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 ? |
8001 | 3*10^95776-59 | 95777 | Robert Price | 10/2017 |
8002 | (82970^19471-1)/82969 | 95772 | Serge Batalov | 06/2022 |
8003 | 193^41901-2 | 95768 | Lelio R Paula | 10/2017 |
8004 | (2^318124+2502)/2 | 95765 | Stefano Morozzi | 09/2020 |
8005 | (12900^23297-1)/12899 | 95761 | Serge Batalov | 06/2022 |
8006 | 4*10^95752-71 | 95753 | Robert Price | 11/2017 |
8007 | (16955^22639-1)/16954 | 95743 | Serge Batalov | 06/2022 |
8008 | (81795^19489-1)/81794 | 95740 | Serge Batalov | 06/2022 |
8009 | (2742^27847-1)/2741 | 95737 | Serge Batalov | 06/2022 |
8010 | (6072^25303-1)/6071 | 95726 | Serge Batalov | 06/2022 |
8011 | 23640^11197+11197^23640 | 95721 | Hans Havermann | 12/2019 |
8012 | 10^95712-4000001 | 95712 | Robert Price | 07/2021 |
8013 | (60657^20011-1)/60656 | 95706 | Serge Batalov | 06/2022 |
8014 | ((10^47850-1)*10000/9+6421)*10^47850+(10^47850-1)/9 | 95704 | Hans Havermann | 05/2019 |
8015 | (509203*2^317881-1)/52805 | 95693 | Lelio R Paula | 04/2015 |
8016 | (26161^21661-1)/26160 | 95687 | Serge Batalov | 06/2022 |
8017 | (2^317853+1086)/2 | 95683 | Stefano Morozzi | 09/2020 |
8018 | 4*10^95672+91 | 95673 | Robert Price | 11/2017 |
8019 | 6*10^95665+43 | 95666 | Robert Price | 07/2020 |
8020 | (20819^22153-1)/20818 | 95663 | Serge Batalov | 06/2022 |
8021 | (64464^19891-1)/64463 | 95658 | Serge Batalov | 06/2022 |
8022 | (934^32203-1)/933 | 95652 | Serge Batalov | 06/2022 |
8023 | (11*10^95641-113)/3 | 95642 | Robert Price | 01/2017 |
8024 | (72330^19681-1)/72329 | 95632 | Serge Batalov | 06/2022 |
8025 | (13224^23203-1)/13223 | 95624 | Serge Batalov | 06/2022 |
8026 | (18211^22441-1)/18210 | 95602 | Serge Batalov | 06/2022 |
8027 | (71975^19681-1)/71974 | 95590 | Serge Batalov | 06/2022 |
8028 | (13*10^95588+11)/3 | 95589 | Robert Price | 05/2015 |
8029 | Phi(191188,10)/191189 | 95587 | Maksym Voznyy | 11/2015 |
8030 | (13368^23167-1)/13367 | 95585 | Serge Batalov | 06/2022 |
8031 | (25874^21661-1)/25873 | 95583 | Serge Batalov | 06/2022 |
8032 | 3^200325+70 | 95580 | Lelio R Paula | 05/2016 |
8033 | (25*10^95574-7)/9 | 95575 | Robert Price | 12/2014 |
8034 | (2^317485-3)/21313 | 95569 | Lelio R Paula | 06/2013 |
8035 | 2^317464+31 | 95567 | Lelio R Paula | 11/2012 |
8036 | (2^317451+2114)/2 | 95562 | Stefano Morozzi | 09/2020 |
8037 | 7^113073+82 | 95558 | Lelio R Paula | 04/2016 |
8038 | (24728^21751-1)/24727 | 95552 | Serge Batalov | 06/2022 |
8039 | (49764^20341-1)/49763 | 95536 | Serge Batalov | 06/2022 |
