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 | (24728^21751-1)/24727 | 95552 | Serge Batalov | 06/2022 |
9752 | (49764^20341-1)/49763 | 95536 | Serge Batalov | 06/2022 |
9753 | (36155^20959-1)/36154 | 95531 | Serge Batalov | 06/2022 |
9754 | (7725^24571-1)/7724 | 95526 | Serge Batalov | 06/2022 |
9755 | (20546^22147-1)/20545 | 95510 | Serge Batalov | 06/2022 |
9756 | 2^317251+465 | 95503 | Lelio R Paula | 03/2016 |
9757 | 19^74684+58 | 95503 | Lelio R Paula | 06/2016 |
9758 | (2^317250-7)/339 | 95500 | Lelio R Paula | 07/2013 |
9759 | 5*10^95498-21 | 95499 | Robert Price | 12/2021 |
9760 | (21615^22031-1)/21614 | 95495 | Serge Batalov | 06/2022 |
9761 | 6^122718-79 | 95494 | Lelio R Paula | 05/2016 |
9762 | 22168^20291+20291^22168 | 95485 | Hans Havermann | 10/2019 |
9763 | (77427^19531-1)/77426 | 95481 | Serge Batalov | 06/2022 |
9764 | (13026^23203-1)/13025 | 95472 | Serge Batalov | 06/2022 |
9765 | 2^317141-110491 | 95469 | Enzo Doria | 05/2014 |
9766 | (2^317097+3014)/2 | 95456 | Stefano Morozzi | 09/2020 |
9767 | 2^317094+76309 | 95455 | Enzo Doria | 05/2016 |
9768 | 2^317085+95 | 95453 | Lelio R Paula | 11/2012 |
9769 | (17*10^95451-377)/9 | 95452 | Robert Price | 11/2017 |
9770 | 2^317057-109551 | 95444 | Enzo Doria | 04/2014 |
9771 | 3^200026+28 | 95437 | Lelio R Paula | 05/2016 |
9772 | 2^317024+93 | 95434 | Donovan Johnson | 06/2008 |
9773 | (86*10^95428-437)/9 | 95429 | Robert Price | 03/2021 |
9774 | 10^95427+192591 | 95428 | Patrick De Geest | 05/2021 |
9777 | 2^316999+1605 | 95427 | Lelio R Paula | 03/2016 |
9776 | (2^317000+3210)/2 | 95427 | Stefano Morozzi | 09/2020 |
9775 | 10^95427-10001 | 95427 | Robert Price | 02/2021 |
9778 | 2^316995-110427 | 95426 | Enzo Doria | 05/2014 |
9779 | (69841^19699-1)/69840 | 95420 | Serge Batalov | 06/2022 |
9780 | (59242^19993-1)/59241 | 95415 | Serge Batalov | 06/2022 |
9782 | 22300^18987+18987^22300 | 95410 | Hans Havermann | 06/2020 |
9781 | (9478^23993-1)/9477 | 95410 | Serge Batalov | 06/2022 |
9784 | 2^316937+177 | 95408 | Lelio R Paula | 11/2012 |
9783 | 4*10^95407+91 | 95408 | Robert Price | 11/2017 |
9785 | (94085^19321-1)/(94085^139-1) | 95403 | Ryan Propper | 11/2023 |
9786 | 2^316908+1915 | 95399 | Lelio R Paula | 03/2016 |
9787 | 2^316894+99 | 95395 | Lelio R Paula | 11/2012 |
9788 | (2^316887+2546)/2 | 95393 | Stefano Morozzi | 09/2020 |
9789 | 2*10^95374-21 | 95375 | Robert Price | 09/2020 |
9790 | 6^122547+11 | 95361 | Lelio R Paula | 05/2016 |
9791 | 2^316771+82875 | 95358 | Lelio R Paula | 03/2016 |
