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 ? |
1001 | 2^974441+1878407 | 293336 | Enzo Doria | 08/2016 |
1002 | 2^974180+1876831 | 293258 | Enzo Doria | 07/2016 |
1003 | 2^974170+453319 | 293255 | Enzo Doria | 01/2017 |
1004 | 2^972744-114419 | 292826 | Enzo Doria | 03/2015 |
1005 | 11^281143+6 | 292781 | Paul Bourdelais | 03/2021 |
1006 | (4*10^292619-91)/3 | 292620 | Robert Price | 12/2022 |
1007 | E(69468)/13247122135 | 292549 | Sergey Batalov | 10/2023 |
1008 | CopelandErdos(292447) | 292447 | Mark Rodenkirch | 12/2015 |
1009 | 3*10^292344-97 | 292345 | Robert Price | 01/2023 |
1010 | 2^970911+1873373 | 292274 | Enzo Doria | 07/2016 |
1011 | 2^970670+383059 | 292201 | Enzo Doria | 12/2016 |
1012 | 2^970141-100363 | 292042 | Enzo Doria | 02/2014 |
1013 | 2^969945+1874717 | 291983 | Enzo Doria | 06/2016 |
1014 | 2^969384+1880701 | 291814 | Enzo Doria | 10/2016 |
1015 | (13*10^291721-43)/3 | 291722 | Robert Price | 08/2023 |
1016 | (3^611333-2^611333)/(99035947*87109884549431) | 291659 | Mike Oakes | 02/2017 |
1017 | 3*2^963892+5 | 290161 | Jon Grantham | 07/2023 |
1018 | (151*10^289877-1)/3 | 289879 | Robert Price | 07/2023 |
1019 | 2^962194+60139 | 289650 | Enzo Doria | 02/2016 |
1020 | 2^959307-1861507 | 288781 | Enzo Doria | 11/2015 |
1021 | (17*10^288692+31)/3 | 288693 | Robert Price | 04/2023 |
1022 | 14^251797-3 | 288592 | Paul Bourdelais | 12/2020 |
1023 | (8745^73441-1)/(8745^271-1) | 288419 | Ryan Propper | 11/2023 |
1024 | (2*10^288371-23)/3 | 288371 | Robert Price | 10/2022 |
1025 | (31*10^288238-1)/3 | 288240 | Robert Price | 02/2023 |
1026 | (7^341063-2^341063)/5 | 288231 | Jon Grantham | 08/2023 |
1027 | 2^957372+283531 | 288198 | Enzo Doria | 02/2017 |
1028 | 2^957192-1864049 | 288144 | Enzo Doria | 01/2016 |
1029 | (95774^58081-1)/(95774^241-1) | 288116 | Ryan Propper | 11/2023 |
1030 | 2^956446+1875199 | 287919 | Enzo Doria | 06/2016 |
1031 | 2^956283+219143 | 287870 | Enzo Doria | 02/2024 |
1032 | 7*10^287854+39 | 287855 | Robert Price | 01/2023 |
1033 | 2^956082+28309 | 287810 | Enzo Doria | 02/2024 |
1034 | (17^233689+1)/18 | 287542 | Paul Bourdelais | 09/2013 |
1035 | 2^955178+1883587 | 287538 | Enzo Doria | 12/2016 |
1036 | 2^954127-954127^2 | 287221 | Henri Lifchitz | 01/2021 |
1037 | 2^954001+1875257 | 287183 | Enzo Doria | 07/2016 |
1038 | 2*10^287175-49 | 287176 | Robert Price | 05/2023 |
1039 | (22*10^287085-19)/3 | 287086 | Robert Price | 07/2023 |
1040 | 2^953488+22411 | 287029 | Enzo Doria | 09/2023 |
1041 | (23891^65536+1)/2 | 286932 | Serge Batalov | 03/2018 |
1042 | 5*3^601336-2^601336 | 286911 | Jon Grantham | 08/2023 |
1043 | 2^952268+1878601 | 286662 | Enzo Doria | 09/2016 |
1044 | 2^950801+81557 | 286220 | Enzo Doria | 08/2016 |
1045 | 2^949403+217523 | 285799 | Enzo Doria | 01/2024 |
1046 | 22845^65536+22844^65536 | 285658 | Kellen Shenton | 03/2021 |
1047 | (11*2^948635-13)/15 | 285568 | Renaud Lifchitz | 11/2011 |
1048 | (54^164839+1)/55 | 285565 | Paul Bourdelais | 07/2019 |
1049 | 2^948510+782569 | 285530 | Enzo Doria | 10/2016 |