8040 | (36155^20959-1)/36154 | 95531 | Serge Batalov | 06/2022 |
8041 | (7725^24571-1)/7724 | 95526 | Serge Batalov | 06/2022 |
8042 | (20546^22147-1)/20545 | 95510 | Serge Batalov | 06/2022 |
8043 | 2^317251+465 | 95503 | Lelio R Paula | 03/2016 |
8044 | 19^74684+58 | 95503 | Lelio R Paula | 06/2016 |
8045 | (2^317250-7)/339 | 95500 | Lelio R Paula | 07/2013 |
8046 | 5*10^95498-21 | 95499 | Robert Price | 12/2021 |
8047 | (21615^22031-1)/21614 | 95495 | Serge Batalov | 06/2022 |
8048 | 6^122718-79 | 95494 | Lelio R Paula | 05/2016 |
8049 | 22168^20291+20291^22168 | 95485 | Hans Havermann | 10/2019 |
8050 | (77427^19531-1)/77426 | 95481 | Serge Batalov | 06/2022 |
8051 | (13026^23203-1)/13025 | 95472 | Serge Batalov | 06/2022 |
8052 | 2^317141-110491 | 95469 | Enzo Doria | 05/2014 |
8053 | (2^317097+3014)/2 | 95456 | Stefano Morozzi | 09/2020 |
8054 | 2^317094+76309 | 95455 | Enzo Doria | 05/2016 |
8055 | 2^317085+95 | 95453 | Lelio R Paula | 11/2012 |
8056 | (17*10^95451-377)/9 | 95452 | Robert Price | 11/2017 |
8057 | 2^317057-109551 | 95444 | Enzo Doria | 04/2014 |
8058 | 3^200026+28 | 95437 | Lelio R Paula | 05/2016 |
8059 | 2^317024+93 | 95434 | Donovan Johnson | 06/2008 |
8060 | (86*10^95428-437)/9 | 95429 | Robert Price | 03/2021 |
8061 | 10^95427+192591 | 95428 | Patrick De Geest | 05/2021 |
8064 | 2^316999+1605 | 95427 | Lelio R Paula | 03/2016 |
8063 | (2^317000+3210)/2 | 95427 | Stefano Morozzi | 09/2020 |
8062 | 10^95427-10001 | 95427 | Robert Price | 02/2021 |
8065 | 2^316995-110427 | 95426 | Enzo Doria | 05/2014 |
8066 | (69841^19699-1)/69840 | 95420 | Serge Batalov | 06/2022 |
8067 | (59242^19993-1)/59241 | 95415 | Serge Batalov | 06/2022 |
8069 | 22300^18987+18987^22300 | 95410 | Hans Havermann | 06/2020 |
8068 | (9478^23993-1)/9477 | 95410 | Serge Batalov | 06/2022 |
8070 | 2^316937+177 | 95408 | Lelio R Paula | 11/2012 |
8071 | 4*10^95407+91 | 95408 | Robert Price | 11/2017 |
8072 | 2^316908+1915 | 95399 | Lelio R Paula | 03/2016 |
8073 | 2^316894+99 | 95395 | Lelio R Paula | 11/2012 |
8074 | (2^316887+2546)/2 | 95393 | Stefano Morozzi | 09/2020 |
8075 | 2*10^95374-21 | 95375 | Robert Price | 09/2020 |
8076 | 6^122547+11 | 95361 | Lelio R Paula | 05/2016 |
8077 | 2^316771+82875 | 95358 | Lelio R Paula | 03/2016 |
8078 | (15609^22741-1)/15608 | 95358 | Serge Batalov | 06/2022 |
8079 | (2^316765+3102)/2 | 95356 | Stefano Morozzi | 09/2020 |
8080 | (2267^28411-1)/2266 | 95329 | Serge Batalov | 06/2022 |
8081 | (824^32693-1)/823 | 95328 | Serge Batalov | 06/2022 |
8082 | (23*10^95326+1)/3 | 95327 | Serge Batalov | 05/2009 |
