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 | 2^238541-112723 | 71808 | Enzo Doria | 06/2014 |
9752 | 2^238529-15949 | 71805 | Enzo Doria | 04/2013 |
9753 | 9*5^102721-8 | 71800 | Lelio R Paula | 10/2014 |
9754 | 2^238494-165 | 71794 | Lelio R Paula | 02/2012 |
9755 | 18035^9562+9562^18035 | 71790 | Hans Havermann | 10/2019 |
9756 | 18390^7993+7993^18390 | 71771 | Hans Havermann | 11/2019 |
9757 | 2^238389-110905 | 71763 | Enzo Doria | 05/2014 |
9758 | (145*10^71760-1)/9 | 71762 | Serge Batalov | 12/2014 |
9759 | 7^84913+5062 | 71760 | Mike Oakes | 02/2010 |
9760 | 21218^2409+2409^21218 | 71756 | Hans Havermann | 10/2019 |
9761 | (F(11446)^31-F(11445)^31)/F(11444) | 71753 | Wayne Halsdorf | 03/2012 |
9762 | (1227^23227-1)/1226 | 71742 | Serge Batalov | 08/2020 |
9763 | 2^238268-97305 | 71726 | Lelio R Paula | 05/2012 |
9764 | gcd(V(1330365),5*F(266073)*(F(266073)-1)+1) | 71724 | Serge Batalov | 04/2015 |
9765 | (2^238249+3770)/2 | 71720 | Stefano Morozzi | 01/2020 |
9766 | 2^238235-113247 | 71716 | Enzo Doria | 07/2014 |
9767 | 2*12^66450-5^66450 | 71712 | Mike Oakes | 11/2010 |
9768 | (2^238213+lucasV(1,2,238213)+1)/4 | 71709 | Mike Oakes | 11/2016 |
9769 | 2^238199-111817 | 71706 | Enzo Doria | 05/2014 |
9770 | lucasU(46,1,43125)+lucasU(46,1,43124) | 71696 | Mike Oakes | 04/2009 |
9771 | 2^238144+98765331 | 71689 | Jason Earls | 01/2009 |
9772 | (2^238137+2810)/2 | 71687 | Stefano Morozzi | 01/2020 |
9773 | 18^57102-4265 | 71679 | Enzo Doria | 03/2013 |
9774 | (73*10^71670-1)/9 | 71671 | Robert Price | 01/2015 |
9775 | 2^238081+1155*238081 | 71670 | Henri Lifchitz | 02/2005 |
9776 | 2^238067-111925 | 71666 | Enzo Doria | 05/2014 |
9777 | (2^238024+3690)/2 | 71653 | Stefano Morozzi | 01/2020 |
9778 | (7388^18523-1)/7387 | 71653 | Serge Batalov | 08/2020 |
9779 | 19143^5528+5528^19143 | 71644 | Hans Havermann | 10/2019 |
9780 | 65401!4+128 | 71638 | Robert Price | 09/2019 |
9781 | 36^46030+35 | 71637 | Norbert Schneider | 04/2014 |
9782 | (74*10^71621+313)/9 | 71622 | Robert Price | 11/2017 |
9783 | 2^237905-91 | 71617 | Lelio R Paula | 02/2012 |
9784 | (31*2^237902-1)/123 | 71616 | Lelio R Paula | 01/2014 |
9785 | (15185*2^237866-1)/43967 | 71605 | Lelio R Paula | 04/2015 |
9786 | 16696^16957+16957^16696 | 71603 | Norbert Schneider | 09/2019 |
9787 | (24737*2^237841+1)/25 | 71601 | Lelio R Paula | 06/2014 |
9788 | 2^237846+1299 | 71599 | Lelio R Paula | 05/2012 |
9789 | 6^92004+199801 | 71594 | Mike Oakes | 02/2011 |
9790 | 159^32516+2 | 71581 | Lelio R Paula | 05/2013 |