9792 | (15609^22741-1)/15608 | 95358 | Serge Batalov | 06/2022 |
9793 | (2^316765+3102)/2 | 95356 | Stefano Morozzi | 09/2020 |
9794 | (2267^28411-1)/2266 | 95329 | Serge Batalov | 06/2022 |
9795 | (824^32693-1)/823 | 95328 | Serge Batalov | 06/2022 |
9796 | (23*10^95326+1)/3 | 95327 | Serge Batalov | 05/2009 |
9797 | 1*10^95326+57 | 95327 | Robert Price | 03/2011 |
9798 | 10^95326-393687 | 95326 | Patrick De Geest | 09/2013 |
9799 | (3^199803+4)/479539 | 95325 | Lelio R Paula | 04/2013 |
9800 | 2^316654-109481 | 95323 | Enzo Doria | 04/2014 |
9801 | 8*10^95316+51 | 95317 | Robert Price | 05/2017 |
9802 | (50^56099+43^56099)/93 | 95309 | Serge Batalov | 12/2016 |
9803 | (23*2^316559+1)/3925 | 95292 | Lelio R Paula | 09/2013 |
9804 | (71*10^95277-539)/9 | 95278 | Robert Price | 11/2017 |
9805 | 23422^11691+11691^23422 | 95278 | Hans Havermann | 10/2019 |
9806 | 19^74503-56 | 95271 | Lelio R Paula | 06/2016 |
9807 | (5038^25733-1)/5037 | 95267 | Serge Batalov | 06/2022 |
9808 | 11^91475-4380 | 95262 | Enzo Doria | 01/2013 |
9809 | (62*10^95251-611)/9 | 95252 | Robert Price | 05/2017 |
9810 | (160090^18301-1)/160089 | 95240 | Serge Batalov | 06/2022 |
9811 | (173205^18181-1)/173204 | 95238 | Serge Batalov | 06/2022 |
9812 | (158710^18313-1)/158709 | 95234 | Serge Batalov | 06/2022 |
9813 | (9979^23813-1)/9978 | 95227 | Serge Batalov | 06/2022 |
9814 | (57376^20011-1)/57375 | 95223 | Serge Batalov | 06/2022 |
9815 | (2^316321-509203)/69377 | 95218 | Lelio R Paula | 06/2013 |
9816 | 26541^3868+3868^26541 | 95216 | Hans Havermann | 07/2020 |
9817 | (79*10^95214-43)/9 | 95215 | Erik Branger | 11/2013 |
9818 | (7176^24691-1)/7175 | 95202 | Anonymous | 11/2020 |
9819 | (9*2^316233-1)/1751 | 95194 | Lelio R Paula | 02/2014 |
9820 | 5*2^316213+9 | 95191 | Lelio R Paula | 08/2014 |
9821 | 2*10^95188-91 | 95189 | Robert Price | 04/2020 |
9822 | 2^316205-1848163 | 95188 | Enzo Doria | 07/2015 |
9823 | 2^316185-2787913 | 95182 | Enzo Doria | 04/2016 |
9824 | (2^316174+3538)/2 | 95178 | Stefano Morozzi | 09/2020 |
9825 | (9787*2^316142-1)/3 | 95172 | Lelio R Paula | 04/2015 |
9826 | 2^316140-5 | 95168 | Henri Lifchitz | 02/2006 |
9827 | (15185*2^316117-1)/3 | 95165 | Lelio R Paula | 04/2015 |
9828 | (34*10^95153-7)/9/19/19157 | 95149 | Makoto Morimoto | 05/2016 |
9829 | 2^316069+76817 | 95147 | Enzo Doria | 05/2016 |
9830 | (199*10^95134-7)/3 | 95136 | Robert Price | 02/2017 |
9831 | 2^316027-97305 | 95134 | Lelio R Paula | 03/2016 |
9832 | 2^316006-2798051 | 95128 | Enzo Doria | 04/2016 |
