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 ? |
8751 | 2^331746-11655 | 99866 | Lelio R Paula | 09/2016 |
8752 | 11^95891-28 | 99861 | Lelio R Paula | 08/2016 |
8753 | (2^331694-5)/11 | 99849 | Lelio R Paula | 10/2013 |
8754 | (3^209246+7)/64 | 99834 | Lelio R Paula | 09/2013 |
8755 | 25062^9617+9617^25062 | 99823 | Hans Havermann | 04/2018 |
8756 | 15^84866-52 | 99811 | Lelio R Paula | 08/2016 |
8757 | 13^89600+30 | 99810 | Lelio R Paula | 08/2016 |
8758 | 2^331527+100280245065 | 99800 | Paul Bourdelais | 10/2021 |
8759 | 22950^22309+22309^22950 | 99798 | Hans Havermann | 03/2018 |
8760 | 5^142772-38 | 99794 | Lelio R Paula | 08/2016 |
8761 | (7482^25759-1)/7481 | 99787 | Serge Batalov | 06/2022 |
8762 | (23*2^331459+1)/475 | 99778 | Lelio R Paula | 11/2013 |
8763 | 2^331416-331416+1 | 99767 | Robert Price | 10/2014 |
8764 | 2^331256-57 | 99718 | Lelio R Paula | 12/2012 |
8765 | 28192^3443+3443^28192 | 99714 | Hans Havermann | 03/2018 |
8766 | 2^331185+737 | 99697 | Lelio R Paula | 01/2013 |
8767 | (4*10^99693+149)/9 | 99693 | Robert Price | 03/2017 |
8768 | 2^331154+1880527 | 99688 | Enzo Doria | 10/2016 |
8770 | (19^77951+1)/20 | 99679 | Paul Bourdelais | 03/2010 |
8769 | 24354^12385+12385^24354 | 99679 | Hans Havermann | 03/2018 |
8771 | 2^331110+1870459 | 99675 | Enzo Doria | 05/2016 |
8772 | 5*2^331075+7 | 99665 | Lelio R Paula | 08/2014 |
8773 | 25776^7351+7351^25776 | 99659 | Hans Havermann | 02/2018 |
8774 | (16*10^99656-31)/3 | 99657 | Robert Price | 03/2016 |
8775 | (6059^26347-1)/6058 | 99652 | Serge Batalov | 06/2022 |
8776 | (15544^23773-1)/15543 | 99642 | Serge Batalov | 06/2022 |
8778 | 2^330968+345 | 99632 | Lelio R Paula | 09/2016 |
8777 | (2^330969+690)/2 | 99632 | Stefano Morozzi | 09/2020 |
8780 | 2*5^142528-7 | 99624 | Lelio R Paula | 11/2014 |
8779 | 2^330942+77329 | 99624 | Enzo Doria | 05/2016 |
8781 | 14^86914-99 | 99615 | Lelio R Paula | 08/2016 |
8782 | (19*2^330902+1)/77 | 99611 | Lelio R Paula | 11/2013 |
8783 | 22971^21692+21692^22971 | 99610 | Hans Havermann | 03/2018 |
8784 | 2^330845+71687 | 99595 | Enzo Doria | 04/2016 |
8785 | (26667^22501-1)/26666 | 99585 | Serge Batalov | 06/2022 |
8786 | (23*2^330770+1)/3 | 99573 | Lelio R Paula | 11/2013 |
8787 | 24494^11613+11613^24494 | 99567 | Hans Havermann | 03/2018 |
8788 | 2^330735+8415 | 99562 | Lelio R Paula | 09/2016 |
8789 | 2^330720+74071 | 99557 | Enzo Doria | 04/2016 |
8790 | (10405^24781-1)/10404 | 99548 | Serge Batalov | 06/2022 |
8792 | 31^66738-2 | 99531 | Lelio R Paula | 09/2013 |