9791 | 2^237781+855 | 71580 | Lelio R Paula | 05/2012 |
9792 | (2^237754+814)/2 | 71571 | Stefano Morozzi | 01/2020 |
9793 | 2^237720-4079 | 71561 | Yves Minet | 06/2011 |
9794 | 2^237698-110945 | 71555 | Enzo Doria | 05/2014 |
9795 | 2^237695-113239 | 71554 | Enzo Doria | 07/2014 |
9796 | 8*5^102362-9 | 71549 | Lelio R Paula | 10/2014 |
9797 | (4*10^71546-31)/9 | 71546 | Erik Branger | 05/2011 |
9798 | 2^237650+37 | 71540 | Lelio R Paula | 02/2012 |
9799 | 18^56977-4391 | 71522 | Enzo Doria | 03/2013 |
9801 | (22699*2^237565+1)/4136637 | 71512 | Lelio R Paula | 06/2014 |
9800 | ((1+sqrt(7)+sqrt(6))^48479-1)*((1+sqrt(7)-sqrt(6))^48479-1)*((1-sqrt(7)+sqrt(6))^48479-1)*((1-sqrt(7 | 71512 | Mike Oakes | 05/2009 |
9802 | (26*10^71503+7)/3 | 71504 | Erik Branger | 11/2013 |
9803 | (4*10^71498+173)/3 | 71499 | Robert Price | 01/2017 |
9804 | 3^149848+110 | 71496 | Lelio R Paula | 11/2014 |
9805 | 2^237498+513 | 71495 | Lelio R Paula | 05/2012 |
9806 | 2^237485+107 | 71491 | Lelio R Paula | 02/2012 |
9807 | (2^237465+9)/2419 | 71481 | Lelio R Paula | 05/2012 |
9808 | 2^237439-112237 | 71477 | Enzo Doria | 06/2014 |
9809 | (2^237427+2510)/2 | 71473 | Stefano Morozzi | 01/2020 |
9810 | 2^237407-9787 | 71467 | Lelio R Paula | 11/2009 |
9811 | 2^237383+72953 | 71460 | Enzo Doria | 04/2016 |
9812 | 19768^4119+4119^19768 | 71458 | Hans Havermann | 11/2019 |
9813 | 82^37337-81^37337 | 71457 | Robert Price | 01/2015 |
9814 | (1017*2^237313-1)/19 | 71441 | Lelio R Paula | 04/2015 |
9815 | 2^237311+191 | 71438 | Lelio R Paula | 02/2012 |
9816 | 11699^17560+17560^11699 | 71437 | Norbert Schneider | 09/2019 |
9817 | (25188^16231-1)/25187 | 71432 | Serge Batalov | 08/2020 |
9818 | 2^237280-59 | 71429 | Lelio R Paula | 02/2012 |
9819 | (2^237275+1038)/2 | 71427 | Stefano Morozzi | 01/2020 |
9820 | (2^237265+2774)/2 | 71424 | Stefano Morozzi | 01/2020 |
9821 | 11^68564-98 | 71403 | Lelio R Paula | 01/2015 |
9822 | 2^237173+74801 | 71397 | Enzo Doria | 04/2016 |
9823 | (2^237150+1458)/2 | 71389 | Stefano Morozzi | 01/2020 |
9824 | 15^60697+56 | 71386 | Lelio R Paula | 01/2015 |
9825 | 2^237118+177 | 71380 | Lelio R Paula | 02/2012 |
9826 | 2^237110+69 | 71378 | Lelio R Paula | 02/2012 |
9827 | 15^60690-58 | 71377 | Lelio R Paula | 01/2015 |
9828 | (5*2^237125-1)/599851191 | 71374 | Lelio R Paula | 01/2014 |
9829 | 20311^3262+3262^20311 | 71363 | Hans Havermann | 11/2019 |
9830 | (3^149567+2)/539 | 71359 | Lelio R Paula | 06/2012 |
9831 | (5*10^71357-143)/3 | 71358 | Robert Price | 04/2016 |
