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 ? |
9751 | 6^127458-17993 | 99182 | Enzo Doria | 03/2013 |
9752 | 27197^4428+4428^27197 | 99166 | Hans Havermann | 10/2017 |
9753 | (46*10^99143-1)/9 | 99144 | Robert Price | 12/2014 |
9754 | (64*10^99135+161)/9 | 99136 | Robert Price | 08/2022 |
9755 | 7*10^99133+69 | 99134 | Robert Price | 10/2017 |
9756 | (21701^22861-1)/21700 | 99132 | Serge Batalov | 06/2022 |
9757 | 18^78968-3065 | 99127 | Enzo Doria | 03/2013 |
9758 | 25927^6654+6654^25927 | 99122 | Hans Havermann | 10/2017 |
9759 | 87915!4+16 | 99122 | Robert Price | 09/2019 |
9760 | 2^329259+189 | 99117 | Lelio R Paula | 12/2012 |
9761 | 8*10^99113+39 | 99114 | Robert Price | 01/2017 |
9762 | 6^127364+77 | 99109 | Lelio R Paula | 08/2016 |
9763 | (82*10^99103+143)/9 | 99104 | Robert Price | 06/2021 |
9764 | (11414^24421-1)/11413 | 99083 | Serge Batalov | 06/2022 |
9765 | (2^329135+1110)/2 | 99080 | Stefano Morozzi | 09/2020 |
9766 | (22089^22807-1)/22088 | 99074 | Serge Batalov | 06/2022 |
9767 | (59*10^99069+391)/9 | 99070 | Robert Price | 03/2021 |
9768 | (797*10^99061-41)/9 | 99063 | Robert Price | 04/2022 |
9769 | 2^329058-287681 | 99057 | Harrison Mesh | 06/2022 |
9770 | 6^127285+13 | 99047 | Lelio R Paula | 08/2016 |
9771 | (228581#+235862)/(2^3*11*71*151) | 99044 | Makoto Morimoto | 10/2017 |
9772 | 2^329007+73283 | 99041 | Enzo Doria | 04/2016 |
9773 | (145413^19321-1)/(145413^139-1) | 99030 | Ryan Propper | 11/2023 |
9774 | 5*3^207543-8 | 99024 | Lelio R Paula | 08/2014 |
9775 | (2^328930+570)/2 | 99018 | Stefano Morozzi | 09/2020 |
9776 | 2^328885+149 | 99005 | Lelio R Paula | 12/2012 |
9777 | 23392^17075+17075^23392 | 99004 | Hans Havermann | 09/2017 |
9778 | (7424^25579-1)/7423 | 99004 | Serge Batalov | 06/2022 |
9779 | (2^328879-9)/119 | 99001 | Lelio R Paula | 10/2013 |
9780 | 5^141618+54 | 98987 | Lelio R Paula | 08/2016 |
9781 | 2^328824+24391 | 98986 | Enzo Doria | 10/2023 |
9782 | 8*10^98982+51 | 98983 | Robert Price | 05/2017 |
9783 | (21*2^328809-1)/7709 | 98979 | Lelio R Paula | 03/2014 |
9784 | 2^328794-2797601 | 98977 | Enzo Doria | 04/2016 |
9785 | (88331^20011-1)/88330 | 98972 | Serge Batalov | 06/2022 |
9786 | 6^127176+5 | 98963 | Lelio R Paula | 11/2014 |
9787 | 2*10^98961-21 | 98962 | Robert Price | 09/2020 |
9788 | 24262^11989+11989^24262 | 98960 | Hans Havermann | 09/2017 |
9789 | (60138^20707-1)/60137 | 98958 | Serge Batalov | 06/2022 |
9790 | 2^328700-109709 | 98949 | Enzo Doria | 04/2014 |
9791 | (2^328698+1810)/2 | 98948 | Stefano Morozzi | 01/2021 |