8791 | (9002^25171-1)/9001 | 99531 | Serge Batalov | 06/2022 |
8793 | (5023^26893-1)/5022 | 99527 | Serge Batalov | 06/2022 |
8794 | 7*3^208567+4 | 99513 | Lelio R Paula | 08/2014 |
8795 | (15881^23689-1)/15880 | 99511 | Serge Batalov | 06/2022 |
8796 | (13*10^99508+47)/3 | 99509 | Robert Price | 01/2017 |
8797 | 2^330553+779 | 99507 | Lelio R Paula | 01/2013 |
8798 | 15^84606-94 | 99505 | Lelio R Paula | 08/2016 |
8799 | (10^99497+173)/3 | 99497 | Robert Price | 05/2021 |
8800 | 9*10^99493-43 | 99494 | Robert Price | 09/2021 |
8801 | 6^127858+47 | 99493 | Lelio R Paula | 08/2016 |
8802 | 5^142337+48 | 99490 | Lelio R Paula | 08/2016 |
8803 | 7^117724-20 | 99489 | Lelio R Paula | 08/2016 |
8804 | (559^36209-1)/558 | 99479 | Serge Batalov | 06/2022 |
8805 | (13470^24091-1)/13469 | 99477 | Serge Batalov | 06/2022 |
8806 | (29*10^99474-17)/3 | 99475 | Erik Branger | 09/2014 |
8807 | (185*10^99466-11)/3 | 99468 | Robert Price | 10/2020 |
8808 | (2^330410+2038)/2 | 99464 | Stefano Morozzi | 09/2020 |
8809 | (9*2^330407-1)/7 | 99463 | Lelio R Paula | 03/2014 |
8810 | (516*32^66077-175)/31 | 99458 | Eric Chen | 04/2023 |
8811 | (2863^28771-1)/2862 | 99453 | Serge Batalov | 06/2022 |
8812 | (2^330373+2114)/2 | 99452 | Stefano Morozzi | 09/2020 |
8813 | (68*10^99441+7)/3 | 99443 | Robert Price | 03/2016 |
8814 | 10^99434+155571 | 99435 | Patrick De Geest | 04/2020 |
8815 | 10^99434-123456789 | 99434 | Robert Price | 02/2020 |
8816 | 10^99434-191237 | 99434 | Patrick De Geest | 04/2020 |
8817 | 46^59791-45^59791 | 99418 | Jean-Louis Charton | 01/2013 |
8818 | 23839^14802+14802^23839 | 99417 | Hans Havermann | 02/2018 |
8819 | (2*10^99408+529)/9 | 99408 | Robert Price | 01/2017 |
8820 | (2^330216+3510)/2 | 99405 | Stefano Morozzi | 09/2020 |
8821 | 53^57646-2 | 99398 | Lelio R Paula | 09/2013 |
8822 | 2^330165+1870025 | 99390 | Enzo Doria | 05/2016 |
8823 | 7*10^99383+97 | 99384 | Robert Price | 09/2021 |
8824 | 2^330142-155 | 99383 | Lelio R Paula | 12/2012 |
8825 | 2^330088-115259 | 99367 | Enzo Doria | 06/2015 |
8826 | 24216^12671+12671^24216 | 99354 | Hans Havermann | 01/2018 |
8827 | Phi(200548,10)/3208769 | 99354 | Makoto Kamada | 04/2019 |
8828 | (746*10^99303+1)/9 | 99305 | Robert Price | 02/2017 |
8829 | 3^208107+40 | 99293 | Lelio R Paula | 08/2016 |
8830 | 2^329843+211343 | 99293 | Enzo Doria | 08/2023 |
8831 | (3939^27617-1)/3938 | 99291 | Serge Batalov | 06/2022 |
8832 | (5*2^329833-1)/369 | 99288 | Lelio R Paula | 02/2014 |
8833 | 8*3^208093+5 | 99287 | Lelio R Paula | 08/2014 |
