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 ? |
4751 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
4752 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
4753 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
4754 | 27764^13065+13065^27764 | 114280 | Hans Havermann | 01/2021 |
4755 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
4756 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
4757 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
4758 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
4759 | (22*10^114178+47)/3 | 114179 | Robert Price | 06/2021 |
4760 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
4761 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
4762 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
4763 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
4764 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
4765 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
4766 | 13643^27572-27572^13643 | 114008 | Norbert Schneider | 12/2022 |
4768 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
4767 | (83*10^113969+133)/9 | 113970 | Robert Price | 08/2021 |
4769 | 32226^3439+3439^32226 | 113966 | Hans Havermann | 02/2021 |
4770 | (77*10^113963-149)/9 | 113964 | Robert Price | 03/2022 |
4771 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
4772 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
4773 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
4776 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
4774 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
4775 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
4777 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
4778 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
4779 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
4781 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
4780 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
4782 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
4783 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
4784 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
4785 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
4786 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
4787 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
4788 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
4789 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
4790 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
4791 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
4792 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
4793 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
4794 | 34596^1949+1949^34596 | 113815 | Hans Havermann | 02/2021 |
4795 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
4796 | (28*10^113810+449)/9 | 113811 | Robert Price | 11/2021 |
4797 | 26273^21448+21448^26273 | 113799 | Hans Havermann | 02/2021 |
4798 | 2^377966+255255 | 113780 | Paul Bourdelais | 10/2021 |
4799 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
4800 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
