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 ? |
6001 | 25777^18798+18798^25777 | 110174 | Hans Havermann | 10/2020 |
6002 | 2^365944+1871341 | 110161 | Enzo Doria | 05/2016 |
6003 | 2^365838-461267 | 110129 | Enzo Doria | 07/2015 |
6004 | 30152^4491+4491^30152 | 110126 | Hans Havermann | 11/2020 |
6005 | 2^365802+23359 | 110118 | Enzo Doria | 10/2023 |
6006 | 10^110111+464019 | 110112 | Patrick De Geest | 05/2021 |
6007 | 10^110111-40001 | 110111 | Robert Price | 01/2021 |
6008 | 2^365763-6555 | 110106 | Lelio R Paula | 04/2017 |
6009 | (2^365689+2^182845+1)/5 | 110083 | | 11/2006 |
6010 | 2^365666-459161 | 110077 | Enzo Doria | 05/2015 |
6011 | 3*10^110076-7 | 110077 | Robert Price | 08/2015 |
6012 | 13^98785-72 | 110041 | Lelio R Paula | 03/2017 |
6013 | (25^78707+1)/26 | 110027 | Robert Price | 06/2013 |
6014 | (7*10^110027-277)/9 | 110027 | Robert Price | 02/2022 |
6015 | (16*10^110014-73)/3 | 110015 | Robert Price | 07/2020 |
6016 | (7*10^110006-151)/9 | 110006 | Robert Price | 02/2018 |
6017 | 6^141361-7 | 110001 | Lelio R Paula | 11/2014 |
6018 | 20^84536+81 | 109984 | Lelio R Paula | 03/2017 |
6019 | 5*2^365347-9 | 109982 | Lelio R Paula | 09/2014 |
6020 | 9*6^141303+7 | 109957 | Lelio R Paula | 11/2014 |
6021 | 27635^9522+9522^27635 | 109953 | Hans Havermann | 11/2020 |
6022 | 78^58099+77 | 109929 | Norbert Schneider | 02/2015 |
6023 | 2*10^109928-7 | 109929 | Robert Price | 10/2015 |
6024 | 30580^3933+3933^30580 | 109927 | Hans Havermann | 11/2020 |
6025 | 26380^14689+14689^26380 | 109926 | Hans Havermann | 11/2020 |
6026 | (2^365161+1130)/2 | 109925 | Stefano Morozzi | 09/2020 |
6027 | (27*2^365137-1)/2021 | 109916 | Lelio R Paula | 03/2014 |
6028 | (4441*29^75152-24)/7 | 109905 | Curtis Bright | 05/2015 |
6029 | (2^365087+7)/6015 | 109899 | Lelio R Paula | 01/2014 |
6030 | (2^365069+3774)/2 | 109897 | Stefano Morozzi | 02/2021 |
6031 | 29652^5083+5083^29652 | 109894 | Mark Rodenkirch | 10/2020 |
6032 | (2^365049+722)/2 | 109891 | Stefano Morozzi | 09/2020 |
6033 | (35*10^109888-89)/9 | 109889 | Robert Price | 02/2022 |
6034 | 18^87533-11 | 109878 | Lelio R Paula | 04/2017 |
6035 | 2^364982-5 | 109871 | Henri Lifchitz | 04/2006 |
6036 | 3*10^109861+59 | 109862 | Robert Price | 08/2018 |
6037 | 5*10^109860-69 | 109861 | Robert Price | 02/2021 |
6038 | 2^364936-1867979 | 109857 | Enzo Doria | 04/2016 |
6039 | 27013^11652+11652^27013 | 109846 | Hans Havermann | 10/2020 |
6040 | 2^364867-112087 | 109836 | Enzo Doria | 06/2014 |
6041 | 4^182415-85061 | 109825 | Mike Oakes | 06/2011 |