8834 | 5^142043+56 | 99284 | Lelio R Paula | 08/2016 |
8835 | (13*10^99282-157)/9 | 99283 | Robert Price | 05/2021 |
8836 | (10^99284+7)/53/67 | 99281 | Alfred Reich | 06/2021 |
8837 | 22*10^99275+7 | 99277 | Robert Price | 04/2016 |
8838 | 2^329778+1876189 | 99274 | Enzo Doria | 07/2016 |
8839 | (6580^25997-1)/6579 | 99259 | Serge Batalov | 06/2022 |
8840 | (2249^29611-1)/2248 | 99253 | Serge Batalov | 06/2022 |
8841 | (10^99239-13)/3 | 99239 | Robert Price | 02/2017 |
8842 | (1199^32233-1)/1198 | 99237 | Serge Batalov | 06/2022 |
8844 | 3^207987+62 | 99236 | Lelio R Paula | 08/2016 |
8843 | (62*10^99235+91)/9 | 99236 | Robert Price | 07/2021 |
8845 | 2^329638+49 | 99231 | Lelio R Paula | 12/2012 |
8846 | 7^117413-18 | 99226 | Lelio R Paula | 08/2016 |
8847 | 2^329586-458903 | 99216 | Enzo Doria | 04/2015 |
8848 | (6290^26119-1)/6289 | 99214 | Serge Batalov | 06/2022 |
8850 | 2^329553+101 | 99206 | Lelio R Paula | 12/2012 |
8849 | (21297^22921-1)/21296 | 99206 | Serge Batalov | 06/2022 |
8851 | 2^329551-112897 | 99205 | Enzo Doria | 06/2014 |
8852 | (3^207925+7)/50 | 99204 | Lelio R Paula | 09/2013 |
8853 | 6^127458-17993 | 99182 | Enzo Doria | 03/2013 |
8854 | 27197^4428+4428^27197 | 99166 | Hans Havermann | 10/2017 |
8855 | (46*10^99143-1)/9 | 99144 | Robert Price | 12/2014 |
8856 | (64*10^99135+161)/9 | 99136 | Robert Price | 08/2022 |
8857 | 7*10^99133+69 | 99134 | Robert Price | 10/2017 |
8858 | (21701^22861-1)/21700 | 99132 | Serge Batalov | 06/2022 |
8859 | 18^78968-3065 | 99127 | Enzo Doria | 03/2013 |
8861 | 25927^6654+6654^25927 | 99122 | Hans Havermann | 10/2017 |
8860 | 87915!4+16 | 99122 | Robert Price | 09/2019 |
8862 | 2^329259+189 | 99117 | Lelio R Paula | 12/2012 |
8863 | 8*10^99113+39 | 99114 | Robert Price | 01/2017 |
8864 | 6^127364+77 | 99109 | Lelio R Paula | 08/2016 |
8865 | (82*10^99103+143)/9 | 99104 | Robert Price | 06/2021 |
8866 | (11414^24421-1)/11413 | 99083 | Serge Batalov | 06/2022 |
8867 | (2^329135+1110)/2 | 99080 | Stefano Morozzi | 09/2020 |
8868 | (22089^22807-1)/22088 | 99074 | Serge Batalov | 06/2022 |
8869 | (59*10^99069+391)/9 | 99070 | Robert Price | 03/2021 |
8870 | (797*10^99061-41)/9 | 99063 | Robert Price | 04/2022 |
8871 | 2^329058-287681 | 99057 | Harrison Mesh | 06/2022 |
8872 | 6^127285+13 | 99047 | Lelio R Paula | 08/2016 |
8873 | (228581#+235862)/(2^3*11*71*151) | 99044 | Makoto Morimoto | 10/2017 |
8874 | 2^329007+73283 | 99041 | Enzo Doria | 04/2016 |
8875 | 5*3^207543-8 | 99024 | Lelio R Paula | 08/2014 |