9792 | (1014^32917-1)/1013 | 98947 | Serge Batalov | 06/2022 |
9793 | 87771!4-128 | 98944 | Robert Price | 11/2019 |
9794 | (4576^27031-1)/4575 | 98943 | Serge Batalov | 06/2022 |
9795 | (2^328658+3042)/2 | 98936 | Stefano Morozzi | 09/2020 |
9796 | (1933^30103-1)/1932 | 98923 | Serge Batalov | 06/2022 |
9797 | 46711!2-2^46711 | 98916 | Robert Price | 09/2015 |
9798 | 9*7^117037+4 | 98909 | Lelio R Paula | 11/2014 |
9799 | 127^47012+126 | 98905 | Norbert Schneider | 01/2019 |
9800 | (93826^19891-1)/93825 | 98900 | Serge Batalov | 06/2022 |
9801 | (2^328525+1250)/2 | 98896 | Stefano Morozzi | 09/2020 |
9802 | (10^98890+149)/3 | 98890 | Serge Batalov | 12/2014 |
9803 | 22775^21978+21978^22775 | 98889 | Hans Havermann | 08/2017 |
9804 | 2^328497-216933 | 98888 | Enzo Doria | 05/2014 |
9805 | 2^328487+32769 | 98885 | Lelio R Paula | 08/2016 |
9806 | 3^207250-112358 | 98884 | Enzo Doria | 06/2014 |
9807 | 13^88754-90 | 98867 | Lelio R Paula | 08/2016 |
9809 | 2^328406+709 | 98861 | Lelio R Paula | 02/2013 |
9808 | 2*10^98860-21 | 98861 | Robert Price | 09/2020 |
9810 | (17*2^328413-1)/16173 | 98860 | Lelio R Paula | 03/2014 |
9811 | 7^116976+76 | 98857 | Lelio R Paula | 08/2016 |
9812 | (2^328378+2710)/2 | 98852 | Stefano Morozzi | 09/2020 |
9813 | (25411^22441-1)/25410 | 98849 | Serge Batalov | 06/2022 |
9814 | (18668^23143-1)/18667 | 98842 | Serge Batalov | 06/2022 |
9816 | 2^328335+1155 | 98839 | Lelio R Paula | 08/2016 |
9815 | 2^328333-349003 | 98839 | Harrison Mesh | 06/2022 |
9818 | (3*2^328311+1)/5 | 98832 | Lelio R Paula | 11/2013 |
9817 | 287^40210+40210^287 | 98832 | Norbert Schneider | 11/2014 |
9819 | (2^328284+9)/1325 | 98821 | Lelio R Paula | 10/2013 |
9820 | (21*2^328258+1)/2125 | 98814 | Lelio R Paula | 11/2013 |
9821 | (28623^22171-1)/28622 | 98806 | Serge Batalov | 06/2022 |
9822 | 24746^9825+9825^24746 | 98795 | Hans Havermann | 07/2020 |
9823 | 26620^5143+5143^26620 | 98793 | Hans Havermann | 07/2020 |
9824 | 65^54491+48 | 98788 | Lelio R Paula | 04/2016 |
9825 | 3^207042+98 | 98785 | Lelio R Paula | 08/2016 |
9826 | (2^328121+3110)/2 | 98774 | Stefano Morozzi | 09/2020 |
9827 | 10^98754+73383 | 98755 | Patrick De Geest | 02/2021 |
9830 | 3^206977-112756 | 98754 | Enzo Doria | 07/2014 |
9829 | 10^98754-40000000001 | 98754 | Robert Price | 09/2020 |
9828 | 10^98754-169343 | 98754 | Patrick De Geest | 02/2021 |
9831 | 7^116846-50 | 98747 | Lelio R Paula | 08/2016 |
9832 | (19152^23059-1)/19151 | 98740 | Serge Batalov | 06/2022 |
9833 | 23267^17512+17512^23267 | 98730 | Hans Havermann | 04/2020 |