6042 | (2^364809+1034)/2 | 109819 | Stefano Morozzi | 09/2020 |
6043 | (53*10^109809-287)/9 | 109810 | Robert Price | 02/2022 |
6044 | (52*10^109807-133)/9 | 109808 | Robert Price | 09/2021 |
6045 | 25708^18675+18675^25708 | 109806 | Hans Havermann | 10/2020 |
6046 | (17^89227+16^89227)/33 | 109788 | Robert Price | 08/2013 |
6047 | 2^364687-187 | 109782 | Lelio R Paula | 01/2012 |
6048 | 2^364644+67 | 109769 | Lelio R Paula | 01/2012 |
6049 | (9787*2^364625-1)/24947 | 109763 | Lelio R Paula | 05/2015 |
6050 | (2^364616+5)/9 | 109760 | Lelio R Paula | 01/2014 |
6051 | (55*10^109759+89)/9 | 109760 | Robert Price | 03/2019 |
6052 | (14*32^72921+4171)/31 | 109757 | Eric Chen | 04/2023 |
6053 | 12^101694+73 | 109747 | Lelio R Paula | 03/2017 |
6054 | 13^98518-66 | 109744 | Lelio R Paula | 03/2017 |
6055 | (8*10^109721-269)/9 | 109721 | Robert Price | 09/2021 |
6056 | 2^364478-113261 | 109719 | Enzo Doria | 07/2014 |
6057 | (2^364513+1)/(3*44290516579) | 109719 | Gord Palameta | 08/2018 |
6058 | (24737*2^364454+1)/3 | 109716 | Lelio R Paula | 07/2014 |
6059 | 27814^8777+8777^27814 | 109681 | Hans Havermann | 10/2020 |
6060 | (62*10^109673+1)/9 | 109674 | Serge Batalov | 11/2010 |
6061 | 3^229824+340 | 109654 | Lelio R Paula | 03/2017 |
6062 | (16623*33^72206-8095)/16 | 109649 | Eric Chen | 05/2023 |
6063 | (2*10^109645-71)/3 | 109645 | Robert Price | 12/2017 |
6064 | 2^364169-112813 | 109626 | Enzo Doria | 06/2014 |
6065 | 28822^6359+6359^28822 | 109622 | Hans Havermann | 11/2020 |
6066 | 10^109615+21 | 109616 | Lelio R Paula | 03/2017 |
6067 | 10^109615-456267 | 109615 | Patrick De Geest | 03/2017 |
6068 | 2^364094-113 | 109604 | Lelio R Paula | 01/2012 |
6069 | 25251^21898+21898^25251 | 109600 | Hans Havermann | 11/2020 |
6070 | (4^182026+11)/3 | 109591 | Robert Price | 02/2017 |
6071 | 2^364041-6883 | 109588 | Lelio R Paula | 09/2011 |
6072 | 12^101545+11 | 109586 | Lelio R Paula | 03/2017 |
6073 | 26536^13475+13475^26536 | 109582 | Hans Havermann | 11/2020 |
6074 | 2^363842-2787101 | 109528 | Enzo Doria | 03/2016 |
6075 | (2^363812+3570)/2 | 109519 | Stefano Morozzi | 09/2020 |
6076 | (22699*2^363782+1)/763 | 109511 | Lelio R Paula | 07/2014 |
6077 | 2^363769+5711 | 109506 | Lelio R Paula | 04/2017 |
6078 | (31*2^363748+1)/49 | 109499 | Lelio R Paula | 03/2014 |
6079 | (2^363670+100280245065) | 109476 | Paul Bourdelais | 05/2022 |
6080 | p(22245)#/2-8 | 109462 | Robert Price | 01/2017 |
6081 | 2^363556+55 | 109442 | Lelio R Paula | 01/2012 |
6082 | (509*10^109414+31)/9 | 109416 | Robert Price | 08/2022 |
6083 | (2^363437+1682)/2 | 109406 | Stefano Morozzi | 09/2020 |
