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 ? |
5751 | 2^364936-1867979 | 109857 | Enzo Doria | 04/2016 |
5752 | 27013^11652+11652^27013 | 109846 | Hans Havermann | 10/2020 |
5753 | 2^364867-112087 | 109836 | Enzo Doria | 06/2014 |
5754 | 4^182415-85061 | 109825 | Mike Oakes | 06/2011 |
5755 | (2^364809+1034)/2 | 109819 | Stefano Morozzi | 09/2020 |
5756 | (53*10^109809-287)/9 | 109810 | Robert Price | 02/2022 |
5757 | (52*10^109807-133)/9 | 109808 | Robert Price | 09/2021 |
5758 | 25708^18675+18675^25708 | 109806 | Hans Havermann | 10/2020 |
5759 | (17^89227+16^89227)/33 | 109788 | Robert Price | 08/2013 |
5760 | 2^364687-187 | 109782 | Lelio R Paula | 01/2012 |
5761 | 2^364644+67 | 109769 | Lelio R Paula | 01/2012 |
5762 | (9787*2^364625-1)/24947 | 109763 | Lelio R Paula | 05/2015 |
5763 | (2^364616+5)/9 | 109760 | Lelio R Paula | 01/2014 |
5764 | (55*10^109759+89)/9 | 109760 | Robert Price | 03/2019 |
5765 | (14*32^72921+4171)/31 | 109757 | Eric Chen | 04/2023 |
5766 | 12^101694+73 | 109747 | Lelio R Paula | 03/2017 |
5767 | 13^98518-66 | 109744 | Lelio R Paula | 03/2017 |
5768 | (8*10^109721-269)/9 | 109721 | Robert Price | 09/2021 |
5769 | 2^364478-113261 | 109719 | Enzo Doria | 07/2014 |
5770 | (2^364513+1)/(3*44290516579) | 109719 | Gord Palameta | 08/2018 |
5771 | (24737*2^364454+1)/3 | 109716 | Lelio R Paula | 07/2014 |
5772 | 27814^8777+8777^27814 | 109681 | Hans Havermann | 10/2020 |
5773 | (62*10^109673+1)/9 | 109674 | Serge Batalov | 11/2010 |
5774 | 3^229824+340 | 109654 | Lelio R Paula | 03/2017 |
5775 | (16623*33^72206-8095)/16 | 109649 | Eric Chen | 05/2023 |
5776 | (2*10^109645-71)/3 | 109645 | Robert Price | 12/2017 |
5777 | 2^364169-112813 | 109626 | Enzo Doria | 06/2014 |
5778 | 28822^6359+6359^28822 | 109622 | Hans Havermann | 11/2020 |
5779 | 10^109615+21 | 109616 | Lelio R Paula | 03/2017 |
5780 | 10^109615-456267 | 109615 | Patrick De Geest | 03/2017 |
5781 | 2^364094-113 | 109604 | Lelio R Paula | 01/2012 |
5782 | 25251^21898+21898^25251 | 109600 | Hans Havermann | 11/2020 |
5783 | (4^182026+11)/3 | 109591 | Robert Price | 02/2017 |
5784 | 2^364041-6883 | 109588 | Lelio R Paula | 09/2011 |
5785 | 12^101545+11 | 109586 | Lelio R Paula | 03/2017 |
5786 | 26536^13475+13475^26536 | 109582 | Hans Havermann | 11/2020 |
5787 | 2^363842-2787101 | 109528 | Enzo Doria | 03/2016 |
5788 | (2^363812+3570)/2 | 109519 | Stefano Morozzi | 09/2020 |
5789 | (22699*2^363782+1)/763 | 109511 | Lelio R Paula | 07/2014 |
5790 | 2^363769+5711 | 109506 | Lelio R Paula | 04/2017 |