9834 | 2^327967-28635 | 98728 | Lelio R Paula | 08/2016 |
9835 | 707^34642+34642^707 | 98710 | Norbert Schneider | 12/2017 |
9836 | (10771^24481-1)/10770 | 98710 | Serge Batalov | 06/2022 |
9837 | 6^126850-47 | 98709 | Lelio R Paula | 08/2016 |
9838 | (2^327886+2170)/2 | 98704 | Stefano Morozzi | 09/2020 |
9839 | (22*10^98697+47)/3 | 98698 | Robert Price | 06/2021 |
9840 | (139706^19321-1)/(139706^139-1) | 98696 | Ryan Propper | 11/2023 |
9841 | 8*6^126830-7 | 98694 | Lelio R Paula | 11/2014 |
9842 | 11^94770-270 | 98693 | Enzo Doria | 08/2012 |
9843 | (2^327846+3250)/2 | 98692 | Stefano Morozzi | 01/2021 |
9844 | 2^327832+43 | 98688 | Lelio R Paula | 12/2012 |
9845 | 2^327795+761 | 98677 | Lelio R Paula | 02/2013 |
9846 | (2^327796+1522)/2 | 98677 | Stefano Morozzi | 09/2020 |
9847 | 10^98669+147411 | 98670 | Patrick De Geest | 11/2016 |
9849 | 10^98669-9^(98669-1) | 98669 | Robert Price | 05/2016 |
9848 | 10^98669-105249 | 98669 | Patrick De Geest | 11/2016 |
9850 | (7911^25309-1)/7910 | 98657 | Serge Batalov | 06/2022 |
9851 | 2^327671-109801 | 98639 | Enzo Doria | 04/2014 |
9852 | 24398^11037+11037^24398 | 98638 | Hans Havermann | 10/2019 |
9853 | 5^141090-14 | 98618 | Lelio R Paula | 08/2016 |
9854 | 20^75799+47 | 98617 | Lelio R Paula | 08/2016 |
9855 | (5452^26393-1)/5451 | 98616 | Serge Batalov | 06/2022 |
9856 | 28101^3226+3226^28101 | 98597 | Hans Havermann | 07/2020 |
9857 | 2^327515-1017 | 98592 | Lelio R Paula | 08/2016 |
9858 | 6^126692+97681 | 98586 | Mike Oakes | 04/2011 |
9859 | 2^327467-741 | 98578 | Lelio R Paula | 08/2016 |
9860 | (2947^28411-1)/2946 | 98566 | Serge Batalov | 06/2022 |
9861 | 22713^21800+21800^22713 | 98540 | Hans Havermann | 05/2020 |
9862 | (2^327329+9)/19 | 98535 | Lelio R Paula | 10/2013 |
9863 | 7^116577-66 | 98519 | Lelio R Paula | 08/2016 |
9865 | (3*2^327250+1)/7 | 98512 | Lelio R Paula | 11/2013 |
9864 | 23220^17479+17479^23220 | 98512 | Hans Havermann | 03/2020 |
9866 | (2^327240+3798)/2 | 98509 | Stefano Morozzi | 02/2021 |
9867 | (28514^22111-1)/28513 | 98502 | Serge Batalov | 06/2022 |
9868 | (19*2^327192+1)/275 | 98494 | Lelio R Paula | 11/2013 |
9869 | 2^327162-66945 | 98486 | Lelio R Paula | 08/2016 |
9870 | 30139^1852+1852^30139 | 98484 | Hans Havermann | 02/2020 |
9871 | 23182^17703+17703^23182 | 98479 | Hans Havermann | 01/2020 |
9872 | 4*10^98472-3 | 98473 | Erik Branger | 11/2013 |
9873 | 26167^5794+5794^26167 | 98466 | Hans Havermann | 01/2020 |
9874 | (6883*2^327047-1)/77 | 98453 | Lelio R Paula | 05/2015 |