4801 | (7*10^113757+11)/9 | 113757 | Tyler Busby | 01/2023 |
4802 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
4803 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
4804 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
4805 | (2^377791+3258)/2 | 113727 | Stefano Morozzi | 02/2021 |
4806 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
4807 | 28206^10739+10739^28206 | 113698 | Hans Havermann | 01/2021 |
4808 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
4809 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
4810 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
4811 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
4812 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |
4813 | (10^113656+7)/40231/6439001 | 113645 | Alfred Reich | 06/2021 |
4814 | (2*10^113631+529)/9 | 113631 | Robert Price | 01/2018 |
4815 | 2^377408-2798279 | 113612 | Enzo Doria | 04/2016 |
4816 | 13706^27459+27459^13706 | 113596 | Norbert Schneider | 12/2020 |
4817 | 5*10^113586+57 | 113587 | Robert Price | 12/2018 |
4818 | (2^377242+2794)/2 | 113561 | Stefano Morozzi | 02/2021 |
4819 | 2^377222+93679 | 113556 | Yves Minet | 02/2011 |
4821 | 262147#+316847 | 113552 | Makoto Morimoto | 02/2016 |
4820 | 27172^15101+15101^27172 | 113552 | Hans Havermann | 01/2021 |
4822 | (2^377131+2018)/2 | 113528 | Stefano Morozzi | 09/2020 |
4823 | 2^377026-458903 | 113497 | Enzo Doria | 04/2015 |
4824 | (2^377014+2202)/2 | 113493 | Stefano Morozzi | 09/2020 |
4825 | 34791^1828+1828^34791 | 113488 | Hans Havermann | 02/2021 |
4826 | (2^376967+258)/2 | 113479 | Stefano Morozzi | 09/2020 |
4827 | (29*10^113468-17)/3 | 113469 | Robert Price | 09/2019 |
4828 | 35^73477-34^73477 | 113454 | Robert Price | 05/2012 |
4829 | (14*10^113451-89)/3 | 113452 | Robert Price | 05/2020 |
4830 | 2^376790+27 | 113426 | Lelio R Paula | 03/2013 |
4831 | 2^376786-112701 | 113424 | Enzo Doria | 06/2014 |
4832 | 2^376731+21 | 113408 | Lelio R Paula | 03/2013 |
4833 | 9*10^113394-73 | 113395 | Robert Price | 01/2021 |
4834 | 29660^6651+6651^29660 | 113387 | Hans Havermann | 01/2021 |
4836 | 2^376634+1871245 | 113379 | Enzo Doria | 05/2016 |
4835 | (4^188318+17)/3 | 113379 | Robert Price | 02/2017 |
4837 | E(29532)/492842728006750534663978950979381235 | 113367 | Thomas Ritschel | 01/2021 |
4838 | 2^376576-344369 | 113361 | Enzo Doria | 07/2015 |
4839 | 2^376549-4543 | 113353 | Yves Minet | 03/2019 |
4840 | (17*2^376526-1)/223 | 113345 | Lelio R Paula | 03/2014 |
4841 | (27*2^376441-1)/811 | 113319 | Lelio R Paula | 03/2014 |
4842 | (2^376350+3538)/2 | 113293 | Stefano Morozzi | 09/2020 |
4843 | 27265^14272+14272^27265 | 113273 | Hans Havermann | 02/2021 |
4844 | 32174^3315+3315^32174 | 113269 | Hans Havermann | 02/2021 |
4845 | (2*10^113248+73)/3 | 113248 | Robert Price | 09/2020 |
4846 | 2^376128-21237 | 113226 | Enzo Doria | 06/2013 |
4847 | 5*10^113224-81 | 113225 | Robert Price | 01/2019 |
4848 | 10^113220+51409 | 113221 | Patrick De Geest | 02/2021 |
4850 | 10^113220-1000001 | 113220 | Robert Price | 08/2020 |
4849 | 10^113220-61779 | 113220 | Patrick De Geest | 02/2021 |
4851 | (2^376102+1578)/2 | 113218 | Stefano Morozzi | 09/2020 |