8083 | 1*10^95326+57 | 95327 | Robert Price | 03/2011 |
8084 | 10^95326-393687 | 95326 | Patrick De Geest | 09/2013 |
8085 | (3^199803+4)/479539 | 95325 | Lelio R Paula | 04/2013 |
8086 | 2^316654-109481 | 95323 | Enzo Doria | 04/2014 |
8087 | 8*10^95316+51 | 95317 | Robert Price | 05/2017 |
8088 | (50^56099+43^56099)/93 | 95309 | Serge Batalov | 12/2016 |
8089 | (23*2^316559+1)/3925 | 95292 | Lelio R Paula | 09/2013 |
8091 | (71*10^95277-539)/9 | 95278 | Robert Price | 11/2017 |
8090 | 23422^11691+11691^23422 | 95278 | Hans Havermann | 10/2019 |
8092 | 19^74503-56 | 95271 | Lelio R Paula | 06/2016 |
8093 | (5038^25733-1)/5037 | 95267 | Serge Batalov | 06/2022 |
8094 | 11^91475-4380 | 95262 | Enzo Doria | 01/2013 |
8095 | (62*10^95251-611)/9 | 95252 | Robert Price | 05/2017 |
8096 | (160090^18301-1)/160089 | 95240 | Serge Batalov | 06/2022 |
8097 | (173205^18181-1)/173204 | 95238 | Serge Batalov | 06/2022 |
8098 | (158710^18313-1)/158709 | 95234 | Serge Batalov | 06/2022 |
8099 | (9979^23813-1)/9978 | 95227 | Serge Batalov | 06/2022 |
8100 | (57376^20011-1)/57375 | 95223 | Serge Batalov | 06/2022 |
8101 | (2^316321-509203)/69377 | 95218 | Lelio R Paula | 06/2013 |
8102 | 26541^3868+3868^26541 | 95216 | Hans Havermann | 07/2020 |
8103 | (79*10^95214-43)/9 | 95215 | Erik Branger | 11/2013 |
8104 | (7176^24691-1)/7175 | 95202 | Anonymous | 11/2020 |
8105 | (9*2^316233-1)/1751 | 95194 | Lelio R Paula | 02/2014 |
8106 | 5*2^316213+9 | 95191 | Lelio R Paula | 08/2014 |
8107 | 2*10^95188-91 | 95189 | Robert Price | 04/2020 |
8108 | 2^316205-1848163 | 95188 | Enzo Doria | 07/2015 |
8109 | 2^316185-2787913 | 95182 | Enzo Doria | 04/2016 |
8110 | (2^316174+3538)/2 | 95178 | Stefano Morozzi | 09/2020 |
8111 | (9787*2^316142-1)/3 | 95172 | Lelio R Paula | 04/2015 |
8112 | 2^316140-5 | 95168 | Henri Lifchitz | 02/2006 |
8113 | (15185*2^316117-1)/3 | 95165 | Lelio R Paula | 04/2015 |
8114 | (34*10^95153-7)/9/19/19157 | 95149 | Makoto Morimoto | 05/2016 |
8115 | 2^316069+76817 | 95147 | Enzo Doria | 05/2016 |
8116 | (199*10^95134-7)/3 | 95136 | Robert Price | 02/2017 |
8117 | 2^316027-97305 | 95134 | Lelio R Paula | 03/2016 |
8118 | 2^316006-2798051 | 95128 | Enzo Doria | 04/2016 |
8119 | 24994^6397+6397^24994 | 95127 | Hans Havermann | 02/2020 |
8121 | 2^315921-2797603 | 95102 | Enzo Doria | 04/2016 |
8120 | (2^315921+1314)/2 | 95102 | Stefano Morozzi | 02/2021 |
8122 | 2*10^95095+63 | 95096 | Robert Price | 10/2017 |
8123 | 2^315857+1299 | 95083 | Lelio R Paula | 03/2016 |