9875 | (71*10^98446+1)/9 | 98447 | Serge Batalov | 11/2010 |
9876 | (2009^29803-1)/2008 | 98436 | Serge Batalov | 06/2022 |
9877 | 13^88363+94 | 98432 | Lelio R Paula | 08/2016 |
9878 | 48^58543-47^58543 | 98425 | Jean-Louis Charton | 11/2005 |
9879 | 6^126481+41 | 98422 | Lelio R Paula | 08/2016 |
9880 | 78*10^98413-7 | 98415 | Robert Price | 05/2017 |
9881 | 22634^22263+22263^22634 | 98404 | Hans Havermann | 07/2020 |
9882 | (47*10^98386+241)/9 | 98387 | Robert Price | 06/2022 |
9883 | 7^116416+46 | 98383 | Lelio R Paula | 08/2016 |
9884 | 7^116407-24 | 98376 | Lelio R Paula | 07/2016 |
9885 | 439^37228+37228^439 | 98374 | Norbert Schneider | 08/2018 |
9886 | 2^326782-1860671 | 98372 | Enzo Doria | 03/2016 |
9887 | 17^79948+58 | 98372 | Lelio R Paula | 08/2016 |
9888 | 22499^22602-22602^22499 | 98368 | Norbert Schneider | 08/2020 |
9889 | 3^206143-98 | 98356 | Lelio R Paula | 08/2016 |
9890 | 8*5^140705+7 | 98350 | Lelio R Paula | 11/2014 |
9891 | (2^326632+1662)/2 | 98326 | Stefano Morozzi | 09/2020 |
9892 | (638^35053-1)/637 | 98315 | Serge Batalov | 06/2022 |
9893 | 2^326585+11 | 98312 | Lelio R Paula | 12/2012 |
9894 | 2^326582-459161 | 98311 | Enzo Doria | 05/2015 |
9895 | (1266^31687-1)/1265 | 98304 | Serge Batalov | 06/2022 |
9897 | (2^326548+9)/5 | 98301 | Lelio R Paula | 10/2013 |
9896 | (2^326549+18)/10 | 98301 | Stefano Morozzi | 03/2020 |
9898 | 2^326538+71659 | 98298 | Enzo Doria | 04/2016 |
9899 | (2^326520+2082)/2 | 98293 | Stefano Morozzi | 09/2020 |
9900 | (2^326521+1002)/2 | 98293 | Stefano Morozzi | 09/2020 |
9901 | 15^83575-58 | 98292 | Lelio R Paula | 08/2016 |
9902 | 10^98288+3 | 98289 | Robert Price | 03/2011 |
9903 | 10^98288-117017 | 98288 | Patrick De Geest | 11/2011 |
9904 | (44*10^98284-719)/9 | 98285 | Robert Price | 12/2017 |
9905 | (5985^26021-1)/5984 | 98280 | Serge Batalov | 06/2022 |
9906 | 23466^15419+15419^23466 | 98277 | Hans Havermann | 02/2020 |
9907 | 2^326421-2797573 | 98263 | Enzo Doria | 04/2016 |
9908 | 2^326401+16095 | 98257 | Lelio R Paula | 07/2016 |
9909 | (6494^25771-1)/6493 | 98249 | Serge Batalov | 06/2022 |
9910 | 7^116240-116240^7 | 98235 | Norbert Schneider | 05/2017 |
9911 | 17^79826-78 | 98222 | Lelio R Paula | 08/2016 |
9912 | 2^326268-809 | 98217 | Yves Minet | 04/2011 |
9913 | 22659^21590+21590^22659 | 98210 | Hans Havermann | 10/2019 |
9914 | 22268^22587-22587^22268 | 98202 | Norbert Schneider | 08/2020 |
9915 | (418*10^98199+23)/9 | 98201 | Robert Price | 08/2017 |
9916 | 12*10^98198-7 | 98200 | Robert Price | 11/2017 |
9917 | 15^83496-76 | 98199 | Lelio R Paula | 08/2016 |