9833 | 24994^6397+6397^24994 | 95127 | Hans Havermann | 02/2020 |
9835 | 2^315921-2797603 | 95102 | Enzo Doria | 04/2016 |
9834 | (2^315921+1314)/2 | 95102 | Stefano Morozzi | 02/2021 |
9836 | 2*10^95095+63 | 95096 | Robert Price | 10/2017 |
9837 | 2^315857+1299 | 95083 | Lelio R Paula | 03/2016 |
9838 | (2^315858+2598)/2 | 95083 | Stefano Morozzi | 09/2020 |
9840 | 14^82955-43 | 95078 | Lelio R Paula | 06/2016 |
9839 | (8*10^95078-269)/9 | 95078 | Robert Price | 09/2021 |
9841 | (23*10^95074-101)/3 | 95075 | Robert Price | 04/2021 |
9842 | 2^315827-933075 | 95074 | Paul Bourdelais | 04/2021 |
9843 | (2*10^95072-17)/3 | 95072 | Robert Price | 08/2014 |
9844 | (286*10^95059-1)/3 | 95061 | Robert Price | 11/2020 |
9845 | (2^315779+1610)/2 | 95059 | Stefano Morozzi | 09/2020 |
9846 | 2^315775-111546435 | 95058 | Paul Bourdelais | 10/2021 |
9847 | 21938^21513+21513^21938 | 95051 | Hans Havermann | 09/2020 |
9848 | 2^315727+65 | 95044 | Lelio R Paula | 11/2012 |
9849 | 2^315725+1885757 | 95043 | Enzo Doria | 03/2017 |
9850 | 10^(2*47509+1)-10^47509-1 | 95019 | Robert Price | 07/2023 |
9851 | 23277^12058+12058^23277 | 95000 | Hans Havermann | 09/2020 |
9852 | 1*10^94983+21 | 94984 | Robert Price | 03/2011 |
9853 | 10^94983-717 | 94983 | Patrick De Geest | 08/2013 |
9854 | 17^77187-82 | 94975 | Lelio R Paula | 07/2016 |
9855 | 3*10^94964+37 | 94965 | Robert Price | 10/2017 |
9856 | (2^315481-1017)/20627285 | 94962 | Lelio R Paula | 06/2013 |
9857 | 2^315423-110329 | 94952 | Enzo Doria | 05/2014 |
9858 | 2^315420+3493 | 94951 | Riley Fisher | 08/2023 |
9859 | 11^91169-7230 | 94943 | Enzo Doria | 02/2013 |
9860 | 7*10^94938+61 | 94939 | Robert Price | 02/2017 |
9861 | 2^315351+1025 | 94931 | Lelio R Paula | 03/2016 |
9862 | 2^315350-112253 | 94930 | Enzo Doria | 06/2014 |
9863 | (2^315343+1766)/2 | 94928 | Stefano Morozzi | 09/2020 |
9864 | (2^315343+3066)/2 | 94928 | Stefano Morozzi | 09/2020 |
9866 | ((1001327^(15817)+1)/1001328) | 94906 | Faysal YAtim | 10/2010 |
9865 | 2^315268+7 | 94906 | Lelio R Paula | 10/2012 |
9867 | 11^91133+62 | 94906 | Lelio R Paula | 05/2016 |
9868 | (2^315264+1558)/2 | 94904 | Stefano Morozzi | 09/2020 |
9869 | 30^64244+41 | 94897 | Lelio R Paula | 08/2012 |
9870 | (151*10^94885-7)/9 | 94887 | Robert Price | 07/2017 |
9871 | fibonacci(453998)+2 | 94880 | Jon Grantham | 08/2023 |
9872 | 4^157583+22147 | 94875 | Mike Oakes | 01/2011 |
9873 | (14*32^63026-190819)/31 | 94864 | Eric Chen | 04/2023 |
9874 | 2^315123+20513 | 94862 | Enzo Doria | 07/2023 |