8124 | (2^315858+2598)/2 | 95083 | Stefano Morozzi | 09/2020 |
8126 | 14^82955-43 | 95078 | Lelio R Paula | 06/2016 |
8125 | (8*10^95078-269)/9 | 95078 | Robert Price | 09/2021 |
8127 | (23*10^95074-101)/3 | 95075 | Robert Price | 04/2021 |
8128 | 2^315827-933075 | 95074 | Paul Bourdelais | 04/2021 |
8129 | (2*10^95072-17)/3 | 95072 | Robert Price | 08/2014 |
8130 | (286*10^95059-1)/3 | 95061 | Robert Price | 11/2020 |
8131 | (2^315779+1610)/2 | 95059 | Stefano Morozzi | 09/2020 |
8132 | 2^315775-111546435 | 95058 | Paul Bourdelais | 10/2021 |
8133 | 21938^21513+21513^21938 | 95051 | Hans Havermann | 09/2020 |
8134 | 2^315727+65 | 95044 | Lelio R Paula | 11/2012 |
8135 | 2^315725+1885757 | 95043 | Enzo Doria | 03/2017 |
8136 | 23277^12058+12058^23277 | 95000 | Hans Havermann | 09/2020 |
8137 | 1*10^94983+21 | 94984 | Robert Price | 03/2011 |
8138 | 10^94983-717 | 94983 | Patrick De Geest | 08/2013 |
8139 | 17^77187-82 | 94975 | Lelio R Paula | 07/2016 |
8140 | 3*10^94964+37 | 94965 | Robert Price | 10/2017 |
8141 | (2^315481-1017)/20627285 | 94962 | Lelio R Paula | 06/2013 |
8142 | 2^315423-110329 | 94952 | Enzo Doria | 05/2014 |
8143 | 11^91169-7230 | 94943 | Enzo Doria | 02/2013 |
8144 | 7*10^94938+61 | 94939 | Robert Price | 02/2017 |
8145 | 2^315351+1025 | 94931 | Lelio R Paula | 03/2016 |
8146 | 2^315350-112253 | 94930 | Enzo Doria | 06/2014 |
8147 | (2^315343+1766)/2 | 94928 | Stefano Morozzi | 09/2020 |
8148 | (2^315343+3066)/2 | 94928 | Stefano Morozzi | 09/2020 |
8151 | ((1001327^(15817)+1)/1001328) | 94906 | Faysal YAtim | 10/2010 |
8150 | 2^315268+7 | 94906 | Lelio R Paula | 10/2012 |
8149 | 11^91133+62 | 94906 | Lelio R Paula | 05/2016 |
8152 | (2^315264+1558)/2 | 94904 | Stefano Morozzi | 09/2020 |
8153 | 30^64244+41 | 94897 | Lelio R Paula | 08/2012 |
8154 | (151*10^94885-7)/9 | 94887 | Robert Price | 07/2017 |
8155 | 4^157583+22147 | 94875 | Mike Oakes | 01/2011 |
8156 | 2^315098+123 | 94854 | Lelio R Paula | 11/2012 |
8157 | 3^198791-34 | 94848 | Robert Price | 11/2013 |
8158 | 3*2^315067+7 | 94846 | Lelio R Paula | 08/2014 |
8159 | (9*2^315057+1)/73 | 94841 | Lelio R Paula | 09/2013 |
8160 | 22576^15875+15875^22576 | 94836 | Hans Havermann | 09/2020 |
8161 | (16*10^94822+53)/3 | 94823 | Robert Price | 12/2016 |
8162 | 8*6^121849-7 | 94818 | Lelio R Paula | 10/2014 |
8163 | (85*10^94802+473)/9 | 94803 | Robert Price | 03/2017 |
8164 | 21767^21852-21852^21767 | 94790 | Norbert Schneider | 06/2020 |
8165 | 2^314870-108761 | 94786 | Enzo Doria | 03/2014 |
8166 | 5*2^314841-3 | 94778 | Lelio R Paula | 08/2014 |