9832 | 2^237039-114621 | 71356 | Enzo Doria | 05/2015 |
9833 | (2^237037+3462)/2 | 71355 | Stefano Morozzi | 01/2020 |
9834 | 12^66116-37 | 71352 | Lelio R Paula | 02/2015 |
9835 | (3^149497+2^149497)/5 | 71328 | Jean-Louis Charton | 10/2010 |
9836 | 2^236941-66945 | 71327 | Lelio R Paula | 05/2012 |
9837 | 2^236911-1147 | 71318 | Yves Minet | 10/2009 |
9838 | 65^39337+42 | 71315 | Lelio R Paula | 01/2016 |
9839 | (13319^17291-1)/13318 | 71313 | Serge Batalov | 08/2020 |
9840 | (2^236863+3234)/2 | 71303 | Stefano Morozzi | 01/2020 |
9841 | 6*10^71299+7 | 71300 | Dmitry Domanov | 11/2010 |
9842 | (25*2^236843+1)/3753 | 71295 | Lelio R Paula | 06/2012 |
9843 | 10^71291+15187 | 71292 | Patrick De Geest | 03/2016 |
9845 | 10^71291-5587309119 | 71291 | Robert Price | 03/2015 |
9844 | 10^71291-273581 | 71291 | Patrick De Geest | 03/2016 |
9846 | -numer(bernoulli(22808)/22808) | 71290 | T. D. Noe | 09/2005 |
9847 | (2^236781+1514)/2 | 71278 | Stefano Morozzi | 01/2020 |
9848 | (2^236775+2886)/2 | 71277 | Stefano Morozzi | 01/2020 |
9849 | 15105^17054+17054^15105 | 71271 | Norbert Schneider | 08/2018 |
9850 | (2^236738+9)/13 | 71265 | Lelio R Paula | 05/2012 |
9851 | 5^101949+42 | 71260 | Lelio R Paula | 07/2014 |
9852 | (9811^17851-1)/9810 | 71253 | Serge Batalov | 08/2020 |
9853 | 7*10^71238-39 | 71239 | Robert Price | 07/2020 |
9854 | 5*10^71237+9 | 71238 | Dmitry Domanov | 11/2010 |
9855 | 2^236634-16025 | 71234 | Enzo Doria | 04/2013 |
9856 | 71^38478+70 | 71233 | Norbert Schneider | 12/2014 |
9857 | 9*7^84280+8 | 71226 | Lelio R Paula | 10/2014 |
9858 | (2^236603+2378)/2 | 71225 | Stefano Morozzi | 01/2020 |
9859 | (10^71223+59)/3 | 71223 | Serge Batalov | 12/2014 |
9860 | 23260^1153+1153^23260 | 71219 | Hans Havermann | 10/2019 |
9861 | (15185*2^236569-1)/453 | 71216 | Lelio R Paula | 04/2015 |
9862 | 51829#+(216091#)/(51829#) | 71208 | Makoto Morimoto | 03/2016 |
9863 | 2^236528-173 | 71203 | Lelio R Paula | 02/2012 |
9864 | 3*2^236511-5 | 71198 | Lelio R Paula | 04/2014 |
9865 | 2^236477-99 | 71187 | Lelio R Paula | 02/2012 |
9866 | 11^68356-2130 | 71186 | Enzo Doria | 12/2012 |
9867 | 15128^17031+17031^15128 | 71186 | Norbert Schneider | 08/2018 |
9868 | 11^68355-3120 | 71185 | Enzo Doria | 12/2012 |
9869 | (2^236463+998)/2 | 71183 | Stefano Morozzi | 01/2020 |
9870 | 36^45737-35^45737 | 71181 | Robert Price | 10/2012 |
9871 | (3^149183-lucasV(1,3,149183)+1)/3 | 71178 | Paul Vanderveen | 12/2017 |
9872 | 2^236432+25 | 71174 | Lelio R Paula | 02/2012 |
9873 | 5^101818-12806 | 71168 | Enzo Doria | 02/2013 |