6084 | (55459*2^363411+1)/81 | 109401 | Lelio R Paula | 07/2014 |
6085 | 2^363406+1871569 | 109397 | Enzo Doria | 05/2016 |
6086 | (2^363370+1338)/2 | 109385 | Stefano Morozzi | 02/2021 |
6087 | (2^363349+7)/9 | 109378 | Lelio R Paula | 01/2014 |
6088 | (13869^26569-1)/(13869^163-1) | 109375 | Ryan Propper | 11/2023 |
6089 | 3^229236+100 | 109374 | Lelio R Paula | 03/2017 |
6090 | (2^363315+3258)/2 | 109369 | Stefano Morozzi | 02/2021 |
6091 | 2^363268-58695 | 109355 | Lelio R Paula | 04/2017 |
6092 | 2^363191-112321 | 109332 | Enzo Doria | 06/2014 |
6093 | 26231^14718+14718^26231 | 109327 | Hans Havermann | 10/2020 |
6094 | 25461^19672+19672^25461 | 109326 | Hans Havermann | 10/2020 |
6095 | 4*10^109316-63 | 109317 | Robert Price | 10/2018 |
6096 | 2^363120+3 | 109311 | Donovan Johnson | 04/2006 |
6097 | (10^109297-1)/9 | 109297 | Paul Bourdelais, Harvey Dubner | 04/2007 |
6098 | 2^363042-114641 | 109287 | Enzo Doria | 04/2015 |
6099 | 2^363020-112793 | 109280 | Enzo Doria | 06/2014 |
6100 | (8732^27889-1)/(8732^167-1) | 109256 | Ryan Propper | 11/2023 |
6101 | (85*10^109253+203)/9 | 109254 | Robert Price | 09/2020 |
6102 | 24990^23431+23431^24990 | 109202 | Hans Havermann | 10/2020 |
6103 | (2^362746+3738)/2 | 109198 | Stefano Morozzi | 09/2020 |
6104 | (2^362716+1282)/2 | 109189 | Stefano Morozzi | 01/2021 |
6105 | (67607*2^362695+1)/70499 | 109183 | Lelio R Paula | 07/2014 |
6106 | 25060^22737+22737^25060 | 109180 | Hans Havermann | 10/2020 |
6107 | (9*2^362644+1)/5 | 109167 | Lelio R Paula | 03/2014 |
6108 | 5^156181+38 | 109166 | Lelio R Paula | 02/2017 |
6109 | (2^362623+3618)/2 | 109161 | Stefano Morozzi | 09/2020 |
6110 | 25138^21993+21993^25138 | 109157 | Hans Havermann | 10/2020 |
6111 | 4*10^109142+91 | 109143 | Robert Price | 08/2018 |
6112 | 3^228750+68 | 109142 | Lelio R Paula | 03/2017 |
6113 | (2^362561+786)/2 | 109142 | Stefano Morozzi | 09/2020 |
6114 | (23*10^109113+61)/3 | 109114 | Robert Price | 01/2022 |
6115 | 2*10^109100-49 | 109101 | Robert Price | 09/2020 |
6116 | (10223*2^362376+1)/9 | 109090 | Lelio R Paula | 07/2014 |
6117 | 65^60167-64^60167 | 109078 | Robert Price | 08/2013 |
6118 | 7*3^228589-2 | 109066 | Lelio R Paula | 09/2014 |
6119 | 24874^24243+24243^24874 | 109063 | Hans Havermann | 11/2020 |
6120 | (17*10^109059-161)/9 | 109060 | Robert Price | 02/2021 |
6121 | 29158^5495+5495^29158 | 109050 | Hans Havermann | 11/2020 |
6122 | (78557*2^362188+1)/189 | 109033 | Lelio R Paula | 07/2014 |
6123 | 2*7^129008+9 | 109025 | Lelio R Paula | 11/2014 |
