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 | 2^342966-16995 | 103244 | Lelio R Paula | 11/2016 |
5752 | 23797^21774+21774^23797 | 103230 | Hans Havermann | 09/2020 |
5753 | 25388^11637+11637^25388 | 103224 | Hans Havermann | 09/2020 |
5754 | (2^342876+2086)/2 | 103216 | Stefano Morozzi | 09/2020 |
5755 | (16*10^103214-31)/3 | 103215 | Robert Price | 03/2019 |
5757 | (37*10^103211+611)/9 | 103212 | Robert Price | 11/2018 |
5756 | (238*10^103210-1)/3 | 103212 | Robert Price | 04/2020 |
5758 | 5^147650+16 | 103203 | Lelio R Paula | 11/2016 |
5759 | gcd(Phi(852060,10),(10^42603+1)^3*(10^42603+1-10^21302)+(10^85206+10^42603)*(10^21302+1)+10^127809) | 103201 | Serge Batalov | 04/2015 |
5760 | 2^342817+2707067 | 103199 | Enzo Doria | 03/2016 |
5761 | 43^63177-42 | 103198 | Theodore Burton | 02/2007 |
5762 | 2^342805-21025 | 103195 | Enzo Doria | 06/2013 |
5763 | 24129^18890+18890^24129 | 103182 | Hans Havermann | 09/2020 |
5764 | (622*10^103171-1)/9 | 103173 | Robert Price | 11/2021 |
5765 | 2^342714+12285 | 103168 | Lelio R Paula | 11/2016 |
5766 | 18^82178-4481 | 103156 | Enzo Doria | 03/2013 |
5767 | 2^342665-17349 | 103153 | Enzo Doria | 05/2013 |
5768 | 30408^2467+2467^30408 | 103150 | Hans Havermann | 09/2020 |
5769 | (5^147571-3^147571)/2 | 103148 | Kellen Shenton | 05/2022 |
5770 | (2^342615+3090)/2 | 103138 | Stefano Morozzi | 09/2020 |
5771 | 2^342612+175 | 103137 | Lelio R Paula | 08/2011 |
5772 | 5*2^342585-3 | 103130 | Lelio R Paula | 09/2014 |
5773 | (16*10^103127+77)/3 | 103128 | Robert Price | 03/2019 |
5774 | (2^342527+3774)/2 | 103111 | Stefano Morozzi | 02/2021 |
5775 | 2^342503-19209 | 103104 | Enzo Doria | 05/2013 |
5776 | (2^342484+2586)/2 | 103098 | Stefano Morozzi | 09/2020 |
5777 | 2^342476+51 | 103096 | Lelio R Paula | 08/2011 |
5778 | 2^342460-18885 | 103091 | Enzo Doria | 05/2013 |
5779 | (2^342442+862)/2 | 103086 | Stefano Morozzi | 02/2021 |
5780 | (194*10^103074-17)/3 | 103076 | Robert Price | 09/2020 |
5781 | 7^121968+82 | 103075 | Lelio R Paula | 10/2016 |
5782 | 2^342392-108863 | 103071 | Enzo Doria | 03/2014 |
5783 | (2^342389+806)/2 | 103070 | Stefano Morozzi | 01/2021 |
5784 | 3^216014+28 | 103065 | Lelio R Paula | 11/2016 |
5785 | 2^342353-108879 | 103059 | Enzo Doria | 03/2014 |
5786 | 10^103055+276081 | 103056 | Patrick De Geest | 12/2016 |
5787 | 10^103055-53 | 103055 | Lelio R Paula | 11/2016 |
5788 | (58*10^103046+419)/9 | 103047 | Robert Price | 05/2019 |
5789 | 7^121933+4 | 103046 | Lelio R Paula | 11/2014 |
5790 | (71*10^103028-143)/9 | 103029 | Robert Price | 10/2021 |
5791 | (2^342235+3618)/2 | 103023 | Stefano Morozzi | 09/2020 |
5792 | 19^80560-68 | 103017 | Lelio R Paula | 10/2016 |
5793 | 2^342209+76607 | 103016 | Enzo Doria | 05/2016 |
5795 | 328^40945+40945^328 | 103013 | Norbert Schneider | 12/2014 |
5794 | 12^95454-91 | 103013 | Lelio R Paula | 10/2016 |
5796 | 25833^9716+9716^25833 | 103009 | Hans Havermann | 01/2019 |
5797 | (893*10^103004+7)/9 | 103006 | Robert Price | 06/2021 |
5798 | (1475^32503-1)/1474 | 102993 | Serge Batalov | 06/2022 |
5799 | 65^56809-64^56809 | 102990 | Robert Price | 08/2013 |
5800 | 5^147338-6 | 102985 | Lelio R Paula | 11/2014 |
5801 | 2^342084-112739 | 102978 | Enzo Doria | 06/2014 |
5802 | 9*6^132330+7 | 102974 | Lelio R Paula | 11/2014 |
5803 | 2^342034-16737 | 102963 | Enzo Doria | 05/2013 |
5804 | (9787*2^342011-1)/25 | 102959 | Lelio R Paula | 05/2015 |