5791 | (31*2^363748+1)/49 | 109499 | Lelio R Paula | 03/2014 |
5792 | (2^363670+100280245065) | 109476 | Paul Bourdelais | 05/2022 |
5793 | p(22245)#/2-8 | 109462 | Robert Price | 01/2017 |
5794 | 2^363556+55 | 109442 | Lelio R Paula | 01/2012 |
5795 | (509*10^109414+31)/9 | 109416 | Robert Price | 08/2022 |
5796 | (2^363437+1682)/2 | 109406 | Stefano Morozzi | 09/2020 |
5797 | (55459*2^363411+1)/81 | 109401 | Lelio R Paula | 07/2014 |
5798 | 2^363406+1871569 | 109397 | Enzo Doria | 05/2016 |
5799 | (2^363370+1338)/2 | 109385 | Stefano Morozzi | 02/2021 |
5800 | (2^363349+7)/9 | 109378 | Lelio R Paula | 01/2014 |
5801 | (13869^26569-1)/(13869^163-1) | 109375 | Ryan Propper | 11/2023 |
5802 | 3^229236+100 | 109374 | Lelio R Paula | 03/2017 |
5803 | (2^363315+3258)/2 | 109369 | Stefano Morozzi | 02/2021 |
5804 | 2^363268-58695 | 109355 | Lelio R Paula | 04/2017 |
5805 | 2^363191-112321 | 109332 | Enzo Doria | 06/2014 |
5806 | 26231^14718+14718^26231 | 109327 | Hans Havermann | 10/2020 |
5807 | 25461^19672+19672^25461 | 109326 | Hans Havermann | 10/2020 |
5808 | 4*10^109316-63 | 109317 | Robert Price | 10/2018 |
5809 | 2^363120+3 | 109311 | Donovan Johnson | 04/2006 |
5810 | (10^109297-1)/9 | 109297 | Paul Bourdelais, Harvey Dubner | 04/2007 |
5811 | 2^363042-114641 | 109287 | Enzo Doria | 04/2015 |
5812 | 2^363020-112793 | 109280 | Enzo Doria | 06/2014 |
5813 | (8732^27889-1)/(8732^167-1) | 109256 | Ryan Propper | 11/2023 |
5814 | (85*10^109253+203)/9 | 109254 | Robert Price | 09/2020 |
5815 | 24990^23431+23431^24990 | 109202 | Hans Havermann | 10/2020 |
5816 | (2^362746+3738)/2 | 109198 | Stefano Morozzi | 09/2020 |
5817 | (2^362716+1282)/2 | 109189 | Stefano Morozzi | 01/2021 |
5818 | (67607*2^362695+1)/70499 | 109183 | Lelio R Paula | 07/2014 |
5819 | 25060^22737+22737^25060 | 109180 | Hans Havermann | 10/2020 |
5820 | (9*2^362644+1)/5 | 109167 | Lelio R Paula | 03/2014 |
5821 | 5^156181+38 | 109166 | Lelio R Paula | 02/2017 |
5822 | (2^362623+3618)/2 | 109161 | Stefano Morozzi | 09/2020 |
5823 | 25138^21993+21993^25138 | 109157 | Hans Havermann | 10/2020 |
5824 | 4*10^109142+91 | 109143 | Robert Price | 08/2018 |
5825 | 3^228750+68 | 109142 | Lelio R Paula | 03/2017 |
5826 | (2^362561+786)/2 | 109142 | Stefano Morozzi | 09/2020 |
5827 | (23*10^109113+61)/3 | 109114 | Robert Price | 01/2022 |
5828 | 2*10^109100-49 | 109101 | Robert Price | 09/2020 |
5829 | (10223*2^362376+1)/9 | 109090 | Lelio R Paula | 07/2014 |
5830 | 65^60167-64^60167 | 109078 | Robert Price | 08/2013 |
5831 | 7*3^228589-2 | 109066 | Lelio R Paula | 09/2014 |