4852 | (2^375996+3102)/2 | 113186 | Stefano Morozzi | 09/2020 |
4853 | 37258^1091+1091^37258 | 113184 | Hans Havermann | 02/2021 |
4854 | (94*10^113176+11)/3 | 113178 | Robert Price | 04/2020 |
4855 | (46*10^113162+521)/9 | 113163 | Robert Price | 02/2019 |
4856 | (2^375885+3614)/2 | 113153 | Stefano Morozzi | 09/2020 |
4857 | (2^375873+2006)/2 | 113149 | Stefano Morozzi | 09/2020 |
4858 | 3*10^113137+23 | 113138 | Robert Price | 03/2022 |
4859 | (8*10^113135-269)/9 | 113135 | Robert Price | 09/2021 |
4860 | (2^375726+694)/2 | 113105 | Stefano Morozzi | 09/2020 |
4861 | 26050^21959+21959^26050 | 113100 | Hans Havermann | 02/2021 |
4862 | 2^375638-1787171 | 113079 | Enzo Doria | 03/2016 |
4863 | (2^375563+2178)/2 | 113056 | Stefano Morozzi | 09/2020 |
4864 | 2^375537+4847 | 113048 | Yves Minet | 02/2011 |
4865 | (17^91837+16^91837)/33 | 113000 | Robert Price | 08/2013 |
4866 | 5*F(270343)*(F(270343)+1)+1 | 112997 | Serge Batalov | 04/2015 |
4867 | 2^375350-3221 | 112992 | Yves Minet | 07/2011 |
4868 | (13*2^375306+1)/7 | 112979 | Lelio R Paula | 03/2014 |
4869 | 2^375305+76757 | 112979 | Enzo Doria | 05/2016 |
4870 | 2^375258+1874989 | 112964 | Enzo Doria | 07/2016 |
4871 | (29^77237-1)/28 | 112950 | Robert Price | 04/2012 |
4872 | 26620^17483+17483^26620 | 112939 | Hans Havermann | 01/2021 |
4873 | 2^375154+74959 | 112933 | Enzo Doria | 04/2016 |
4874 | (266*10^112930-17)/3 | 112932 | Robert Price | 07/2020 |
4875 | (2^375134+778)/2 | 112927 | Stefano Morozzi | 09/2020 |
4876 | (85*10^112919+293)/9 | 112920 | Robert Price | 12/2019 |
4877 | 29412^6899+6899^29412 | 112907 | Hans Havermann | 02/2021 |
4878 | (191*10^112903-17)/3 | 112905 | Robert Price | 11/2021 |
4879 | (2^375022+1210)/2 | 112893 | Stefano Morozzi | 09/2020 |
4880 | 25992^22039+22039^25992 | 112889 | Hans Havermann | 02/2021 |
4881 | 2^374993-2787163 | 112885 | Enzo Doria | 11/2015 |
4882 | (697*10^112869+23)/9 | 112871 | Robert Price | 04/2021 |
4883 | (19*10^112851+377)/9 | 112852 | Alfred Reich | 11/2022 |
4884 | 30420^5111+5111^30420 | 112813 | Hans Havermann | 02/2021 |
4885 | 13699^27268+27268^13699 | 112800 | Norbert Schneider | 12/2020 |
4886 | 123^53970+2 | 112793 | Lelio R Paula | 06/2013 |
4887 | 28942^7887+7887^28942 | 112785 | Hans Havermann | 01/2021 |
4888 | 2^374640-4685 | 112778 | Yves Minet | 05/2019 |
4889 | 57290^93+93^57290 | 112775 | Hans Havermann | 01/2021 |
4890 | 13608^27277-27277^13608 | 112758 | Norbert Schneider | 12/2022 |
4891 | (2^374556+682)/2 | 112753 | Stefano Morozzi | 09/2020 |
4892 | p(22852)#/2+512 | 112745 | Robert Price | 05/2017 |
4893 | (11*10^112742-131)/3 | 112743 | Robert Price | 09/2018 |
4894 | 2^374441-21049 | 112718 | Enzo Doria | 06/2013 |
4895 | 4*10^112706-93 | 112707 | Robert Price | 07/2020 |
4896 | (2^374321+1)/3 | 112682 | H.&R. Lifchitz | 12/2000 |
4897 | (2^374213+1766)/2 | 112650 | Stefano Morozzi | 09/2020 |
4898 | (2^374094+2674)/2 | 112614 | Stefano Morozzi | 09/2020 |
4899 | (13^101089-1)/12 | 112607 | Paul Bourdelais | 03/2010 |
4900 | 3^236011-8828 | 112606 | Enzo Doria | 05/2014 |