5805 | (3^215802-5)/124343756 | 102956 | Lelio R Paula | 12/2013 |
5806 | 2^341991-17421 | 102950 | Enzo Doria | 05/2013 |
5807 | 2^341969-112225 | 102943 | Enzo Doria | 06/2014 |
5808 | (9787*2^341864-1)/3 | 102915 | Lelio R Paula | 05/2015 |
5809 | 2^341859+1869413 | 102910 | Enzo Doria | 05/2016 |
5810 | 10^102909+61 | 102910 | Lelio R Paula | 11/2016 |
5811 | 10^102909-67049 | 102909 | Patrick De Geest | 12/2016 |
5812 | 23814^20935+20935^23814 | 102898 | Hans Havermann | 02/2019 |
5813 | 28345^4264+4264^28345 | 102888 | Hans Havermann | 02/2019 |
5814 | 25665^10204+10204^25665 | 102886 | Hans Havermann | 02/2019 |
5815 | 26288^8181+8181^26288 | 102860 | Hans Havermann | 02/2019 |
5816 | 2^341659+1836773 | 102850 | Enzo Doria | 03/2016 |
5817 | 2^341608-110639 | 102835 | Enzo Doria | 05/2014 |
5818 | 10^102824+49 | 102825 | Lelio R Paula | 11/2016 |
5819 | 10^102824-168011 | 102824 | Patrick De Geest | 12/2016 |
5820 | 11^98708-8 | 102794 | Lelio R Paula | 10/2016 |
5821 | 2^341466+4429 | 102792 | Lelio R Paula | 09/2011 |
5822 | 2^341460-107 | 102790 | Lelio R Paula | 08/2011 |
5823 | (2^341388+47)/21014943 | 102761 | Lelio R Paula | 12/2013 |
5824 | 2^341355+21 | 102759 | Lelio R Paula | 08/2011 |
5825 | (6496^26947-1)/6495 | 102736 | Serge Batalov | 06/2022 |
5826 | 2^341246+40755 | 102726 | Lelio R Paula | 11/2016 |
5827 | (2^341256-5)/991699 | 102723 | Lelio R Paula | 12/2013 |
5828 | 7^121547+78 | 102720 | Lelio R Paula | 10/2016 |
5829 | 2^341223-461197 | 102719 | Enzo Doria | 06/2015 |
5830 | 27838^4893+4893^27838 | 102711 | Hans Havermann | 03/2019 |
5832 | 2^341190-20577 | 102709 | Enzo Doria | 06/2013 |
5831 | (27*2^341212+1)/84674449 | 102709 | Lelio R Paula | 12/2013 |
5833 | 6^131976-65 | 102698 | Lelio R Paula | 10/2016 |
5834 | 2^341134+2533 | 102692 | Henri Lifchitz | 09/2005 |
5835 | 13^92184-54 | 102688 | Lelio R Paula | 10/2016 |
5836 | 2^341116-20873 | 102687 | Enzo Doria | 06/2013 |
5837 | 2^341094-19953 | 102680 | Enzo Doria | 06/2013 |
5838 | 23761^20950+20950^23761 | 102676 | Hans Havermann | 03/2019 |
5840 | 2^341053+129 | 102668 | Lelio R Paula | 08/2011 |
5839 | (4724^27943-1)/4723 | 102668 | Serge Batalov | 06/2022 |
5842 | (31*2^341006-1)/3 | 102655 | Lelio R Paula | 03/2014 |
5841 | 7^121471+94 | 102655 | Lelio R Paula | 10/2016 |
5844 | 2^341004-173 | 102653 | Lelio R Paula | 08/2011 |
5843 | 20^78901+3 | 102653 | Lelio R Paula | 10/2016 |
5845 | (2^341001+1674)/2 | 102652 | Stefano Morozzi | 09/2020 |
5846 | 5^146853-52 | 102646 | Lelio R Paula | 10/2016 |
5847 | 2^340969-216931 | 102642 | Enzo Doria | 05/2014 |
5848 | 122^49196+121 | 102641 | Norbert Schneider | 10/2018 |
5849 | 2^340952+1870021 | 102637 | Enzo Doria | 05/2016 |
5850 | 5*2^340940-3 | 102634 | Lelio R Paula | 08/2014 |
5851 | (6^131891+1)/7 | 102631 | Paul Bourdelais | 01/2010 |
5852 | (11*10^102629-23)/3 | 102630 | Robert Price | 09/2021 |
5853 | 25876^9239+9239^25876 | 102615 | Hans Havermann | 02/2019 |
5854 | (4*10^102612+83)/3 | 102613 | Robert Price | 02/2018 |
5855 | 75^54717+74 | 102598 | Norbert Schneider | 01/2015 |
5856 | (17*10^102597+13)/3 | 102598 | Robert Price | 01/2019 |
5857 | 2^340806-20393 | 102593 | Enzo Doria | 06/2013 |
5858 | (2^340790+5)/138267 | 102583 | Lelio R Paula | 12/2013 |
5859 | 6^131821+77 | 102577 | Lelio R Paula | 10/2016 |
5860 | (203*10^102561-11)/3 | 102563 | Robert Price | 12/2020 |