9874 | (7*10^71166-457)/9 | 71166 | Robert Price | 09/2017 |
9875 | 65^39253+62 | 71163 | Lelio R Paula | 01/2016 |
9876 | 39^44718+38 | 71150 | Norbert Schneider | 05/2014 |
9877 | 18996^5557+5557^18996 | 71137 | Hans Havermann | 10/2019 |
9878 | 64983!4-256 | 71135 | Robert Price | 11/2019 |
9879 | 2^236296-407 | 71133 | Yves Minet | 02/2012 |
9880 | 2^236286-117 | 71130 | Lelio R Paula | 02/2012 |
9881 | (2^236283+1074)/2 | 71128 | Stefano Morozzi | 01/2020 |
9882 | 2^236257-111193 | 71121 | Enzo Doria | 05/2014 |
9883 | 7^84149-36 | 71115 | Lelio R Paula | 08/2014 |
9885 | 164176#-p(164176) | 71108 | Henri Lifchitz | 12/2018 |
9884 | (2^236214+2934)/2 | 71108 | Stefano Morozzi | 01/2020 |
9886 | 2^236209-112123 | 71106 | Enzo Doria | 06/2014 |
9887 | 18^56644-4985 | 71104 | Enzo Doria | 03/2013 |
9888 | (46*10^71077+521)/9 | 71078 | Robert Price | 08/2017 |
9889 | 21^53753+2 | 71074 | Ray Chandler | 08/2011 |
9890 | (5482^19009-1)/5481 | 71070 | Serge Batalov | 08/2020 |
9891 | (86*10^71068-311)/9 | 71069 | Robert Price | 10/2017 |
9892 | 2^236081-110923 | 71068 | Enzo Doria | 05/2014 |
9893 | (293*27^49642-1736)/13 | 71058 | Curtis Bright | 05/2015 |
9894 | 65^39192+22 | 71052 | Lelio R Paula | 01/2016 |
9895 | (2^236016+1246)/2 | 71048 | Stefano Morozzi | 01/2020 |
9896 | (439#+1)^397-439#^397 | 71038 | Mike Oakes | 11/2010 |
9897 | 7^84050-2 | 71031 | Lelio R Paula | 09/2013 |
9898 | 15813^16916+16916^15813 | 71031 | Norbert Schneider | 09/2019 |
9899 | (2^235956+3978)/2 | 71030 | Stefano Morozzi | 01/2020 |
9900 | (3568^19993-1)/3567 | 71021 | Serge Batalov | 08/2020 |
9901 | 64877!4+128 | 71007 | Robert Price | 09/2019 |
9902 | 18562^6681+6681^18562 | 70997 | Hans Havermann | 10/2019 |
9903 | (2^235802+474)/2 | 70984 | Stefano Morozzi | 01/2020 |
9904 | 20^54558-23 | 70982 | Lelio R Paula | 01/2015 |
9905 | 2^235793-112555 | 70981 | Enzo Doria | 06/2014 |
9906 | 49936!3-3^49936 | 70981 | Robert Price | 08/2015 |
9907 | 2^235787-78795 | 70979 | Lelio R Paula | 05/2012 |
9908 | (2^235761+2930)/2 | 70971 | Stefano Morozzi | 01/2020 |
9909 | Perrin(581132) | 70970 | Eric W. Weisstein | 02/2011 |
9910 | (2^235748+3478)/2 | 70967 | Stefano Morozzi | 01/2020 |
9911 | 14^61913-97 | 70961 | Lelio R Paula | 01/2015 |
9913 | 2^235720+16321 | 70959 | Yves Minet | 03/2010 |
9912 | 2^235718+16095 | 70959 | Lelio R Paula | 05/2012 |
9914 | 163841#+349933 | 70957 | Nicolas PALAU | 11/2016 |
9915 | 2^235710+7 | 70956 | Lelio R Paula | 02/2012 |