8876 | (2^328930+570)/2 | 99018 | Stefano Morozzi | 09/2020 |
8877 | 2^328885+149 | 99005 | Lelio R Paula | 12/2012 |
8879 | 23392^17075+17075^23392 | 99004 | Hans Havermann | 09/2017 |
8878 | (7424^25579-1)/7423 | 99004 | Serge Batalov | 06/2022 |
8880 | (2^328879-9)/119 | 99001 | Lelio R Paula | 10/2013 |
8881 | 5^141618+54 | 98987 | Lelio R Paula | 08/2016 |
8882 | 8*10^98982+51 | 98983 | Robert Price | 05/2017 |
8883 | (21*2^328809-1)/7709 | 98979 | Lelio R Paula | 03/2014 |
8884 | 2^328794-2797601 | 98977 | Enzo Doria | 04/2016 |
8885 | (88331^20011-1)/88330 | 98972 | Serge Batalov | 06/2022 |
8886 | 6^127176+5 | 98963 | Lelio R Paula | 11/2014 |
8887 | 2*10^98961-21 | 98962 | Robert Price | 09/2020 |
8888 | 24262^11989+11989^24262 | 98960 | Hans Havermann | 09/2017 |
8889 | (60138^20707-1)/60137 | 98958 | Serge Batalov | 06/2022 |
8890 | 2^328700-109709 | 98949 | Enzo Doria | 04/2014 |
8891 | (2^328698+1810)/2 | 98948 | Stefano Morozzi | 01/2021 |
8892 | (1014^32917-1)/1013 | 98947 | Serge Batalov | 06/2022 |
8893 | 87771!4-128 | 98944 | Robert Price | 11/2019 |
8894 | (4576^27031-1)/4575 | 98943 | Serge Batalov | 06/2022 |
8895 | (2^328658+3042)/2 | 98936 | Stefano Morozzi | 09/2020 |
8896 | (1933^30103-1)/1932 | 98923 | Serge Batalov | 06/2022 |
8897 | 46711!2-2^46711 | 98916 | Robert Price | 09/2015 |
8898 | 9*7^117037+4 | 98909 | Lelio R Paula | 11/2014 |
8899 | 127^47012+126 | 98905 | Norbert Schneider | 01/2019 |
8900 | (93826^19891-1)/93825 | 98900 | Serge Batalov | 06/2022 |
8901 | (2^328525+1250)/2 | 98896 | Stefano Morozzi | 09/2020 |
8902 | (10^98890+149)/3 | 98890 | Serge Batalov | 12/2014 |
8903 | 22775^21978+21978^22775 | 98889 | Hans Havermann | 08/2017 |
8904 | 2^328497-216933 | 98888 | Enzo Doria | 05/2014 |
8905 | 2^328487+32769 | 98885 | Lelio R Paula | 08/2016 |
8906 | 3^207250-112358 | 98884 | Enzo Doria | 06/2014 |
8907 | 13^88754-90 | 98867 | Lelio R Paula | 08/2016 |
8908 | 2^328406+709 | 98861 | Lelio R Paula | 02/2013 |
8909 | 2*10^98860-21 | 98861 | Robert Price | 09/2020 |
8910 | (17*2^328413-1)/16173 | 98860 | Lelio R Paula | 03/2014 |
8911 | 7^116976+76 | 98857 | Lelio R Paula | 08/2016 |
8912 | (2^328378+2710)/2 | 98852 | Stefano Morozzi | 09/2020 |
8913 | (25411^22441-1)/25410 | 98849 | Serge Batalov | 06/2022 |
8914 | (18668^23143-1)/18667 | 98842 | Serge Batalov | 06/2022 |
8916 | 2^328335+1155 | 98839 | Lelio R Paula | 08/2016 |
8915 | 2^328333-349003 | 98839 | Harrison Mesh | 06/2022 |
8918 | (3*2^328311+1)/5 | 98832 | Lelio R Paula | 11/2013 |