1050 | 2^948374-114137 | 285490 | Enzo Doria | 01/2015 |
1051 | 2^947739+1872923 | 285298 | Enzo Doria | 06/2016 |
1052 | 2^946274+218407 | 284857 | Enzo Doria | 03/2024 |
1053 | 7*10^284751+9 | 284752 | Robert Price | 04/2023 |
1054 | 2^944561-21 | 284342 | Stefano Morozzi | 07/2022 |
1055 | (4*10^284330-31)/9 | 284330 | Serge Batalov | 11/2022 |
1056 | (32*10^283826-23)/9 | 283827 | Serge Batalov | 01/2023 |
1057 | 2^942533+82067 | 283731 | Enzo Doria | 09/2016 |
1058 | 2^941469+39 | 283411 | Jon Grantham | 08/2023 |
1059 | (10^283342-31)/3 | 283342 | Robert Price | 03/2023 |
1060 | 2^940950+1874989 | 283255 | Enzo Doria | 06/2016 |
1061 | 2^940426-2786141 | 283097 | Enzo Doria | 11/2015 |
1062 | (2*10^282948-11)/9 | 282948 | Serge Batalov | 11/2022 |
1063 | 2^939881-114115 | 282933 | Enzo Doria | 12/2014 |
1064 | (14*10^282673-143)/3 | 282674 | Robert Price | 05/2023 |
1065 | 10^282493+632551 | 282494 | Patrick De Geest | 02/2017 |
1067 | 10^282493-9^282493 | 282493 | Jean-Louis Charton | 03/2007 |
1066 | 10^282493-132327 | 282493 | Patrick De Geest | 02/2017 |
1068 | 3^591827-2^591827 | 282374 | Mike Oakes | 08/2009 |
1069 | Tau(509^18946) | 282048 | Nik Lygeros / Olivier Rozier | 05/2015 |
1070 | 2^936716-1835519 | 281980 | Enzo Doria | 04/2015 |
1071 | 2^936588+79411 | 281942 | Enzo Doria | 06/2016 |
1072 | 2^936047+1877423 | 281779 | Enzo Doria | 08/2016 |
1073 | 2^935713+1869377 | 281678 | Enzo Doria | 05/2016 |
1074 | 2^933082+1877059 | 280886 | Enzo Doria | 08/2016 |
1075 | (5*10^280710+1)/3 | 280711 | Serge Batalov | 10/2014 |
1076 | 2^932064+26521 | 280580 | Enzo Doria | 12/2023 |
1077 | 2^931976+25 | 280553 | Stefano Morozzi | 05/2021 |
1078 | (42^172663+1)/43 | 280274 | Paul Bourdelais | 07/2019 |
1079 | (10^279978+11)/3 | 279978 | Serge Batalov | 07/2015 |
1080 | 2^929524-1864829 | 279815 | Enzo Doria | 01/2016 |
1081 | 2^929329+22697 | 279756 | Enzo Doria | 09/2023 |
1082 | 2^929152+60181 | 279703 | Enzo Doria | 02/2016 |
1083 | 2^928621-2787883 | 279543 | Enzo Doria | 11/2015 |
1084 | (2^927763+lucasV(-1,2,927763)+1)/2 | 279285 | Mike Oakes | 04/2017 |
1085 | 2^927429+1882277 | 279184 | Enzo Doria | 12/2016 |
1086 | 2^926766-11 | 278985 | Stefano Morozzi | 07/2022 |
1087 | (71^150697-1)/70 | 278978 | Paul Bourdelais | 07/2019 |
1088 | (33^183661-1)/32 | 278891 | Paul Bourdelais | 12/2018 |
1089 | (166*10^278633-7)/3 | 278635 | Robert Price | 02/2023 |
1090 | (2^925391+lucasV(-1,2,925391)+1)/2 | 278571 | Mike Oakes | 03/2017 |
1091 | 2^925330-1301 | 278553 | Yves Minet | 06/2015 |
1092 | (299*10^278546+1)/3 | 278548 | Robert Price | 01/2023 |
1093 | 2^925299+216053 | 278543 | Enzo Doria | 11/2023 |
1094 | (21^210599+1)/22 | 278457 | Paul Bourdelais | 08/2013 |
1095 | (266*10^278325+1)/3 | 278327 | Robert Price | 01/2023 |
1096 | 2^923261-69175 | 277930 | Enzo Doria | 11/2015 |
1097 | 757^96487-756^96487 | 277796 | Jean-Louis Charton | 12/2012 |
1098 | 2^922301+57 | 277641 | Stefano Morozzi | 05/2021 |