9875 | 2^315098+123 | 94854 | Lelio R Paula | 11/2012 |
9876 | 3^198791-34 | 94848 | Robert Price | 11/2013 |
9877 | 3*2^315067+7 | 94846 | Lelio R Paula | 08/2014 |
9878 | (9*2^315057+1)/73 | 94841 | Lelio R Paula | 09/2013 |
9879 | 22576^15875+15875^22576 | 94836 | Hans Havermann | 09/2020 |
9880 | (16*10^94822+53)/3 | 94823 | Robert Price | 12/2016 |
9881 | 8*6^121849-7 | 94818 | Lelio R Paula | 10/2014 |
9882 | (85*10^94802+473)/9 | 94803 | Robert Price | 03/2017 |
9883 | 21767^21852-21852^21767 | 94790 | Norbert Schneider | 06/2020 |
9884 | (2*34^61893-647)/3 | 94788 | Eric Chen | 03/2023 |
9885 | 2^314870-108761 | 94786 | Enzo Doria | 03/2014 |
9886 | 5*2^314841-3 | 94778 | Lelio R Paula | 08/2014 |
9888 | (2^314842+550)/2 | 94777 | Stefano Morozzi | 09/2020 |
9887 | 11738^23289-23289^11738 | 94777 | Norbert Schneider | 05/2022 |
9889 | (2^314787+2630)/2 | 94761 | Stefano Morozzi | 09/2020 |
9890 | 6^121772+113 | 94758 | Lelio R Paula | 05/2016 |
9891 | (19812^22201-1)/(19812^149-1) | 94756 | Ryan Propper | 11/2023 |
9892 | 2^314751+221873 | 94750 | Enzo Doria | 09/2023 |
9893 | (2^314740+3642)/2 | 94746 | Stefano Morozzi | 09/2020 |
9894 | 6^121756-16253 | 94745 | Enzo Doria | 02/2013 |
9895 | (5*25^67773-2267)/6 | 94743 | Eric Chen | 04/2023 |
9896 | 2^314718-112793 | 94740 | Enzo Doria | 06/2014 |
9897 | 2^314697-1847293 | 94734 | Enzo Doria | 07/2015 |
9898 | 22836^14065+14065^22836 | 94727 | Hans Havermann | 09/2020 |
9899 | 2^314548-249 | 94689 | Yves Minet | 02/2012 |
9900 | 5^135469+42 | 94689 | Lelio R Paula | 04/2016 |
9901 | 2^314545+22697 | 94688 | Enzo Doria | 09/2023 |
9902 | gcd(V(1137755),5*F(227551)*(F(227551)-1)+1) | 94676 | Serge Batalov | 04/2015 |
9903 | 25773^4712+4712^25773 | 94670 | Hans Havermann | 08/2020 |
9904 | (2^314485+3614)/2 | 94670 | Stefano Morozzi | 09/2020 |
9905 | (2*10^94604+547)/9 | 94604 | Robert Price | 10/2017 |
9906 | (2^314164+2562)/2 | 94573 | Stefano Morozzi | 09/2020 |
9907 | 3*2^314152+5 | 94570 | Lelio R Paula | 08/2014 |
9908 | 2*6^121530-5 | 94570 | Lelio R Paula | 10/2014 |
9909 | 7*10^94563+87 | 94564 | Robert Price | 11/2016 |
9910 | 2^314114-109677 | 94558 | Enzo Doria | 04/2014 |
9911 | (218389#+240867)/(3*11*811) | 94557 | Makoto Morimoto | 10/2017 |
9912 | 2^314059+76583 | 94542 | Enzo Doria | 05/2016 |
9913 | 2^314041-4035 | 94536 | Lelio R Paula | 03/2016 |
9914 | (2^314028+3102)/2 | 94532 | Stefano Morozzi | 09/2020 |
9915 | (807*33^62250-7)/32 | 94529 | Eric Chen | 05/2023 |
9916 | (2^314012+618)/2 | 94527 | Stefano Morozzi | 09/2020 |