9916 | (175^31627+1)/176 | 70939 | Eric Chen | 10/2017 |
9917 | (2^235644+670)/2 | 70936 | Stefano Morozzi | 01/2020 |
9918 | 2^235635-112087 | 70934 | Enzo Doria | 06/2014 |
9919 | 153^32467+152 | 70931 | Norbert Schneider | 11/2018 |
9921 | 2^235575-1965 | 70916 | Lelio R Paula | 05/2012 |
9920 | 6^91133+101 | 70916 | Lelio R Paula | 01/2015 |
9922 | (24755^16141-1)/24754 | 70914 | Serge Batalov | 08/2020 |
9923 | (1102^23311-1)/1101 | 70914 | Serge Batalov | 08/2020 |
9924 | (21181*2^235505+1)/3 | 70898 | Lelio R Paula | 06/2014 |
9925 | (14*10^70896+211)/9 | 70897 | Robert Price | 11/2017 |
9926 | 5^101425+84 | 70894 | Lelio R Paula | 07/2014 |
9927 | 177^31536+2 | 70893 | Lelio R Paula | 05/2013 |
9928 | 43*71^38292+4 | 70891 | Serge Batalov | 03/2016 |
9929 | (29*2^235506-1)/112029785 | 70888 | Lelio R Paula | 01/2014 |
9930 | 11^68045-1530 | 70862 | Enzo Doria | 12/2012 |
9931 | (6969^18439-1)/6968 | 70861 | Serge Batalov | 08/2020 |
9932 | 5*10^70858+21 | 70859 | Robert Price | 01/2017 |
9934 | 2^235326+3 | 70841 | Donovan Johnson | 02/2006 |
9933 | 34523!2+4 | 70841 | Robert Price | 02/2015 |
9935 | (31*2^235307+1)/3 | 70836 | Lelio R Paula | 06/2012 |
9936 | (2^235310+1038)/2 | 70836 | Stefano Morozzi | 01/2020 |
9937 | 2^235308+135 | 70835 | Lelio R Paula | 02/2012 |
9938 | r(35414,90)*1000+911 | 70831 | Patrick De Geest | 07/2004 |
9939 | 2^235274+159 | 70825 | Lelio R Paula | 02/2012 |
9940 | 3^148409+70 | 70810 | Lelio R Paula | 11/2014 |
9941 | (19*10^70794+17)/9 | 70795 | Robert Price | 03/2015 |
9942 | (2^235144+1350)/2 | 70786 | Stefano Morozzi | 01/2020 |
9943 | 2^235108-232709 | 70775 | Paul Bourdelais | 03/2013 |
9944 | 22995^1196+1196^22995 | 70773 | Hans Havermann | 11/2019 |
9945 | (2^235099-2^117550+1)/5 | 70772 | | 11/2006 |
9946 | 19^55325-6 | 70748 | Lelio R Paula | 08/2014 |
9947 | 257^29356+29356^257 | 70746 | Norbert Schneider | 10/2014 |
9948 | 22*10^70729+3 | 70731 | Robert Price | 10/2017 |
9949 | 2^234956+11715 | 70729 | Lelio R Paula | 05/2012 |
9950 | (7*10^70727-43)/9 | 70727 | Robert Price | 10/2014 |
9951 | (2^234897+3906)/2 | 70711 | Stefano Morozzi | 01/2020 |
9952 | 2^234890-30341 | 70709 | Enzo Doria | 11/2012 |
9953 | 17^57461+14 | 70703 | Lelio R Paula | 01/2015 |
9954 | 18281^7366+7366^18281 | 70697 | Hans Havermann | 09/2019 |
9955 | (9179*2^234834-1)/5 | 70696 | Lelio R Paula | 04/2015 |
9956 | 2*10^70688+51 | 70689 | Robert Price | 10/2017 |
9957 | 123^33822-122 | 70685 | Norbert Schneider | 01/2015 |
9958 | 7*10^70680-99 | 70681 | Robert Price | 11/2017 |