1099 | 7*10^277478-39 | 277479 | Robert Price | 07/2023 |
1100 | ((((((((((((((291487327^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2 | 277369 | Serge Batalov | 07/2016 |
1101 | (5^396413-1)/4 | 277081 | Paul Bourdelais | 06/2010 |
1102 | 2^920318+2792299 | 277044 | Enzo Doria | 02/2016 |
1103 | (16*10^276537-31)/3 | 276538 | Robert Price | 03/2023 |
1104 | 2^918596-2786939 | 276525 | Enzo Doria | 11/2015 |
1105 | 2^917910+28369 | 276319 | Enzo Doria | 01/2024 |
1106 | (6737^72361-1)/(6737^269-1) | 276002 | Ryan Propper | 11/2023 |
1107 | 2^916639+60563 | 275936 | Enzo Doria | 04/2016 |
1108 | (2*10^275922+1)/3 | 275922 | Serge Batalov | 11/2022 |
1109 | 2^916385-2786383 | 275860 | Enzo Doria | 11/2015 |
1110 | 2^915928+82081 | 275722 | Enzo Doria | 09/2016 |
1111 | (4*10^275671+11)/3 | 275672 | Tyler Busby | 02/2023 |
1112 | 2^915576-114137 | 275616 | Enzo Doria | 01/2015 |
1113 | 2^914828+1876741 | 275391 | Enzo Doria | 07/2016 |
1114 | 2^914757-2203 | 275370 | Yves Minet | 12/2015 |
1115 | 7*10^275157-27 | 275158 | Robert Price | 11/2022 |
1116 | 2^912744+20011 | 274764 | Enzo Doria | 06/2023 |
1117 | 2^911900+1871245 | 274510 | Enzo Doria | 05/2016 |
1118 | (16*10^274251-91)/3 | 274252 | Robert Price | 11/2022 |
1119 | 2^910999+9 | 274239 | Stefano Morozzi | 05/2021 |
1120 | (2^910869-7)/5 | 274199 | Stefano Morozzi | 08/2022 |
1121 | 2^910490+283429 | 274085 | Enzo Doria | 01/2017 |
1122 | 7*10^273983+39 | 273984 | Robert Price | 01/2023 |
1123 | 2^909988-1862249 | 273934 | Enzo Doria | 12/2015 |
1124 | 2^909297+1879157 | 273726 | Enzo Doria | 08/2016 |
1125 | 2^909257-1864423 | 273714 | Enzo Doria | 01/2016 |
1126 | 2^909254+31309 | 273713 | Enzo Doria | 03/2024 |
1127 | 2^908739-7 | 273558 | Stefano Morozzi | 07/2022 |
1128 | 2^908579-1861747 | 273510 | Enzo Doria | 12/2015 |
1129 | 3*2^908005-5 | 273338 | Jon Grantham | 07/2023 |
1130 | 2^907635-2791597 | 273226 | Enzo Doria | 01/2016 |
1131 | 2^907059+211193 | 273052 | Enzo Doria | 08/2023 |
1132 | (53^158341+1)/54 | 273022 | Paul Bourdelais | 07/2019 |
1133 | 2^906594-113201 | 272912 | Enzo Doria | 08/2014 |
1134 | 9*10^272800+91 | 272801 | Robert Price | 05/2023 |
1135 | 2*10^272710-49 | 272711 | Robert Price | 05/2023 |
1136 | 2*10^272403-91 | 272404 | Robert Price | 02/2023 |
1137 | 2^904773+1872167 | 272364 | Enzo Doria | 06/2016 |
1138 | 6^349903+5 | 272278 | Paul Bourdelais | 02/2021 |
1139 | 2^904149-1862473 | 272176 | Enzo Doria | 12/2015 |
1140 | (7*10^271757-61)/9 | 271757 | Erik Branger | 03/2012 |
1141 | 2^902497+1880117 | 271679 | Enzo Doria | 09/2016 |
1142 | 2^901858+77689 | 271487 | Enzo Doria | 06/2016 |
1143 | 2^901372+1875301 | 271341 | Enzo Doria | 06/2016 |
1144 | 2^901146+219049 | 271272 | Enzo Doria | 02/2024 |
1145 | (8*10^271039-77)/3 | 271040 | Robert Price | 04/2023 |
1146 | 2^899999+1144031 | 270927 | Norman Luhn | 09/2023 |
1147 | 2*10^270802-91 | 270803 | Robert Price | 02/2023 |
1148 | 23*10^270799-7 | 270801 | Robert Price | 10/2022 |
1149 | 2^899121-115033 | 270663 | Enzo Doria | 07/2015 |