9918 | (10223*2^326179+1)/6125 | 98190 | Lelio R Paula | 07/2014 |
9919 | 9*5^140471-4 | 98186 | Lelio R Paula | 11/2014 |
9920 | (35103^21601-1)/35102 | 98180 | Serge Batalov | 06/2022 |
9922 | 22774^20455+20455^22774 | 98175 | Hans Havermann | 07/2020 |
9921 | (4516^26863-1)/4515 | 98175 | Serge Batalov | 06/2022 |
9923 | 2^326069-2383 | 98157 | Yves Minet | 10/2013 |
9924 | (9661^24631-1)/9660 | 98152 | Serge Batalov | 06/2022 |
9925 | (4175^27109-1)/4174 | 98149 | Serge Batalov | 06/2022 |
9926 | (57801^20611-1)/57800 | 98144 | Serge Batalov | 06/2022 |
9927 | 2^325973+1185 | 98128 | Lelio R Paula | 07/2016 |
9928 | (25*2^325949-1)/17 | 98121 | Lelio R Paula | 02/2014 |
9929 | 2^325945-1847269 | 98120 | Enzo Doria | 07/2015 |
9930 | 8*10^98111-67 | 98112 | Robert Price | 10/2017 |
9931 | (2^325902+1138)/2 | 98106 | Stefano Morozzi | 09/2020 |
9932 | (215^42061-1)/214 | 98103 | Serge Batalov | 06/2022 |
9933 | (6775^25609-1)/6774 | 98102 | Serge Batalov | 06/2022 |
9934 | (6718^25633-1)/6717 | 98100 | Serge Batalov | 06/2022 |
9935 | 2^325866+15 | 98096 | Lelio R Paula | 12/2012 |
9936 | 22591^21952+21952^22591 | 98079 | Hans Havermann | 02/2020 |
9937 | 2^325765+1974305 | 98066 | Enzo Doria | 06/2016 |
9938 | 2^325733-58695 | 98056 | Lelio R Paula | 07/2016 |
9939 | (7*10^98055-529)/9 | 98055 | Robert Price | 08/2021 |
9940 | 2^325717-3645 | 98051 | Lelio R Paula | 07/2016 |
9941 | (2^325704+718)/2 | 98047 | Stefano Morozzi | 09/2020 |
9942 | (15679^23371-1)/15678 | 98045 | Serge Batalov | 06/2022 |
9943 | (23342^22441-1)/23341 | 98022 | Serge Batalov | 06/2022 |
9944 | 23968^12293+12293^23968 | 98021 | Hans Havermann | 11/2019 |
9946 | (23*2^325583-1)/21 | 98011 | Lelio R Paula | 02/2014 |
9945 | (128678^19321-1)/(128678^139-1) | 98011 | Ryan Propper | 11/2023 |
9947 | (7*10^98007-13)/3 | 98008 | Robert Price | 06/2016 |
9948 | (2^325544+1110)/2 | 97999 | Stefano Morozzi | 09/2020 |
9949 | 2^325512-459959 | 97989 | Enzo Doria | 05/2015 |
9950 | (2^325492+682)/2 | 97983 | Stefano Morozzi | 09/2020 |
9951 | (3^205354-8)/17 | 97978 | Lelio R Paula | 09/2013 |
9952 | 2^325433+76817 | 97966 | Enzo Doria | 05/2016 |
9953 | 5*10^97963+81 | 97964 | Robert Price | 08/2017 |
9954 | (9996^24649-1)/(9996^157-1) | 97964 | Ryan Propper | 11/2023 |
9955 | (13208^23773-1)/13207 | 97961 | Serge Batalov | 06/2022 |
9956 | 2^325390-109061 | 97953 | Enzo Doria | 04/2014 |
9957 | (55*10^97940+323)/9 | 97941 | Robert Price | 10/2021 |
9958 | 13^87921+24 | 97940 | Lelio R Paula | 08/2016 |