5832 | 24874^24243+24243^24874 | 109063 | Hans Havermann | 11/2020 |
5833 | (17*10^109059-161)/9 | 109060 | Robert Price | 02/2021 |
5834 | 29158^5495+5495^29158 | 109050 | Hans Havermann | 11/2020 |
5835 | (78557*2^362188+1)/189 | 109033 | Lelio R Paula | 07/2014 |
5836 | 2*7^129008+9 | 109025 | Lelio R Paula | 11/2014 |
5837 | 88*10^109022+7 | 109024 | Robert Price | 03/2020 |
5838 | 28194^7337+7337^28194 | 108985 | Hans Havermann | 10/2020 |
5839 | (5*10^108960+91)/3 | 108961 | Robert Price | 04/2018 |
5840 | 2^361957+76367 | 108960 | Enzo Doria | 05/2016 |
5841 | 2^361873-1411 | 108935 | Yves Minet | 01/2013 |
5842 | (24737*2^361857+1)/35 | 108933 | Lelio R Paula | 07/2014 |
5843 | 115^52842-114 | 108892 | Norbert Schneider | 07/2018 |
5844 | 13^97745-24 | 108883 | Lelio R Paula | 03/2017 |
5845 | (2^361696+2562)/2 | 108882 | Stefano Morozzi | 09/2020 |
5846 | 17^88475+8 | 108864 | Lelio R Paula | 04/2017 |
5847 | 2^361567+51 | 108843 | Lelio R Paula | 01/2012 |
5848 | 2^361563-109501 | 108842 | Enzo Doria | 04/2014 |
5849 | 12^100852+83 | 108838 | Lelio R Paula | 03/2017 |
5850 | 2^361497+767 | 108822 | Lelio R Paula | 01/2012 |
5851 | 28019^7630+7630^28019 | 108785 | Hans Havermann | 10/2020 |
5852 | 24723^24758-24758^24723 | 108765 | Norbert Schneider | 11/2020 |
5853 | 2^361280-112349 | 108757 | Enzo Doria | 06/2014 |
5854 | 2^361271-5955 | 108754 | Lelio R Paula | 04/2017 |
5855 | 12^100757+19 | 108736 | Lelio R Paula | 03/2017 |
5856 | 25368^19333+19333^25368 | 108735 | Hans Havermann | 11/2020 |
5857 | (2^361195+3618)/2 | 108731 | Stefano Morozzi | 09/2020 |
5858 | 2^361153-111546435 | 108718 | Paul Bourdelais | 10/2021 |
5859 | 14^94849-61 | 108710 | Lelio R Paula | 05/2017 |
5860 | 27267^9700+9700^27267 | 108708 | Hans Havermann | 11/2020 |
5861 | (19^85009-5^85009)/14 | 108705 | Robert Price | 07/2013 |
5862 | 24832^23853+23853^24832 | 108704 | Hans Havermann | 11/2020 |
5863 | 27502^8963+8963^27502 | 108701 | Hans Havermann | 11/2020 |
5864 | 25807^16278+16278^25807 | 108689 | Hans Havermann | 11/2020 |
5865 | (11*2^361027+1)/40463 | 108677 | Lelio R Paula | 03/2014 |
5866 | 2^360973+1299 | 108664 | Lelio R Paula | 04/2017 |
5867 | 2^360959-109101 | 108660 | Enzo Doria | 04/2014 |
5868 | 6*5^155454+7 | 108659 | Lelio R Paula | 11/2014 |
5869 | 10^108656+79 | 108657 | Lelio R Paula | 02/2017 |
5870 | 10^108656-50001 | 108656 | Patrick De Geest | 03/2017 |
5871 | 2^360918-35 | 108648 | Lelio R Paula | 01/2012 |
5872 | 4*10^108646-47 | 108647 | Robert Price | 10/2021 |
5873 | (11*2^360886-1)/7 | 108638 | Lelio R Paula | 03/2014 |