8917 | 287^40210+40210^287 | 98832 | Norbert Schneider | 11/2014 |
8919 | (2^328284+9)/1325 | 98821 | Lelio R Paula | 10/2013 |
8920 | (21*2^328258+1)/2125 | 98814 | Lelio R Paula | 11/2013 |
8921 | (28623^22171-1)/28622 | 98806 | Serge Batalov | 06/2022 |
8922 | 24746^9825+9825^24746 | 98795 | Hans Havermann | 07/2020 |
8923 | 26620^5143+5143^26620 | 98793 | Hans Havermann | 07/2020 |
8924 | 65^54491+48 | 98788 | Lelio R Paula | 04/2016 |
8925 | 3^207042+98 | 98785 | Lelio R Paula | 08/2016 |
8926 | (2^328121+3110)/2 | 98774 | Stefano Morozzi | 09/2020 |
8927 | 10^98754+73383 | 98755 | Patrick De Geest | 02/2021 |
8930 | 3^206977-112756 | 98754 | Enzo Doria | 07/2014 |
8929 | 10^98754-40000000001 | 98754 | Robert Price | 09/2020 |
8928 | 10^98754-169343 | 98754 | Patrick De Geest | 02/2021 |
8931 | 7^116846-50 | 98747 | Lelio R Paula | 08/2016 |
8932 | (19152^23059-1)/19151 | 98740 | Serge Batalov | 06/2022 |
8933 | 23267^17512+17512^23267 | 98730 | Hans Havermann | 04/2020 |
8934 | 2^327967-28635 | 98728 | Lelio R Paula | 08/2016 |
8935 | 707^34642+34642^707 | 98710 | Norbert Schneider | 12/2017 |
8936 | (10771^24481-1)/10770 | 98710 | Serge Batalov | 06/2022 |
8937 | 6^126850-47 | 98709 | Lelio R Paula | 08/2016 |
8938 | (2^327886+2170)/2 | 98704 | Stefano Morozzi | 09/2020 |
8939 | (22*10^98697+47)/3 | 98698 | Robert Price | 06/2021 |
8940 | 8*6^126830-7 | 98694 | Lelio R Paula | 11/2014 |
8941 | 11^94770-270 | 98693 | Enzo Doria | 08/2012 |
8942 | (2^327846+3250)/2 | 98692 | Stefano Morozzi | 01/2021 |
8943 | 2^327832+43 | 98688 | Lelio R Paula | 12/2012 |
8944 | 2^327795+761 | 98677 | Lelio R Paula | 02/2013 |
8945 | (2^327796+1522)/2 | 98677 | Stefano Morozzi | 09/2020 |
8946 | 10^98669+147411 | 98670 | Patrick De Geest | 11/2016 |
8948 | 10^98669-9^(98669-1) | 98669 | Robert Price | 05/2016 |
8947 | 10^98669-105249 | 98669 | Patrick De Geest | 11/2016 |
8949 | (7911^25309-1)/7910 | 98657 | Serge Batalov | 06/2022 |
8950 | 2^327671-109801 | 98639 | Enzo Doria | 04/2014 |
8951 | 24398^11037+11037^24398 | 98638 | Hans Havermann | 10/2019 |
8952 | 5^141090-14 | 98618 | Lelio R Paula | 08/2016 |
8953 | 20^75799+47 | 98617 | Lelio R Paula | 08/2016 |
8954 | (5452^26393-1)/5451 | 98616 | Serge Batalov | 06/2022 |
8955 | 28101^3226+3226^28101 | 98597 | Hans Havermann | 07/2020 |
8956 | 2^327515-1017 | 98592 | Lelio R Paula | 08/2016 |
8957 | 6^126692+97681 | 98586 | Mike Oakes | 04/2011 |
8958 | 2^327467-741 | 98578 | Lelio R Paula | 08/2016 |