8167 | (2^314842+550)/2 | 94777 | Stefano Morozzi | 09/2020 |
8168 | 11738^23289-23289^11738 | 94777 | Norbert Schneider | 05/2022 |
8169 | (2^314787+2630)/2 | 94761 | Stefano Morozzi | 09/2020 |
8170 | 6^121772+113 | 94758 | Lelio R Paula | 05/2016 |
8171 | (2^314740+3642)/2 | 94746 | Stefano Morozzi | 09/2020 |
8172 | 6^121756-16253 | 94745 | Enzo Doria | 02/2013 |
8173 | 2^314718-112793 | 94740 | Enzo Doria | 06/2014 |
8174 | 2^314697-1847293 | 94734 | Enzo Doria | 07/2015 |
8175 | 22836^14065+14065^22836 | 94727 | Hans Havermann | 09/2020 |
8177 | 2^314548-249 | 94689 | Yves Minet | 02/2012 |
8176 | 5^135469+42 | 94689 | Lelio R Paula | 04/2016 |
8178 | gcd(V(1137755),5*F(227551)*(F(227551)-1)+1) | 94676 | Serge Batalov | 04/2015 |
8179 | 25773^4712+4712^25773 | 94670 | Hans Havermann | 08/2020 |
8180 | (2^314485+3614)/2 | 94670 | Stefano Morozzi | 09/2020 |
8181 | (2*10^94604+547)/9 | 94604 | Robert Price | 10/2017 |
8182 | (2^314164+2562)/2 | 94573 | Stefano Morozzi | 09/2020 |
8183 | 3*2^314152+5 | 94570 | Lelio R Paula | 08/2014 |
8184 | 2*6^121530-5 | 94570 | Lelio R Paula | 10/2014 |
8185 | 7*10^94563+87 | 94564 | Robert Price | 11/2016 |
8186 | 2^314114-109677 | 94558 | Enzo Doria | 04/2014 |
8187 | (218389#+240867)/(3*11*811) | 94557 | Makoto Morimoto | 10/2017 |
8188 | 2^314059+76583 | 94542 | Enzo Doria | 05/2016 |
8189 | 2^314041-4035 | 94536 | Lelio R Paula | 03/2016 |
8190 | (2^314028+3102)/2 | 94532 | Stefano Morozzi | 09/2020 |
8191 | (2^314012+618)/2 | 94527 | Stefano Morozzi | 09/2020 |
8192 | (7*2^314003-11)/15 | 94524 | Renaud Lifchitz | 03/2011 |
8193 | 11^90760-4710 | 94517 | Enzo Doria | 01/2013 |
8194 | 21675^21796+21796^21675 | 94507 | Norbert Schneider | 04/2020 |
8195 | 2^313942+100280245065 | 94506 | Paul Bourdelais | 10/2021 |
8196 | (7*2^313926-1)/148821 | 94497 | Lelio R Paula | 02/2014 |
8197 | 94*10^94493-3 | 94495 | Robert Price | 11/2016 |
8198 | (1000547^15749+1)/1000548 | 94492 | Faysal YAtim | 12/2008 |
8199 | (2^313859+2706)/2 | 94481 | Stefano Morozzi | 09/2020 |
8200 | (F(3*75345)^2+L(3*75345)^2)/4 | 94477 | Henri Lifchitz | 10/2017 |
8201 | 7^111792-32 | 94476 | Lelio R Paula | 04/2016 |
8202 | (2^313800+822)/2 | 94463 | Stefano Morozzi | 01/2021 |
8203 | 9*10^94462-47 | 94463 | Robert Price | 11/2021 |
8204 | 11*10^94460+3 | 94462 | Robert Price | 07/2021 |
8205 | (202*10^94441-1)/3 | 94443 | Robert Price | 05/2020 |
8206 | 11^90680-5850 | 94434 | Enzo Doria | 01/2013 |
8207 | (752^32833-1)/751 | 94432 | Michael Stocker | 05/2020 |