5874 | 3*10^108624-49 | 108625 | Robert Price | 07/2018 |
5875 | (2^360831+666)/2 | 108621 | Stefano Morozzi | 09/2020 |
5876 | (134*10^108610+7)/3 | 108612 | Robert Price | 02/2020 |
5877 | 2*10^108609+29 | 108610 | Robert Price | 02/2021 |
5878 | (10^108604+7)/71/191 | 108600 | Alfred Reich | 06/2021 |
5879 | (2^360742+3222)/2 | 108594 | Stefano Morozzi | 09/2020 |
5880 | 2^360706-458903 | 108584 | Enzo Doria | 04/2015 |
5881 | 31112^3087+3087^31112 | 108567 | Hans Havermann | 11/2020 |
5882 | 2^360641-113185 | 108564 | Enzo Doria | 07/2014 |
5883 | (29*10^108559-71)/3 | 108560 | Robert Price | 07/2021 |
5884 | 2^360618+75019 | 108557 | Enzo Doria | 04/2016 |
5885 | 2^360607-216367 | 108554 | Harrison Mesh | 06/2022 |
5886 | (14*10^108545-221)/9 | 108546 | Robert Price | 04/2018 |
5887 | (44*10^108544-233)/9 | 108545 | Robert Price | 01/2019 |
5888 | 3*5^155286+2 | 108541 | Lelio R Paula | 11/2014 |
5889 | 7^128422+4 | 108530 | Lelio R Paula | 11/2014 |
5890 | 7^128421-80 | 108529 | Lelio R Paula | 03/2017 |
5891 | 2^360497+53807 | 108521 | Yves Minet | 07/2010 |
5892 | 2^360468-4559 | 108512 | Yves Minet | 03/2019 |
5893 | (2^360451+578)/2 | 108507 | Stefano Morozzi | 09/2020 |
5894 | 2^360366+1459 | 108481 | Lelio R Paula | 09/2011 |
5895 | 2^360364-1499 | 108481 | Yves Minet | 05/2013 |
5896 | 25662^16871+16871^25662 | 108477 | Hans Havermann | 10/2020 |
5897 | 25009^21748+21748^25009 | 108475 | Hans Havermann | 10/2020 |
5898 | 14^94642-23 | 108472 | Lelio R Paula | 05/2017 |
5899 | (197*10^108446-11)/3 | 108448 | Robert Price | 03/2020 |
5900 | (2^360229+1290)/2 | 108440 | Stefano Morozzi | 01/2021 |
5901 | 4*10^108412+9 | 108413 | Robert Price | 05/2015 |
5902 | (259*10^108409-1)/3 | 108411 | Robert Price | 12/2020 |
5903 | (2^360122+1194)/2 | 108408 | Stefano Morozzi | 09/2020 |
5904 | 133^51042-132 | 108406 | Norbert Schneider | 02/2019 |
5905 | 2^360093-351103 | 108399 | Enzo Doria | 06/2015 |
5906 | 2^360042+723 | 108384 | Lelio R Paula | 01/2012 |
5907 | 85^56164-2 | 108364 | Lelio R Paula | 10/2013 |
5908 | 2^359937-29323 | 108352 | Enzo Doria | 07/2015 |
5909 | (2^359922+1762)/2 | 108348 | Stefano Morozzi | 09/2020 |
5910 | (2^359903+758)/2 | 108342 | Stefano Morozzi | 09/2020 |
5911 | 2^359898-19997 | 108341 | Enzo Doria | 06/2013 |
5912 | (2^359886+2122)/2 | 108337 | Stefano Morozzi | 09/2020 |
5913 | 2*5^154948-7 | 108305 | Lelio R Paula | 11/2014 |
5914 | 10^108303+13 | 108304 | Lelio R Paula | 02/2017 |
5915 | 10^108303-207959 | 108303 | Patrick De Geest | 03/2017 |