9959 | (2^234784+3646)/2 | 70677 | Stefano Morozzi | 01/2020 |
9960 | (21*2^234772-1)/2095 | 70672 | Lelio R Paula | 01/2014 |
9961 | (2^234762+1498)/2 | 70671 | Stefano Morozzi | 01/2020 |
9962 | (24355^16111-1)/24354 | 70668 | Serge Batalov | 08/2020 |
9963 | (2^234723+3690)/2 | 70659 | Stefano Morozzi | 01/2020 |
9964 | (2^234718+3798)/2 | 70657 | Stefano Morozzi | 01/2020 |
9965 | (2^234680+2902)/2 | 70646 | Stefano Morozzi | 01/2020 |
9966 | (29*2^234686-1)/21145 | 70645 | Lelio R Paula | 01/2014 |
9967 | (15*2^234688-1)/5278141 | 70643 | Lelio R Paula | 01/2014 |
9969 | 2^234655-67 | 70639 | Lelio R Paula | 02/2012 |
9968 | (79*10^70638-61)/9 | 70639 | Robert Price | 10/2015 |
9970 | 2^234652-173 | 70638 | Lelio R Paula | 02/2012 |
9971 | 2^234618-3645 | 70628 | Lelio R Paula | 05/2012 |
9972 | (2^234610-5)/19 | 70624 | Lelio R Paula | 05/2012 |
9973 | 4*5^101038+9 | 70624 | Lelio R Paula | 10/2014 |
9974 | 2^234603+71963 | 70623 | Enzo Doria | 04/2016 |
9975 | 2^234600-113129 | 70622 | Enzo Doria | 07/2014 |
9976 | (2^234601+1446)/2 | 70622 | Stefano Morozzi | 01/2020 |
9977 | 2^234586+9 | 70618 | Lelio R Paula | 02/2012 |
9978 | 3^148008+26 | 70618 | Robert Price | 11/2013 |
9979 | (7*10^70616-151)/9 | 70616 | Robert Price | 09/2017 |
9980 | 65^38947+32 | 70608 | Lelio R Paula | 01/2016 |
9981 | (2^234517+2750)/2 | 70597 | Stefano Morozzi | 01/2020 |
9982 | 18363^6988+6988^18363 | 70594 | Hans Havermann | 10/2019 |
9983 | (2^234488-3)/11 | 70587 | Lelio R Paula | 05/2012 |
9984 | (3*2^234451+1)/35 | 70576 | Lelio R Paula | 08/2012 |
9985 | (8848^17881-1)/8847 | 70570 | Serge Batalov | 08/2020 |
9986 | Phi(159495,10) | 70560 | Maksym Voznyy | 09/2015 |
9987 | 2^234383-4035 | 70557 | Lelio R Paula | 05/2012 |
9988 | (2^234364+3618)/2 | 70551 | Stefano Morozzi | 01/2020 |
9989 | (2^234333+3482)/2 | 70541 | Stefano Morozzi | 01/2020 |
9990 | 2^234315-111241 | 70536 | Enzo Doria | 05/2014 |
9991 | 14^61526+97 | 70517 | Lelio R Paula | 01/2015 |
9992 | 14742^16915+16915^14742 | 70512 | Norbert Schneider | 10/2019 |
9993 | (2^234203+3306)/2 | 70502 | Stefano Morozzi | 01/2020 |
9994 | (3*2^234170+1)/13 | 70492 | Lelio R Paula | 08/2012 |
9995 | 51^41281-50^41281 | 70491 | Robert Price | 02/2013 |
9996 | 2^234141-112123 | 70484 | Enzo Doria | 06/2014 |
9997 | 2^234139+12285 | 70483 | Lelio R Paula | 04/2012 |
9998 | 2^234133-3163 | 70482 | Yves Minet | 04/2009 |
9999 | 2^234115+83 | 70476 | Lelio R Paula | 02/2012 |
10000 | 2^234114-1595 | 70476 | Lelio R Paula | 05/2012 |