6124 | 88*10^109022+7 | 109024 | Robert Price | 03/2020 |
6125 | 28194^7337+7337^28194 | 108985 | Hans Havermann | 10/2020 |
6126 | (5*10^108960+91)/3 | 108961 | Robert Price | 04/2018 |
6127 | 2^361957+76367 | 108960 | Enzo Doria | 05/2016 |
6128 | 2^361873-1411 | 108935 | Yves Minet | 01/2013 |
6129 | (24737*2^361857+1)/35 | 108933 | Lelio R Paula | 07/2014 |
6130 | 115^52842-114 | 108892 | Norbert Schneider | 07/2018 |
6131 | 13^97745-24 | 108883 | Lelio R Paula | 03/2017 |
6132 | (2^361696+2562)/2 | 108882 | Stefano Morozzi | 09/2020 |
6133 | 17^88475+8 | 108864 | Lelio R Paula | 04/2017 |
6134 | 2^361567+51 | 108843 | Lelio R Paula | 01/2012 |
6135 | 2^361563-109501 | 108842 | Enzo Doria | 04/2014 |
6136 | 12^100852+83 | 108838 | Lelio R Paula | 03/2017 |
6137 | 2^361497+767 | 108822 | Lelio R Paula | 01/2012 |
6138 | 28019^7630+7630^28019 | 108785 | Hans Havermann | 10/2020 |
6139 | 24723^24758-24758^24723 | 108765 | Norbert Schneider | 11/2020 |
6140 | 2^361280-112349 | 108757 | Enzo Doria | 06/2014 |
6141 | 2^361271-5955 | 108754 | Lelio R Paula | 04/2017 |
6142 | 12^100757+19 | 108736 | Lelio R Paula | 03/2017 |
6143 | 25368^19333+19333^25368 | 108735 | Hans Havermann | 11/2020 |
6144 | (2^361195+3618)/2 | 108731 | Stefano Morozzi | 09/2020 |
6145 | 2^361153-111546435 | 108718 | Paul Bourdelais | 10/2021 |
6146 | 14^94849-61 | 108710 | Lelio R Paula | 05/2017 |
6147 | 27267^9700+9700^27267 | 108708 | Hans Havermann | 11/2020 |
6148 | (19^85009-5^85009)/14 | 108705 | Robert Price | 07/2013 |
6149 | 24832^23853+23853^24832 | 108704 | Hans Havermann | 11/2020 |
6150 | 27502^8963+8963^27502 | 108701 | Hans Havermann | 11/2020 |
6151 | 25807^16278+16278^25807 | 108689 | Hans Havermann | 11/2020 |
6152 | (11*2^361027+1)/40463 | 108677 | Lelio R Paula | 03/2014 |
6153 | 2^360973+1299 | 108664 | Lelio R Paula | 04/2017 |
6154 | 2^360959-109101 | 108660 | Enzo Doria | 04/2014 |
6155 | 6*5^155454+7 | 108659 | Lelio R Paula | 11/2014 |
6156 | 10^108656+79 | 108657 | Lelio R Paula | 02/2017 |
6157 | 10^108656-50001 | 108656 | Patrick De Geest | 03/2017 |
6158 | 2^360918-35 | 108648 | Lelio R Paula | 01/2012 |
6159 | 4*10^108646-47 | 108647 | Robert Price | 10/2021 |
6160 | (11*2^360886-1)/7 | 108638 | Lelio R Paula | 03/2014 |
6161 | 3*10^108624-49 | 108625 | Robert Price | 07/2018 |
6162 | (2^360831+666)/2 | 108621 | Stefano Morozzi | 09/2020 |
6163 | (134*10^108610+7)/3 | 108612 | Robert Price | 02/2020 |
6164 | 2*10^108609+29 | 108610 | Robert Price | 02/2021 |
6165 | (10^108604+7)/71/191 | 108600 | Alfred Reich | 06/2021 |