5916 | 7*10^108300+69 | 108301 | Robert Price | 07/2019 |
5917 | 2^359730+753 | 108290 | Lelio R Paula | 12/2011 |
5918 | (8*10^108273+13)/3 | 108274 | Robert Price | 06/2018 |
5919 | 2^359650-19797 | 108266 | Enzo Doria | 06/2013 |
5920 | (47*10^108261+259)/9 | 108262 | Robert Price | 03/2022 |
5921 | 91^55259-90^55259 | 108255 | Robert Price | 07/2015 |
5922 | 10^108251+85749 | 108252 | Patrick De Geest | 02/2021 |
5923 | 10^108251-50000000001 | 108251 | Robert Price | 12/2020 |
5924 | 10^108251-252081 | 108251 | Patrick De Geest | 02/2021 |
5925 | 4*10^108238-71 | 108239 | Robert Price | 12/2018 |
5926 | 3^226845+68 | 108233 | Lelio R Paula | 03/2017 |
5927 | 31836^2509+2509^31836 | 108227 | Hans Havermann | 10/2020 |
5928 | 5^154822-12866 | 108216 | Enzo Doria | 05/2014 |
5929 | (9*2^359480+1)/5 | 108215 | Lelio R Paula | 03/2014 |
5930 | 2^359454-177 | 108207 | Lelio R Paula | 12/2011 |
5931 | (19*10^108184+191)/3 | 108185 | Robert Price | 05/2019 |
5932 | 6^139022-59 | 108181 | Lelio R Paula | 02/2017 |
5933 | (19*10^108176-43)/3 | 108177 | Robert Price | 05/2019 |
5934 | 2^359352-179 | 108176 | Lelio R Paula | 12/2011 |
5935 | 2^359338-1858781 | 108172 | Enzo Doria | 04/2016 |
5936 | 12^100230-97 | 108167 | Lelio R Paula | 03/2017 |
5937 | 27028^10043+10043^27028 | 108163 | Hans Havermann | 11/2020 |
5938 | (25*2^359297-1)/137 | 108159 | Lelio R Paula | 03/2014 |
5939 | (2^359262-509203)/9 | 108148 | Lelio R Paula | 12/2013 |
5940 | 26583^11702+11702^26583 | 108147 | Hans Havermann | 11/2020 |
5941 | (3^226648-8)/352715633 | 108131 | Lelio R Paula | 12/2013 |
5942 | 6^138956-59 | 108129 | Lelio R Paula | 02/2017 |
5943 | 2^359184+30285 | 108126 | Lelio R Paula | 04/2017 |
5944 | 13^97059+10 | 108119 | Lelio R Paula | 03/2017 |
5945 | 2^359137-81 | 108112 | Lelio R Paula | 12/2011 |
5947 | (15185*2^359105-1)/567 | 108103 | Lelio R Paula | 05/2015 |
5946 | 2^359110-11655 | 108103 | Lelio R Paula | 04/2017 |
5948 | 2^359069-933075 | 108091 | Paul Bourdelais | 06/2021 |
5949 | 26227^13222+13222^26227 | 108090 | Hans Havermann | 10/2020 |
5950 | 8*6^138875+5 | 108067 | Lelio R Paula | 11/2014 |
5951 | 2^358987-216919 | 108066 | Enzo Doria | 05/2014 |
5952 | (2^358977+2714)/2 | 108063 | Stefano Morozzi | 09/2020 |
5953 | 39^67918-67918^39 | 108062 | Norbert Schneider | 02/2017 |
5954 | 24^78283-1*78283^24 | 108048 | Dylan Delgado | 12/2019 |
5955 | 25282^18767+18767^25282 | 108040 | Hans Havermann | 11/2020 |
5956 | 2^358785-16933 | 108006 | Enzo Doria | 05/2013 |
5957 | (2^358783+5)/33383 | 108000 | Lelio R Paula | 12/2013 |