9959 | 10^97930+188277 | 97931 | Patrick De Geest | 09/2022 |
9961 | 10^97930-7000001 | 97930 | Robert Price | 06/2021 |
9960 | 10^97930-66951 | 97930 | Patrick De Geest | 09/2022 |
9962 | 961^32831-960^32831 | 97926 | Jean-Louis Charton | 02/2011 |
9963 | (17968^23017-1)/17967 | 97922 | Serge Batalov | 06/2022 |
9964 | (3337^27793-1)/3336 | 97922 | Serge Batalov | 06/2022 |
9966 | 22600^21499+21499^22600 | 97913 | Hans Havermann | 05/2020 |
9965 | (18703^22921-1)/18702 | 97913 | Serge Batalov | 06/2022 |
9967 | 2^325215+12909 | 97900 | Lelio R Paula | 07/2016 |
9968 | 22619^21286+21286^22619 | 97898 | Hans Havermann | 04/2020 |
9969 | (126852^19321-1)/(126852^139-1) | 97892 | Ryan Propper | 11/2023 |
9970 | (85*10^97876-49)/9 | 97877 | Robert Price | 12/2020 |
9971 | 7*10^97874-69 | 97875 | Robert Price | 01/2017 |
9972 | 5*10^97869-63 | 97870 | Robert Price | 04/2021 |
9973 | (643*32^65022-23)/31 | 97870 | Eric Chen | 04/2023 |
9975 | (17^79537+5^79537)/22 | 97865 | Robert Price | 06/2013 |
9974 | 15^83212+22 | 97865 | Lelio R Paula | 08/2016 |
9976 | 2^325087-109057 | 97861 | Enzo Doria | 04/2014 |
9977 | 2^325074+1870657 | 97858 | Enzo Doria | 05/2016 |
9979 | 2^325066-2111 | 97855 | Yves Minet | 08/2013 |
9978 | CopelandErdos(97855) | 97855 | Eric W. Weisstein | 10/2015 |
9980 | (8694^24841-1)/8693 | 97851 | Serge Batalov | 06/2022 |
9981 | (21159^22621-1)/21158 | 97843 | Serge Batalov | 06/2022 |
9982 | 26660^4677+4677^26660 | 97842 | Hans Havermann | 12/2019 |
9983 | 18^77944-89 | 97841 | Lelio R Paula | 07/2016 |
9984 | 119^47136+118 | 97833 | Norbert Schneider | 09/2018 |
9985 | 6^125720-17663 | 97830 | Enzo Doria | 02/2013 |
9986 | (73640^20101-1)/73639 | 97829 | Serge Batalov | 06/2022 |
9987 | 2^324972-2069 | 97827 | Yves Minet | 05/2013 |
9988 | 2^324958-4691 | 97823 | Yves Minet | 05/2019 |
9989 | (8087^25033-1)/8086 | 97820 | Serge Batalov | 06/2022 |
9990 | 2^324944-2788319 | 97818 | Enzo Doria | 03/2016 |
9992 | 22845^19112+19112^22845 | 97807 | Hans Havermann | 10/2019 |
9991 | (317^39107-1)/316 | 97807 | Anonymous | 04/2024 |
9993 | (2^324898+1270)/2 | 97804 | Stefano Morozzi | 09/2020 |
9995 | 4*10^97799-9 | 97800 | Erik Branger | 11/2013 |
9994 | 2^324882+577294575 | 97800 | Lelio R Paula | 09/2016 |
9996 | 2^324863-1867921 | 97794 | Enzo Doria | 04/2016 |
9997 | 6^125662-18473 | 97785 | Enzo Doria | 03/2013 |
9998 | 12^90604+11 | 97779 | Lelio R Paula | 07/2016 |
9999 | (29*10^97751+49)/3 | 97752 | Robert Price | 09/2021 |
10000 | (37108^21391-1)/37107 | 97741 | Serge Batalov | 06/2022 |