1150 | 2^898900+883021 | 270596 | Enzo Doria | 12/2016 |
1151 | 2^898762-251511 | 270555 | Enzo Doria | 12/2013 |
1152 | 2^898589+47 | 270503 | Lelio R Paula | 03/2016 |
1153 | 187322*10^270343-187321 | 270349 | Danilo Nitsche | 09/2013 |
1154 | 48358*10^270343-48357 | 270348 | Danilo Nitsche | 09/2013 |
1155 | (10^270343-1)/9 | 270343 | Maksym Voznyy | 07/2007 |
1156 | 2^897867-3217 | 270285 | Yves Minet | 12/2016 |
1157 | 2^897849-115273 | 270280 | Enzo Doria | 07/2015 |
1158 | 2^897367+26289 | 270135 | Enzo Doria | 03/2024 |
1159 | 2^896616-114179 | 269909 | Enzo Doria | 12/2014 |
1160 | (12648^66049-1)/(12648^257-1) | 269881 | Ryan Propper | 11/2023 |
1161 | (22*10^269517-19)/3 | 269518 | Robert Price | 07/2023 |
1162 | 10^(2*134739+1)-7*10^134739-1 | 269479 | Robert Price | 07/2023 |
1163 | 2^894928+583561 | 269401 | Enzo Doria | 04/2017 |
1164 | 2^894673-39343 | 269324 | Enzo Doria | 08/2015 |
1165 | 2^893836+23221 | 269072 | Enzo Doria | 10/2023 |
1166 | 2^892718+29539 | 268735 | Enzo Doria | 03/2024 |
1167 | 2^892660+213901 | 268718 | Enzo Doria | 10/2023 |
1168 | F(1285607) | 268676 | Henri Lifchitz | 11/2008 |
1169 | 2^892234+22189 | 268590 | Enzo Doria | 09/2023 |
1170 | 2^892151+24863 | 268565 | Enzo Doria | 11/2023 |
1171 | 2^892112+187663 | 268553 | Enzo Doria | 08/2016 |
1172 | (266*10^268407-17)/3 | 268409 | Robert Price | 02/2023 |
1173 | (10^268207+1)/11 | 268206 | Paul Bourdelais | 02/2010 |
1174 | 2^890957-69583 | 268205 | Enzo Doria | 12/2015 |
1175 | 32841!^2+32840!^2+...+2!^2+1!^2 | 268128 | Serge Batalov | 07/2017 |
1176 | 2^890122-351401 | 267954 | Enzo Doria | 07/2015 |
1177 | (19^209359-1)/18 | 267718 | Paul Bourdelais | 08/2010 |
1178 | (85^138647+1)/86 | 267507 | Paul Bourdelais | 07/2019 |
1179 | (1005^89087-1)/1004 | 267451 | Jinyuan Wang | 01/2024 |
1180 | (485^99523-1)/484 | 267291 | Michael Stocker | 05/2020 |
1181 | 2^886324+77791 | 266811 | Enzo Doria | 06/2016 |
1182 | 7*10^266731+87 | 266732 | Robert Price | 11/2022 |
1183 | 2^886040+213901 | 266725 | Enzo Doria | 10/2023 |
1184 | 2^885242+217909 | 266485 | Enzo Doria | 01/2024 |
1185 | 2^884738+26809 | 266333 | Enzo Doria | 12/2023 |
1186 | 2^884714+1880527 | 266326 | Enzo Doria | 10/2016 |
1187 | 7*10^266024-47 | 266025 | Robert Price | 08/2023 |
1188 | 11^255426+255426^11 | 265999 | Serge Batalov | 05/2014 |
1189 | 2^883462-343421 | 265949 | Enzo Doria | 07/2015 |
1190 | 2^883442+1875019 | 265943 | Enzo Doria | 06/2016 |
1191 | 2^883333-291583 | 265910 | Enzo Doria | 07/2015 |
1192 | 2^883132+30421 | 265850 | Enzo Doria | 03/2024 |
1193 | 10^265566+3593037 | 265567 | Paul Bourdelais | 08/2023 |
1194 | (2*10^265321-17)/3 | 265321 | Serge Batalov | 11/2022 |
1195 | (16*10^265052-91)/3 | 265053 | Robert Price | 11/2022 |
1196 | 2^880411+1882673 | 265031 | Enzo Doria | 12/2016 |
1197 | (14*10^264791-11)/3 | 264792 | Robert Price | 08/2022 |
1198 | 2^879530-2788031 | 264765 | Enzo Doria | 12/2015 |
1199 | 2^879292+25171 | 264694 | Enzo Doria | 11/2023 |