5958 | 6^138770-77 | 107985 | Lelio R Paula | 02/2017 |
5959 | (2^358727-1017)/94178479 | 107980 | Lelio R Paula | 12/2013 |
5960 | 10^107976+19 | 107977 | Lelio R Paula | 02/2017 |
5961 | 10^107976-202103 | 107976 | Patrick De Geest | 03/2017 |
5962 | 2^358684+75781 | 107975 | Enzo Doria | 04/2016 |
5963 | 24928^21411+21411^24928 | 107955 | Hans Havermann | 10/2020 |
5964 | 99^54094+2 | 107952 | Lelio R Paula | 06/2013 |
5965 | (2^358583-9)/391 | 107942 | Lelio R Paula | 12/2013 |
5966 | (2^358562+838)/2 | 107938 | Stefano Morozzi | 02/2021 |
5967 | 2^358538+55 | 107931 | Lelio R Paula | 12/2011 |
5968 | 2^358517-109009 | 107925 | Enzo Doria | 03/2014 |
5969 | 26395^12258+12258^26395 | 107914 | Hans Havermann | 10/2020 |
5970 | 3^226159+104 | 107906 | Lelio R Paula | 03/2017 |
5971 | 7*10^107870+39 | 107871 | Robert Price | 01/2023 |
5972 | 2^358310-11655 | 107863 | Lelio R Paula | 03/2017 |
5973 | (68*10^107852-329)/9 | 107853 | Robert Price | 03/2021 |
5974 | 2^358275-112987 | 107852 | Enzo Doria | 07/2014 |
5975 | 3^226038+4 | 107848 | Tyler NeSmith | 01/2021 |
5976 | 2^358233-20005 | 107839 | Enzo Doria | 06/2013 |
5977 | 2^358194+175 | 107828 | Lelio R Paula | 12/2011 |
5978 | 107^53125-106 | 107812 | Norbert Schneider | 12/2016 |
5979 | (25*10^107804+161)/3 | 107805 | Robert Price | 09/2019 |
5980 | 3^225922-2 | 107793 | Lelio R Paula | 10/2013 |
5981 | 12^99874-95 | 107783 | Lelio R Paula | 03/2017 |
5982 | (2^358042+3042)/2 | 107782 | Stefano Morozzi | 09/2020 |
5983 | 2^358030-171 | 107778 | Lelio R Paula | 12/2011 |
5984 | 11^103488-4170 | 107772 | Enzo Doria | 12/2012 |
5985 | 2^357930-4547 | 107748 | Yves Minet | 03/2019 |
5986 | 2^357911-20089 | 107742 | Enzo Doria | 06/2013 |
5987 | 2^357885+12857 | 107735 | Yves Minet | 07/2010 |
5988 | (248587#+9289)/(7*1327*111512899) | 107733 | Makoto Morimoto | 06/2019 |
5989 | 7^127473+40 | 107728 | Lelio R Paula | 03/2017 |
5990 | 5*2^357850+9 | 107725 | Lelio R Paula | 09/2014 |
5991 | 2^357786-291191 | 107705 | Enzo Doria | 06/2015 |
5992 | 28907^5320+5320^28907 | 107705 | Hans Havermann | 10/2020 |
5993 | 18^85798-23 | 107700 | Lelio R Paula | 04/2017 |
5994 | 3*10^107697+61 | 107698 | Robert Price | 09/2020 |
5995 | (14*10^107694-83)/3 | 107695 | Robert Price | 12/2018 |
5996 | 2^357688-1854269 | 107675 | Enzo Doria | 09/2015 |
5997 | 2^357671-112225 | 107670 | Enzo Doria | 06/2014 |
5998 | 34323^1370+1370^34323 | 107662 | Hans Havermann | 11/2020 |
5999 | 2^357618-21005 | 107654 | Enzo Doria | 06/2013 |
6000 | 2^357615-17505 | 107653 | Enzo Doria | 05/2013 |