9917 | (7*2^314003-11)/15 | 94524 | Renaud Lifchitz | 03/2011 |
9918 | 11^90760-4710 | 94517 | Enzo Doria | 01/2013 |
9919 | 21675^21796+21796^21675 | 94507 | Norbert Schneider | 04/2020 |
9920 | 2^313942+100280245065 | 94506 | Paul Bourdelais | 10/2021 |
9921 | (457829*27^66017-385)/26 | 94499 | Eric Chen | 02/2023 |
9922 | (7*2^313926-1)/148821 | 94497 | Lelio R Paula | 02/2014 |
9923 | 94*10^94493-3 | 94495 | Robert Price | 11/2016 |
9924 | (1000547^15749+1)/1000548 | 94492 | Faysal YAtim | 12/2008 |
9925 | (2^313859+2706)/2 | 94481 | Stefano Morozzi | 09/2020 |
9926 | (F(3*75345)^2+L(3*75345)^2)/4 | 94477 | Henri Lifchitz | 10/2017 |
9927 | 7^111792-32 | 94476 | Lelio R Paula | 04/2016 |
9928 | (2^313800+822)/2 | 94463 | Stefano Morozzi | 01/2021 |
9929 | 9*10^94462-47 | 94463 | Robert Price | 11/2021 |
9930 | 11*10^94460+3 | 94462 | Robert Price | 07/2021 |
9931 | (202*10^94441-1)/3 | 94443 | Robert Price | 05/2020 |
9932 | 11^90680-5850 | 94434 | Enzo Doria | 01/2013 |
9933 | (752^32833-1)/751 | 94432 | Michael Stocker | 05/2020 |
9934 | 2^313675-110125 | 94426 | Enzo Doria | 04/2014 |
9935 | (2999*29^64556-2427)/4 | 94410 | Curtis Bright | 05/2015 |
9936 | 2^313609-112225 | 94406 | Enzo Doria | 06/2014 |
9937 | 2^313584+72901 | 94399 | Enzo Doria | 04/2016 |
9938 | (2^313567+1310)/2 | 94393 | Stefano Morozzi | 02/2021 |
9939 | (2^313562+730)/2 | 94392 | Stefano Morozzi | 09/2020 |
9940 | 2^313510-112865 | 94376 | Enzo Doria | 06/2014 |
9941 | 7*10^94373+87 | 94374 | Robert Price | 11/2016 |
9942 | (9^98897+8^98897)/17 | 94371 | Robert Price | 12/2012 |
9943 | 2^313453-114703 | 94359 | Enzo Doria | 04/2015 |
9944 | 8*10^94357-51 | 94358 | Robert Price | 10/2017 |
9945 | 25903^4392+4392^25903 | 94356 | Hans Havermann | 09/2020 |
9946 | 31706^945+945^31706 | 94340 | Hans Havermann | 09/2020 |
9947 | (379*10^94337-1)/9 | 94339 | Robert Price | 08/2017 |
9948 | 2^313344-313344+1 | 94326 | Robert Price | 10/2014 |
9949 | (2^313313+3662)/2 | 94317 | Stefano Morozzi | 09/2020 |
9950 | 20^72481-49 | 94300 | Lelio R Paula | 05/2016 |
9951 | (34*10^94293-259)/9 | 94294 | Robert Price | 12/2021 |
9952 | 44725!2+128 | 94289 | Robert Price | 03/2015 |
9953 | 4*10^94286+9 | 94287 | Robert Price | 05/2015 |
9954 | 17^76621+24 | 94279 | Lelio R Paula | 07/2016 |
9955 | 26274^3875+3875^26274 | 94279 | Hans Havermann | 09/2020 |
9956 | 4*10^94270-33 | 94271 | Robert Price | 04/2021 |
9957 | 2^313156-112859 | 94270 | Enzo Doria | 06/2014 |
9958 | (49*10^94262-211)/9 | 94263 | Robert Price | 06/2022 |