4901 | 31854^3427+3427^31854 | 112602 | Hans Havermann | 02/2021 |
4902 | (85*10^112530-373)/9 | 112531 | Robert Price | 05/2021 |
4903 | (2^373748+1090)/2 | 112510 | Stefano Morozzi | 09/2020 |
4904 | (2^373727+1806)/2 | 112503 | Stefano Morozzi | 01/2021 |
4905 | 27618^11821+11821^27618 | 112479 | Hans Havermann | 01/2021 |
4906 | 26343^18580+18580^26343 | 112460 | Hans Havermann | 01/2021 |
4907 | 39^70666+2 | 112435 | Lelio R Paula | 11/2013 |
4908 | (134*10^112400+7)/3 | 112402 | Robert Price | 02/2020 |
4909 | (2^373365+326)/2 | 112394 | Stefano Morozzi | 09/2020 |
4910 | (2^373350-7)/3 | 112390 | Lelio R Paula | 03/2014 |
4911 | 2^373306-1840931 | 112377 | Enzo Doria | 06/2015 |
4912 | 8*10^112371-37 | 112372 | Robert Price | 03/2022 |
4913 | (2^373262+270)/2 | 112363 | Stefano Morozzi | 09/2020 |
4914 | 6*10^112359-79 | 112360 | Robert Price | 03/2022 |
4915 | 34409^1842+1842^34409 | 112356 | Hans Havermann | 02/2021 |
4916 | (142*10^112352-7)/9 | 112354 | Robert Price | 12/2019 |
4917 | 25580^24633+24633^25580 | 112336 | Hans Havermann | 01/2021 |
4918 | 2^373117+1872107 | 112320 | Enzo Doria | 05/2016 |
4919 | (2^373083+2226)/2 | 112309 | Stefano Morozzi | 01/2021 |
4920 | (35*10^112308+37)/9 | 112309 | Robert Price | 05/2022 |
4921 | 27963^10376+10376^27963 | 112301 | Hans Havermann | 01/2021 |
4922 | 2^373040+2792761 | 112297 | Enzo Doria | 03/2016 |
4923 | EPrime(112280) | 112280 | Eric W. Weisstein | 10/2010 |
4924 | 2^372950+7 | 112270 | Lelio R Paula | 03/2013 |
4925 | 2^372945+1872167 | 112268 | Enzo Doria | 05/2016 |
4926 | 36274^1243+1243^36274 | 112249 | Hans Havermann | 02/2021 |
4927 | (55*10^112246+629)/9 | 112247 | Robert Price | 01/2022 |
4928 | (2^372872+3798)/2 | 112246 | Stefano Morozzi | 02/2021 |
4929 | 25513^25060+25060^25513 | 112232 | Hans Havermann | 02/2021 |
4930 | (53*10^112229-611)/9 | 112230 | Robert Price | 02/2022 |
4931 | (2^372754+3094)/2 | 112210 | Stefano Morozzi | 09/2020 |
4932 | 6*10^112208+43 | 112209 | Robert Price | 07/2020 |
4933 | (2^372728+3058)/2 | 112203 | Stefano Morozzi | 09/2020 |
4934 | 32061^3158+3158^32061 | 112195 | Hans Havermann | 02/2021 |
4935 | 2^372695+72113 | 112193 | Enzo Doria | 04/2016 |
4936 | 2^372689-2563 | 112191 | Yves Minet | 07/2014 |
4937 | 2^372638-1859441 | 112176 | Enzo Doria | 03/2016 |
4938 | 3^235074-113056 | 112159 | Enzo Doria | 07/2014 |
4939 | -E(29242)/489848365125303139750087 | 112140 | Thomas Ritschel | 01/2021 |
4940 | (2^372514+1822)/2 | 112138 | Stefano Morozzi | 02/2021 |
4941 | 29468^6387+6387^29468 | 112135 | Hans Havermann | 02/2021 |
4942 | (2^372428+3742)/2 | 112112 | Stefano Morozzi | 01/2021 |
4943 | 25375^25454-25454^25375 | 112110 | Norbert Schneider | 01/2021 |
4944 | 131^52949-130 | 112108 | Norbert Schneider | 02/2019 |
4945 | (91*10^112098-1)/9 | 112100 | Robert Price | 04/2015 |
4946 | (68*10^112087-149)/9 | 112088 | Robert Price | 07/2022 |
4947 | 2^372310-85061 | 112077 | Paul Bourdelais | 02/2012 |
4948 | (10^112067+53)/9 | 112067 | Serge Batalov | 07/2015 |
4949 | (2^372237+2082)/2 | 112055 | Stefano Morozzi | 09/2020 |
4950 | 75^59760+2 | 112054 | Lelio R Paula | 08/2013 |