1200 | (2*10^264539+61)/9 | 264539 | Serge Batalov | 11/2022 |
1201 | 2^878118+2793079 | 264340 | Enzo Doria | 02/2016 |
1202 | (2^878073-7)/5 | 264326 | Stefano Morozzi | 08/2022 |
1203 | (37^168527-1)/36 | 264283 | Paul Bourdelais | 08/2019 |
1204 | 5^377865+3^377865-1 | 264117 | Jon Grantham | 08/2023 |
1205 | (19*10^263971+131)/3 | 263972 | Robert Price | 08/2023 |
1206 | 2^876202-17 | 263764 | Stefano Morozzi | 07/2022 |
1207 | (14*10^263720-11)/3 | 263721 | Robert Price | 08/2022 |
1208 | 2^875748-461129 | 263627 | Enzo Doria | 07/2015 |
1209 | 2^875518-15 | 263558 | Stefano Morozzi | 07/2022 |
1210 | (29*10^263523-17)/3 | 263524 | Robert Price | 01/2023 |
1211 | 2^875181-2789923 | 263456 | Enzo Doria | 12/2015 |
1212 | 2^875101-350083 | 263432 | Enzo Doria | 08/2015 |
1213 | 2^874684+1883587 | 263307 | Enzo Doria | 01/2017 |
1214 | 2^874616+1872211 | 263286 | Enzo Doria | 05/2016 |
1215 | 2^874566+30049 | 263271 | Enzo Doria | 03/2024 |
1216 | 3^551491+16 | 263129 | Robert Price | 04/2020 |
1217 | 2^874073+2792957 | 263123 | Enzo Doria | 02/2016 |
1218 | (7*10^263006-19)/3 | 263007 | Robert Price | 10/2023 |
1219 | 2^873628+1878751 | 262989 | Enzo Doria | 08/2016 |
1220 | 2^873497+20837 | 262949 | Enzo Doria | 08/2023 |
1221 | 2^873288-269 | 262886 | Yves Minet | 01/2015 |
1222 | (2*10^262874+7)/9 | 262874 | Rytis Slatkevicius | 01/2023 |
1223 | (28*10^262655-43)/3 | 262656 | Robert Price | 04/2023 |
1224 | (2*10^262603-11)/9 | 262603 | Serge Batalov | 11/2022 |
1225 | 2^871991-4207 | 262496 | Yves Minet | 03/2021 |
1226 | (23*10^262324+7)/3 | 262325 | Robert Price | 05/2023 |
1227 | 2^871146-3173 | 262242 | Yves Minet | 07/2016 |
1228 | 2^869601-163 | 261776 | Yves Minet | 01/2015 |
1229 | (2*10^261704+61)/9 | 261704 | Serge Batalov | 11/2022 |
1230 | 2^869326+282859 | 261694 | Enzo Doria | 11/2016 |
1231 | 2^869035-114181 | 261606 | Enzo Doria | 12/2014 |
1232 | 2^868817+26777 | 261540 | Enzo Doria | 12/2023 |
1233 | 3^547850+9000004 | 261391 | Enzo Doria | 10/2023 |
1234 | 4*10^261248+91 | 261249 | Robert Price | 11/2022 |
1235 | 2^865715+80303 | 260607 | Enzo Doria | 07/2016 |
1236 | 2^865474+23209 | 260534 | Enzo Doria | 10/2023 |
1237 | 2^865238+214759 | 260463 | Enzo Doria | 11/2023 |
1238 | 2^865048+1882411 | 260406 | Enzo Doria | 12/2016 |
1239 | 2^864899-113871 | 260361 | Enzo Doria | 10/2014 |
1240 | (16*10^260256-91)/3 | 260257 | Robert Price | 11/2022 |
1241 | (7*10^260198-61)/9 | 260198 | Erik Branger | 11/2011 |
1242 | 2^864298-2788421 | 260180 | Enzo Doria | 12/2015 |
1243 | 2^864039-2791357 | 260102 | Enzo Doria | 01/2016 |
1244 | 2^863621-113833 | 259976 | Enzo Doria | 10/2014 |
1245 | 10^259945-11 | 259945 | Mischa Rodermond | 04/2023 |
1246 | (28*10^259897+113)/3 | 259898 | Robert Price | 06/2023 |
1247 | 2^862837+75767 | 259740 | Enzo Doria | 05/2016 |
1248 | 2^862637-31 | 259680 | Jorge Menendez | 05/2019 |
1249 | 2^862088+1874941 | 259515 | Enzo Doria | 06/2016 |
1250 | 2^861987-134557 | 259484 | Hans Havermann | 10/2018 |