8959 | (2947^28411-1)/2946 | 98566 | Serge Batalov | 06/2022 |
8960 | 22713^21800+21800^22713 | 98540 | Hans Havermann | 05/2020 |
8961 | (2^327329+9)/19 | 98535 | Lelio R Paula | 10/2013 |
8962 | 7^116577-66 | 98519 | Lelio R Paula | 08/2016 |
8963 | (3*2^327250+1)/7 | 98512 | Lelio R Paula | 11/2013 |
8964 | 23220^17479+17479^23220 | 98512 | Hans Havermann | 03/2020 |
8965 | (2^327240+3798)/2 | 98509 | Stefano Morozzi | 02/2021 |
8966 | (28514^22111-1)/28513 | 98502 | Serge Batalov | 06/2022 |
8967 | (19*2^327192+1)/275 | 98494 | Lelio R Paula | 11/2013 |
8968 | 2^327162-66945 | 98486 | Lelio R Paula | 08/2016 |
8969 | 30139^1852+1852^30139 | 98484 | Hans Havermann | 02/2020 |
8970 | 23182^17703+17703^23182 | 98479 | Hans Havermann | 01/2020 |
8971 | 4*10^98472-3 | 98473 | Erik Branger | 11/2013 |
8972 | 26167^5794+5794^26167 | 98466 | Hans Havermann | 01/2020 |
8973 | (6883*2^327047-1)/77 | 98453 | Lelio R Paula | 05/2015 |
8974 | (71*10^98446+1)/9 | 98447 | Serge Batalov | 11/2010 |
8975 | (2009^29803-1)/2008 | 98436 | Serge Batalov | 06/2022 |
8976 | 13^88363+94 | 98432 | Lelio R Paula | 08/2016 |
8977 | 48^58543-47^58543 | 98425 | Jean-Louis Charton | 11/2005 |
8978 | 6^126481+41 | 98422 | Lelio R Paula | 08/2016 |
8979 | 78*10^98413-7 | 98415 | Robert Price | 05/2017 |
8980 | 22634^22263+22263^22634 | 98404 | Hans Havermann | 07/2020 |
8981 | (47*10^98386+241)/9 | 98387 | Robert Price | 06/2022 |
8982 | 7^116416+46 | 98383 | Lelio R Paula | 08/2016 |
8983 | 7^116407-24 | 98376 | Lelio R Paula | 07/2016 |
8984 | 439^37228+37228^439 | 98374 | Norbert Schneider | 08/2018 |
8985 | 2^326782-1860671 | 98372 | Enzo Doria | 03/2016 |
8986 | 17^79948+58 | 98372 | Lelio R Paula | 08/2016 |
8987 | 22499^22602-22602^22499 | 98368 | Norbert Schneider | 08/2020 |
8988 | 3^206143-98 | 98356 | Lelio R Paula | 08/2016 |
8989 | 8*5^140705+7 | 98350 | Lelio R Paula | 11/2014 |
8990 | (2^326632+1662)/2 | 98326 | Stefano Morozzi | 09/2020 |
8991 | (638^35053-1)/637 | 98315 | Serge Batalov | 06/2022 |
8992 | 2^326585+11 | 98312 | Lelio R Paula | 12/2012 |
8993 | 2^326582-459161 | 98311 | Enzo Doria | 05/2015 |
8994 | (1266^31687-1)/1265 | 98304 | Serge Batalov | 06/2022 |
8996 | (2^326548+9)/5 | 98301 | Lelio R Paula | 10/2013 |
8995 | (2^326549+18)/10 | 98301 | Stefano Morozzi | 03/2020 |
8997 | 2^326538+71659 | 98298 | Enzo Doria | 04/2016 |
8998 | (2^326520+2082)/2 | 98293 | Stefano Morozzi | 09/2020 |
8999 | (2^326521+1002)/2 | 98293 | Stefano Morozzi | 09/2020 |
9000 | 15^83575-58 | 98292 | Lelio R Paula | 08/2016 |