 2^909988-1862249 | 273934 | Enzo Doria | 12/2015 |
876 | 2^909297+1879157 | 273726 | Enzo Doria | 08/2016 |
877 | 2^909257-1864423 | 273714 | Enzo Doria | 01/2016 |
878 | 2^908739-7 | 273558 | Stefano Morozzi | 07/2022 |
879 | 2^908579-1861747 | 273510 | Enzo Doria | 12/2015 |
880 | 3*2^908005-5 | 273338 | Jon Grantham | 07/2023 |
881 | 2^907635-2791597 | 273226 | Enzo Doria | 01/2016 |
882 | 2^907059+211193 | 273052 | Enzo Doria | 08/2023 |
883 | (53^158341+1)/54 | 273022 | Paul Bourdelais | 07/2019 |
884 | 2^906594-113201 | 272912 | Enzo Doria | 08/2014 |
885 | 9*10^272800+91 | 272801 | Robert Price | 05/2023 |
886 | 2*10^272710-49 | 272711 | Robert Price | 05/2023 |
887 | 2*10^272403-91 | 272404 | Robert Price | 02/2023 |
888 | 2^904773+1872167 | 272364 | Enzo Doria | 06/2016 |
889 | 6^349903+5 | 272278 | Paul Bourdelais | 02/2021 |
890 | 2^904149-1862473 | 272176 | Enzo Doria | 12/2015 |
891 | (7*10^271757-61)/9 | 271757 | Erik Branger | 03/2012 |
892 | 2^902497+1880117 | 271679 | Enzo Doria | 09/2016 |
893 | 2^901858+77689 | 271487 | Enzo Doria | 06/2016 |
894 | 2^901372+1875301 | 271341 | Enzo Doria | 06/2016 |
895 | (8*10^271039-77)/3 | 271040 | Robert Price | 04/2023 |
896 | 2^899999+1144031 | 270927 | Norman Luhn | 09/2023 |
897 | 2*10^270802-91 | 270803 | Robert Price | 02/2023 |
898 | 23*10^270799-7 | 270801 | Robert Price | 10/2022 |
899 | 2^899121-115033 | 270663 | Enzo Doria | 07/2015 |
900 | 2^898900+883021 | 270596 | Enzo Doria | 12/2016 |
901 | 2^898762-251511 | 270555 | Enzo Doria | 12/2013 |
902 | 2^898589+47 | 270503 | Lelio R Paula | 03/2016 |
903 | 187322*10^270343-187321 | 270349 | Danilo Nitsche | 09/2013 |
904 | 48358*10^270343-48357 | 270348 | Danilo Nitsche | 09/2013 |
905 | (10^270343-1)/9 | 270343 | Maksym Voznyy | 07/2007 |
906 | 2^897867-3217 | 270285 | Yves Minet | 12/2016 |
907 | 2^897849-115273 | 270280 | Enzo Doria | 07/2015 |
908 | 2^896616-114179 | 269909 | Enzo Doria | 12/2014 |
909 | (22*10^269517-19)/3 | 269518 | Robert Price | 07/2023 |
910 | 10^(2*134739+1)-7*10^134739-1 | 269479 | Robert Price | 07/2023 |
911 | 2^894928+583561 | 269401 | Enzo Doria | 04/2017 |
912 | 2^894673-39343 | 269324 | Enzo Doria | 08/2015 |
913 | F(1285607) | 268676 | Henri Lifchitz | 11/2008 |
914 | 2^892234+22189 | 268590 | Enzo Doria | 09/2023 |
915 | 2^892112+187663 | 268553 | Enzo Doria | 08/2016 |
916 | (266*10^268407-17)/3 | 268409 | Robert Price | 02/2023 |
917 | (10^268207+1)/11 | 268206 | Paul Bourdelais | 02/2010 |
918 | 2^890957-69583 | 268205 | Enzo Doria | 12/2015 |
919 | 32841!^2+32840!^2+...+2!^2+1!^2 | 268128 | Serge Batalov | 07/2017 |
920 | 2^890122-351401 | 267954 | Enzo Doria | 07/2015 |
921 | (19^209359-1)/18 | 267718 | Paul Bourdelais | 08/2010 |
922 | (85^138647+1)/86 | 267507 | Paul Bourdelais | 07/2019 |
923 | (485^99523-1)/484 | 267291 | Michael Stocker | 05/2020 |
924 | 2^886324+77791 | 266811 | Enzo Doria | 06/2016 |
925 | 7*10^266731+87 | 266732 | Robert Price | 11/2022 |
926 | 2^884714+1880527 | 266326 | Enzo Doria | 10/2016 |
927 | 7*10^266024-47 | 266025 | Robert Price | 08/2023 |
928 | 11^255426+255426^11 | 265999 | Serge Batalov | 05/2014 |
929 | 2^883462-343421 | 265949 | Enzo Doria | 07/2015 |
930 | 2^883442+1875019 | 265943 | Enzo Doria | 06/2016 |
931 | 2^883333-291583 | 265910 | Enzo Doria | 07/2015 |
932 | 10^265566+3593037 | 265567 | Paul Bourdelais | 08/2023 |
933 | (2*10^265321-17)/3 | 265321 | Serge Batalov | 11/2022 |
934 | (16*10^265052-91)/3 | 265053 | Robert Price | 11/2022 |
935 | 2^880411+1882673 | 265031 | Enzo Doria | 12/2016 |
936 | (14*10^264791-11)/3 | 264792 | Robert Price | 08/2022 |
937 | 2^879530-2788031 | 264765 | Enzo Doria | 12/2015 |