6166 | (2^360742+3222)/2 | 108594 | Stefano Morozzi | 09/2020 |
6167 | 2^360706-458903 | 108584 | Enzo Doria | 04/2015 |
6168 | 31112^3087+3087^31112 | 108567 | Hans Havermann | 11/2020 |
6169 | 2^360641-113185 | 108564 | Enzo Doria | 07/2014 |
6170 | (29*10^108559-71)/3 | 108560 | Robert Price | 07/2021 |
6171 | 2^360618+75019 | 108557 | Enzo Doria | 04/2016 |
6172 | 2^360607-216367 | 108554 | Harrison Mesh | 06/2022 |
6173 | (14*10^108545-221)/9 | 108546 | Robert Price | 04/2018 |
6174 | (44*10^108544-233)/9 | 108545 | Robert Price | 01/2019 |
6175 | 3*5^155286+2 | 108541 | Lelio R Paula | 11/2014 |
6176 | 7^128422+4 | 108530 | Lelio R Paula | 11/2014 |
6177 | 7^128421-80 | 108529 | Lelio R Paula | 03/2017 |
6178 | 2^360497+53807 | 108521 | Yves Minet | 07/2010 |
6179 | 2^360468-4559 | 108512 | Yves Minet | 03/2019 |
6180 | (2^360451+578)/2 | 108507 | Stefano Morozzi | 09/2020 |
6181 | 2^360366+1459 | 108481 | Lelio R Paula | 09/2011 |
6182 | 2^360364-1499 | 108481 | Yves Minet | 05/2013 |
6183 | 25662^16871+16871^25662 | 108477 | Hans Havermann | 10/2020 |
6184 | 25009^21748+21748^25009 | 108475 | Hans Havermann | 10/2020 |
6185 | 14^94642-23 | 108472 | Lelio R Paula | 05/2017 |
6186 | (197*10^108446-11)/3 | 108448 | Robert Price | 03/2020 |
6187 | (2^360229+1290)/2 | 108440 | Stefano Morozzi | 01/2021 |
6188 | 4*10^108412+9 | 108413 | Robert Price | 05/2015 |
6189 | (259*10^108409-1)/3 | 108411 | Robert Price | 12/2020 |
6190 | (2^360122+1194)/2 | 108408 | Stefano Morozzi | 09/2020 |
6191 | 133^51042-132 | 108406 | Norbert Schneider | 02/2019 |
6192 | 2^360093-351103 | 108399 | Enzo Doria | 06/2015 |
6193 | 2^360042+723 | 108384 | Lelio R Paula | 01/2012 |
6194 | 85^56164-2 | 108364 | Lelio R Paula | 10/2013 |
6195 | 2^359937-29323 | 108352 | Enzo Doria | 07/2015 |
6196 | (2^359922+1762)/2 | 108348 | Stefano Morozzi | 09/2020 |
6197 | (2^359903+758)/2 | 108342 | Stefano Morozzi | 09/2020 |
6198 | 2^359898-19997 | 108341 | Enzo Doria | 06/2013 |
6199 | (2^359886+2122)/2 | 108337 | Stefano Morozzi | 09/2020 |
6200 | 2*5^154948-7 | 108305 | Lelio R Paula | 11/2014 |
6201 | 10^108303+13 | 108304 | Lelio R Paula | 02/2017 |
6202 | 10^108303-207959 | 108303 | Patrick De Geest | 03/2017 |
6203 | 7*10^108300+69 | 108301 | Robert Price | 07/2019 |
6204 | 2^359730+753 | 108290 | Lelio R Paula | 12/2011 |
6205 | (8*10^108273+13)/3 | 108274 | Robert Price | 06/2018 |
6206 | 2^359650-19797 | 108266 | Enzo Doria | 06/2013 |
6207 | (47*10^108261+259)/9 | 108262 | Robert Price | 03/2022 |
6208 | 91^55259-90^55259 | 108255 | Robert Price | 07/2015 |