8208 | 2^313675-110125 | 94426 | Enzo Doria | 04/2014 |
8209 | (2999*29^64556-2427)/4 | 94410 | Curtis Bright | 05/2015 |
8210 | 2^313609-112225 | 94406 | Enzo Doria | 06/2014 |
8211 | 2^313584+72901 | 94399 | Enzo Doria | 04/2016 |
8212 | (2^313567+1310)/2 | 94393 | Stefano Morozzi | 02/2021 |
8213 | (2^313562+730)/2 | 94392 | Stefano Morozzi | 09/2020 |
8214 | 2^313510-112865 | 94376 | Enzo Doria | 06/2014 |
8215 | 7*10^94373+87 | 94374 | Robert Price | 11/2016 |
8216 | (9^98897+8^98897)/17 | 94371 | Robert Price | 12/2012 |
8217 | 2^313453-114703 | 94359 | Enzo Doria | 04/2015 |
8218 | 8*10^94357-51 | 94358 | Robert Price | 10/2017 |
8219 | 25903^4392+4392^25903 | 94356 | Hans Havermann | 09/2020 |
8220 | 31706^945+945^31706 | 94340 | Hans Havermann | 09/2020 |
8221 | (379*10^94337-1)/9 | 94339 | Robert Price | 08/2017 |
8222 | 2^313344-313344+1 | 94326 | Robert Price | 10/2014 |
8223 | (2^313313+3662)/2 | 94317 | Stefano Morozzi | 09/2020 |
8224 | 20^72481-49 | 94300 | Lelio R Paula | 05/2016 |
8225 | (34*10^94293-259)/9 | 94294 | Robert Price | 12/2021 |
8226 | 44725!2+128 | 94289 | Robert Price | 03/2015 |
8227 | 4*10^94286+9 | 94287 | Robert Price | 05/2015 |
8228 | 17^76621+24 | 94279 | Lelio R Paula | 07/2016 |
8229 | 26274^3875+3875^26274 | 94279 | Hans Havermann | 09/2020 |
8230 | 4*10^94270-33 | 94271 | Robert Price | 04/2021 |
8231 | 2^313156-112859 | 94270 | Enzo Doria | 06/2014 |
8232 | (49*10^94262-211)/9 | 94263 | Robert Price | 06/2022 |
8233 | 6^121130-59 | 94258 | Lelio R Paula | 05/2016 |
8234 | 3^197542+28 | 94252 | Robert Price | 11/2013 |
8235 | (2^313076+3550)/2 | 94245 | Stefano Morozzi | 09/2020 |
8236 | (3^197521+2)/51655 | 94237 | Lelio R Paula | 04/2013 |
8237 | 2^313045-5955 | 94236 | Lelio R Paula | 03/2016 |
8238 | 22099^18358+18358^22099 | 94227 | Hans Havermann | 08/2020 |
8239 | (67*10^94212+23)/9 | 94213 | Ray Chandler | 02/2011 |
8240 | 2^312909-1865203 | 94195 | Enzo Doria | 03/2016 |
8241 | 25013^5830+5830^25013 | 94191 | Hans Havermann | 09/2020 |
8242 | 2^312848-149 | 94177 | Lelio R Paula | 11/2012 |
8243 | (2^312819+3278)/2 | 94168 | Stefano Morozzi | 02/2021 |
8244 | 4^156403-77597 | 94164 | Mike Oakes | 04/2011 |
8245 | 18^75004-85 | 94151 | Lelio R Paula | 06/2016 |
8246 | 19^73622-48 | 94145 | Lelio R Paula | 06/2016 |
8247 | (11^90397-3^90397)/8 | 94138 | Robert Price | 04/2011 |
8248 | (3^197287+5)/16 | 94129 | Lelio R Paula | 04/2013 |
8249 | (23*10^94123-29)/3 | 94124 | Robert Price | 03/2022 |
8250 | 24220^7691+7691^24220 | 94119 | Hans Havermann | 09/2020 |