9959 | 6^121130-59 | 94258 | Lelio R Paula | 05/2016 |
9960 | 3^197542+28 | 94252 | Robert Price | 11/2013 |
9961 | (2^313076+3550)/2 | 94245 | Stefano Morozzi | 09/2020 |
9962 | (81822^19321-1)/(81822^139-1) | 94239 | Ryan Propper | 11/2023 |
9963 | (3^197521+2)/51655 | 94237 | Lelio R Paula | 04/2013 |
9964 | 2^313045-5955 | 94236 | Lelio R Paula | 03/2016 |
9965 | 22099^18358+18358^22099 | 94227 | Hans Havermann | 08/2020 |
9966 | (9*33^62046+53831)/16 | 94218 | Eric Chen | 05/2023 |
9967 | (67*10^94212+23)/9 | 94213 | Ray Chandler | 02/2011 |
9968 | 2^312909-1865203 | 94195 | Enzo Doria | 03/2016 |
9969 | 25013^5830+5830^25013 | 94191 | Hans Havermann | 09/2020 |
9970 | 2^312848-149 | 94177 | Lelio R Paula | 11/2012 |
9971 | (2^312819+3278)/2 | 94168 | Stefano Morozzi | 02/2021 |
9972 | 4^156403-77597 | 94164 | Mike Oakes | 04/2011 |
9973 | 18^75004-85 | 94151 | Lelio R Paula | 06/2016 |
9974 | (19573*10^94146-7)/9 | 94150 | Patrick De Geest | 05/2023 |
9975 | 19^73622-48 | 94145 | Lelio R Paula | 06/2016 |
9976 | (11^90397-3^90397)/8 | 94138 | Robert Price | 04/2011 |
9977 | (3^197287+5)/16 | 94129 | Lelio R Paula | 04/2013 |
9978 | (23*10^94123-29)/3 | 94124 | Robert Price | 03/2022 |
9979 | 24220^7691+7691^24220 | 94119 | Hans Havermann | 09/2020 |
9980 | 12^87212-37 | 94118 | Lelio R Paula | 06/2016 |
9981 | 11^90376+76 | 94117 | Lelio R Paula | 05/2016 |
9982 | (6^120946-11)/5 | 94114 | Paul Bourdelais | 11/2011 |
9983 | (2^312622+534)/2 | 94109 | Stefano Morozzi | 09/2020 |
9984 | (10^94096+191)/3 | 94096 | Robert Price | 11/2017 |
9985 | 23794^9005+9005^23794 | 94093 | Hans Havermann | 09/2020 |
9986 | 2^312562-112685 | 94091 | Enzo Doria | 06/2014 |
9987 | 137^44026-2 | 94072 | Lelio R Paula | 10/2017 |
9988 | s(164441) | 94052 | Robert Price | 05/2020 |
9989 | 2^312404-869688105 | 94043 | Lelio R Paula | 09/2016 |
9990 | 2^312387+2793533 | 94038 | Enzo Doria | 03/2016 |
9992 | 10^94016+91 | 94017 | Robert Price | 04/2011 |
9991 | (15*2^312325-1)/7201 | 94017 | Lelio R Paula | 02/2014 |
9993 | 10^94016-188963 | 94016 | Patrick De Geest | 08/2013 |
9994 | (17*2^312315-1)/45 | 94016 | Lelio R Paula | 02/2014 |
9995 | (401*10^94010+13)/9 | 94012 | Robert Price | 09/2021 |
9996 | 23866^8691+8691^23866 | 94010 | Hans Havermann | 08/2020 |
9997 | 6^120802+7661 | 94003 | Mike Oakes | 04/2011 |
9998 | (67607*2^312254+1)/177 | 94001 | Lelio R Paula | 06/2014 |
9999 | (2^312245+702)/2 | 93995 | Stefano Morozzi | 09/2020 |
10000 | (56*10^93993+493)/9 | 93994 | Robert Price | 10/2021 |