4951 | (2^372200+1710)/2 | 112044 | Stefano Morozzi | 09/2020 |
4952 | (10^112048+7)/8747/373157 | 112039 | Alfred Reich | 06/2021 |
4953 | (2^372181+2250)/2 | 112038 | Stefano Morozzi | 01/2021 |
4954 | (2^372161+2574)/2 | 112032 | Stefano Morozzi | 09/2020 |
4955 | 8*10^112018-49 | 112019 | Robert Price | 08/2019 |
4957 | 2^372108+74221 | 112016 | Enzo Doria | 04/2016 |
4956 | (2^372109+2046)/2 | 112016 | Stefano Morozzi | 09/2020 |
4958 | 8*10^112011-73 | 112012 | Robert Price | 01/2022 |
4959 | (184*10^112007-1)/3 | 112009 | Robert Price | 02/2020 |
4960 | 5*10^112003+81 | 112004 | Robert Price | 03/2019 |
4961 | 215^48018+48018^215 | 112000 | Norbert Schneider | 10/2018 |
4962 | 25549^24148+24148^25549 | 111979 | Hans Havermann | 02/2021 |
4963 | 2^371970-459491 | 111975 | Enzo Doria | 05/2015 |
4964 | 41^69427-40^69427 | 111971 | Robert Price | 07/2012 |
4965 | (2^371782+358)/2 | 111918 | Stefano Morozzi | 01/2021 |
4966 | (34*10^111902-547)/9 | 111903 | Robert Price | 03/2021 |
4967 | (10^111902+71)/9 | 111902 | Serge Batalov | 07/2015 |
4968 | (2^371625+1610)/2 | 111870 | Stefano Morozzi | 09/2020 |
4969 | 29080^7027+7027^29080 | 111865 | Hans Havermann | 01/2021 |
4970 | 31307^3730+3730^31307 | 111820 | Hans Havermann | 02/2021 |
4971 | 2^371440-112865 | 111815 | Enzo Doria | 06/2014 |
4972 | 2^371426-459161 | 111811 | Enzo Doria | 05/2015 |
4973 | 25768^21771+21771^25768 | 111779 | Hans Havermann | 02/2021 |
4974 | 28022^9747+9747^28022 | 111777 | Hans Havermann | 02/2021 |
4975 | (94*10^111770+11)/3 | 111772 | Robert Price | 04/2020 |
4976 | 2^371276-461267 | 111766 | Enzo Doria | 07/2015 |
4977 | 6*10^111760+43 | 111761 | Robert Price | 07/2020 |
4978 | 98079!4-1024 | 111747 | Robert Price | 11/2019 |
4979 | (2^371186+3162)/2 | 111738 | Stefano Morozzi | 09/2020 |
4980 | 2^371098-113261 | 111712 | Enzo Doria | 07/2014 |
4981 | (7*10^111706-61)/9 | 111706 | Erik Branger | 01/2011 |
4982 | 2^371044+1869391 | 111696 | Enzo Doria | 05/2016 |
4983 | 26310^17581+17581^26310 | 111688 | Hans Havermann | 01/2021 |
4984 | 88^57416+87 | 111645 | Norbert Schneider | 11/2015 |
4985 | 2^370829-109693 | 111631 | Enzo Doria | 04/2014 |
4986 | 15^94906-2 | 111619 | Lelio R Paula | 11/2013 |
4987 | 25835^20904+20904^25835 | 111614 | Hans Havermann | 02/2021 |
4988 | (2^370726+1198)/2 | 111600 | Stefano Morozzi | 09/2020 |
4989 | 27758^10479+10479^27758 | 111597 | Hans Havermann | 02/2021 |
4990 | 2^370695-4627 | 111591 | Yves Minet | 05/2019 |
4991 | 7*10^111579-87 | 111580 | Robert Price | 06/2020 |
4993 | 2^370643-37 | 111575 | Yves Minet | 01/2012 |
4992 | (2^370655-5)/2439 | 111575 | Lelio R Paula | 03/2014 |
4994 | 2^370610-3911 | 111565 | Yves Minet | 04/2017 |
4995 | (154*10^111549+17)/9 | 111551 | Robert Price | 06/2022 |
4996 | 26200^18097+18097^26200 | 111550 | Hans Havermann | 01/2021 |
4997 | 286^45405+45405^286 | 111532 | Norbert Schneider | 04/2015 |
4998 | 4*5^159563-9 | 111531 | Lelio R Paula | 11/2014 |
4999 | 5*10^111528-21 | 111529 | Robert Price | 12/2021 |
5000 | (4^185225+23)/3 | 111517 | Robert Price | 10/2018 |