6209 | 10^108251+85749 | 108252 | Patrick De Geest | 02/2021 |
6210 | 10^108251-50000000001 | 108251 | Robert Price | 12/2020 |
6211 | 10^108251-252081 | 108251 | Patrick De Geest | 02/2021 |
6212 | 4*10^108238-71 | 108239 | Robert Price | 12/2018 |
6213 | 3^226845+68 | 108233 | Lelio R Paula | 03/2017 |
6214 | 31836^2509+2509^31836 | 108227 | Hans Havermann | 10/2020 |
6215 | 5^154822-12866 | 108216 | Enzo Doria | 05/2014 |
6216 | (9*2^359480+1)/5 | 108215 | Lelio R Paula | 03/2014 |
6217 | 2^359454-177 | 108207 | Lelio R Paula | 12/2011 |
6218 | (19*10^108184+191)/3 | 108185 | Robert Price | 05/2019 |
6219 | 6^139022-59 | 108181 | Lelio R Paula | 02/2017 |
6220 | (19*10^108176-43)/3 | 108177 | Robert Price | 05/2019 |
6221 | 2^359352-179 | 108176 | Lelio R Paula | 12/2011 |
6222 | 2^359338-1858781 | 108172 | Enzo Doria | 04/2016 |
6223 | 12^100230-97 | 108167 | Lelio R Paula | 03/2017 |
6224 | 27028^10043+10043^27028 | 108163 | Hans Havermann | 11/2020 |
6225 | (25*2^359297-1)/137 | 108159 | Lelio R Paula | 03/2014 |
6226 | (2^359262-509203)/9 | 108148 | Lelio R Paula | 12/2013 |
6227 | 26583^11702+11702^26583 | 108147 | Hans Havermann | 11/2020 |
6228 | (3^226648-8)/352715633 | 108131 | Lelio R Paula | 12/2013 |
6229 | 6^138956-59 | 108129 | Lelio R Paula | 02/2017 |
6230 | 2^359184+30285 | 108126 | Lelio R Paula | 04/2017 |
6231 | 13^97059+10 | 108119 | Lelio R Paula | 03/2017 |
6232 | 2^359137-81 | 108112 | Lelio R Paula | 12/2011 |
6234 | (15185*2^359105-1)/567 | 108103 | Lelio R Paula | 05/2015 |
6233 | 2^359110-11655 | 108103 | Lelio R Paula | 04/2017 |
6235 | 2^359069-933075 | 108091 | Paul Bourdelais | 06/2021 |
6236 | 26227^13222+13222^26227 | 108090 | Hans Havermann | 10/2020 |
6237 | 8*6^138875+5 | 108067 | Lelio R Paula | 11/2014 |
6238 | 2^358987-216919 | 108066 | Enzo Doria | 05/2014 |
6239 | (2^358977+2714)/2 | 108063 | Stefano Morozzi | 09/2020 |
6240 | 39^67918-67918^39 | 108062 | Norbert Schneider | 02/2017 |
6241 | 24^78283-1*78283^24 | 108048 | Dylan Delgado | 12/2019 |
6242 | 25282^18767+18767^25282 | 108040 | Hans Havermann | 11/2020 |
6243 | 2^358785-16933 | 108006 | Enzo Doria | 05/2013 |
6244 | (2^358783+5)/33383 | 108000 | Lelio R Paula | 12/2013 |
6245 | 6^138770-77 | 107985 | Lelio R Paula | 02/2017 |
6246 | (2^358727-1017)/94178479 | 107980 | Lelio R Paula | 12/2013 |
6247 | 10^107976+19 | 107977 | Lelio R Paula | 02/2017 |
6248 | 10^107976-202103 | 107976 | Patrick De Geest | 03/2017 |
6249 | 2^358684+75781 | 107975 | Enzo Doria | 04/2016 |
6250 | 24928^21411+21411^24928 | 107955 | Hans Havermann | 10/2020 |