938 | (2*10^264539+61)/9 | 264539 | Serge Batalov | 11/2022 |
939 | 2^878118+2793079 | 264340 | Enzo Doria | 02/2016 |
940 | (2^878073-7)/5 | 264326 | Stefano Morozzi | 08/2022 |
941 | (37^168527-1)/36 | 264283 | Paul Bourdelais | 08/2019 |
942 | 5^377865+3^377865-1 | 264117 | Jon Grantham | 08/2023 |
943 | (19*10^263971+131)/3 | 263972 | Robert Price | 08/2023 |
944 | 2^876202-17 | 263764 | Stefano Morozzi | 07/2022 |
945 | (14*10^263720-11)/3 | 263721 | Robert Price | 08/2022 |
946 | 2^875748-461129 | 263627 | Enzo Doria | 07/2015 |
947 | 2^875518-15 | 263558 | Stefano Morozzi | 07/2022 |
948 | (29*10^263523-17)/3 | 263524 | Robert Price | 01/2023 |
949 | 2^875181-2789923 | 263456 | Enzo Doria | 12/2015 |
950 | 2^875101-350083 | 263432 | Enzo Doria | 08/2015 |
951 | 2^874684+1883587 | 263307 | Enzo Doria | 01/2017 |
952 | 2^874616+1872211 | 263286 | Enzo Doria | 05/2016 |
953 | 3^551491+16 | 263129 | Robert Price | 04/2020 |
954 | 2^874073+2792957 | 263123 | Enzo Doria | 02/2016 |
955 | 2^873628+1878751 | 262989 | Enzo Doria | 08/2016 |
956 | 2^873497+20837 | 262949 | Enzo Doria | 08/2023 |
957 | 2^873288-269 | 262886 | Yves Minet | 01/2015 |
958 | (2*10^262874+7)/9 | 262874 | Rytis Slatkevicius | 01/2023 |
959 | (28*10^262655-43)/3 | 262656 | Robert Price | 04/2023 |
960 | (2*10^262603-11)/9 | 262603 | Serge Batalov | 11/2022 |
961 | 2^871991-4207 | 262496 | Yves Minet | 03/2021 |
962 | (23*10^262324+7)/3 | 262325 | Robert Price | 05/2023 |
963 | 2^871146-3173 | 262242 | Yves Minet | 07/2016 |
964 | 2^869601-163 | 261776 | Yves Minet | 01/2015 |
965 | (2*10^261704+61)/9 | 261704 | Serge Batalov | 11/2022 |
966 | 2^869326+282859 | 261694 | Enzo Doria | 11/2016 |
967 | 2^869035-114181 | 261606 | Enzo Doria | 12/2014 |
968 | 4*10^261248+91 | 261249 | Robert Price | 11/2022 |
969 | 2^865715+80303 | 260607 | Enzo Doria | 07/2016 |
970 | 2^865048+1882411 | 260406 | Enzo Doria | 12/2016 |
971 | 2^864899-113871 | 260361 | Enzo Doria | 10/2014 |
972 | (16*10^260256-91)/3 | 260257 | Robert Price | 11/2022 |
973 | (7*10^260198-61)/9 | 260198 | Erik Branger | 11/2011 |
974 | 2^864298-2788421 | 260180 | Enzo Doria | 12/2015 |
975 | 2^864039-2791357 | 260102 | Enzo Doria | 01/2016 |
976 | 2^863621-113833 | 259976 | Enzo Doria | 10/2014 |
977 | 10^259945-11 | 259945 | Mischa Rodermond | 04/2023 |
978 | (28*10^259897+113)/3 | 259898 | Robert Price | 06/2023 |
979 | 2^862837+75767 | 259740 | Enzo Doria | 05/2016 |
980 | 2^862637-31 | 259680 | Jorge Menendez | 05/2019 |
981 | 2^862088+1874941 | 259515 | Enzo Doria | 06/2016 |
982 | 2^861987-134557 | 259484 | Hans Havermann | 10/2018 |
983 | (14*10^259431-143)/3 | 259432 | Robert Price | 05/2023 |
984 | 2^861516-1854029 | 259343 | Enzo Doria | 09/2015 |
985 | 2^860975-1853977 | 259180 | Enzo Doria | 09/2015 |
986 | 3^543128+16 | 259138 | Robert Price | 04/2020 |
987 | 2^860625-1858843 | 259074 | Enzo Doria | 10/2015 |
988 | 2^860110+77689 | 258919 | Enzo Doria | 05/2016 |
989 | 2^859600+1880371 | 258766 | Enzo Doria | 10/2016 |
990 | 2^859433+494711 | 258716 | Makoto Morimoto | 10/2013 |
991 | (2^859433-44)/(2^2*3) | 258715 | Makoto Morimoto | 08/2019 |
992 | (2^859431-11)/3 | 258715 | Stefano Morozzi | 08/2022 |
993 | (13*10^258710-43)/3 | 258711 | Robert Price | 08/2023 |
994 | (2^859433+39899)/(73*3907*46301) | 258705 | Makoto Morimoto | 10/2016 |
995 | 2^859240+1875271 | 258658 | Enzo Doria | 06/2016 |
996 | Tau(971^15738) | 258571 | Nik Lygeros / Olivier Rozier | 05/2015 |
997 | (23*10^258494-47)/3 | 258495 | Robert Price | 09/2023 |
998 | (144^119671+1)/145 | 258292 | Paul Bourdelais | 08/2019 |
999 | 2^857608+105 | 258166 | Stefano Morozzi | 05/2021 |
1000 | (11*2^857159-13)/15 | 258031 | Renaud Lifchitz | 08/2011 |