5861 | 2^340688-29 | 102558 | Henri Lifchitz | 05/2005 |
5862 | 2^340685-109101 | 102557 | Enzo Doria | 04/2014 |
5863 | 65^56567-64 | 102552 | Norbert Schneider | 11/2014 |
5864 | 26230^8121+8121^26230 | 102550 | Hans Havermann | 03/2019 |
5865 | (8*10^102541+49)/3 | 102542 | Robert Price | 05/2020 |
5866 | (536*10^102533-23)/9 | 102535 | Robert Price | 02/2020 |
5867 | 2^340547-108897 | 102515 | Enzo Doria | 03/2014 |
5868 | (2^340536+778)/2 | 102512 | Stefano Morozzi | 09/2020 |
5869 | 25592^10119+10119^25592 | 102500 | Hans Havermann | 02/2019 |
5870 | 7^121268+82 | 102484 | Lelio R Paula | 10/2016 |
5871 | (6883*2^340443-1)/162401 | 102483 | Lelio R Paula | 05/2015 |
5872 | 2^340434-5955 | 102481 | Lelio R Paula | 11/2016 |
5873 | (7*10^102480-19)/3 | 102481 | Robert Price | 10/2020 |
5874 | (2^340381+47)/7 | 102465 | Lelio R Paula | 12/2013 |
5875 | 111^50095+110 | 102461 | Norbert Schneider | 04/2017 |
5876 | 2^340291-20301 | 102438 | Enzo Doria | 06/2013 |
5877 | 12^94915+89 | 102431 | Lelio R Paula | 10/2016 |
5878 | (106*10^102422+17)/3 | 102424 | Robert Price | 04/2021 |
5879 | 24466^15349+15349^24466 | 102417 | Hans Havermann | 03/2019 |
5880 | 2^340212+111546435 | 102415 | Paul Bourdelais | 10/2021 |
5881 | 2^340208-20373 | 102413 | Enzo Doria | 06/2013 |
5882 | 6^131598-17273 | 102404 | Enzo Doria | 02/2013 |
5883 | (2^340132+1182)/2 | 102390 | Stefano Morozzi | 09/2020 |
5884 | (2^340133+1166)/2 | 102390 | Stefano Morozzi | 09/2020 |
5885 | 2^340115-21181 | 102385 | Enzo Doria | 06/2013 |
5886 | (55459*2^340090+1)/1591 | 102379 | Lelio R Paula | 07/2014 |
5887 | (2^340064+3630)/2 | 102370 | Stefano Morozzi | 09/2020 |
5888 | 89^52511-88^52511 | 102365 | Robert Price | 06/2015 |
5889 | 3^214539-112756 | 102362 | Enzo Doria | 07/2014 |
5890 | 23159^23450-23450^23159 | 102353 | Norbert Schneider | 12/2019 |
5891 | 2^340000-19877 | 102351 | Enzo Doria | 06/2013 |
5892 | 2^339990-114845 | 102348 | Enzo Doria | 05/2015 |
5893 | (3^214516-2)/51383 | 102346 | Lelio R Paula | 12/2013 |
5894 | 20^78661-99 | 102341 | Lelio R Paula | 09/2016 |
5895 | 2^339951-19585 | 102336 | Enzo Doria | 05/2013 |
5896 | (17*10^102335-167)/3 | 102336 | Robert Price | 05/2022 |
5897 | (13*2^339943+1)/45 | 102333 | Lelio R Paula | 12/2013 |
5898 | (236282#+121441)/(121441*25499429) | 102332 | Makoto Morimoto | 08/2020 |
5899 | 5*10^102323-91 | 102324 | Robert Price | 03/2021 |
5901 | 2^339889+29327 | 102317 | Yves Minet | 06/2010 |
5900 | (167*10^102315-17)/3 | 102317 | Robert Price | 03/2020 |
5902 | 26348^7641+7641^26348 | 102314 | Hans Havermann | 03/2019 |
5903 | Phi(239414,10)/6333218543 | 102303 | Makoto Kamada | 08/2019 |
5904 | 2^339834-109493 | 102301 | Enzo Doria | 04/2014 |
5905 | (16*10^102299-19)/3 | 102300 | Robert Price | 02/2019 |
5906 | (11^98221-5^98221)/6 | 102286 | Robert Price | 01/2013 |
5907 | 3^214365+1820 | 102279 | Lelio R Paula | 10/2016 |
5908 | 31978^1579+1579^31978 | 102278 | Hans Havermann | 03/2019 |
5909 | (89*10^102262-467)/9 | 102263 | Robert Price | 04/2022 |
5910 | (11*2^339705-1)/183 | 102261 | Lelio R Paula | 03/2014 |
5911 | 2^339694-16995 | 102259 | Lelio R Paula | 09/2016 |
5912 | (56*10^102255+691)/9 | 102256 | Robert Price | 05/2019 |
5913 | 122^49008+121 | 102249 | Norbert Schneider | 10/2018 |
5914 | (2^339652+3178)/2 | 102246 | Stefano Morozzi | 09/2020 |
5915 | 7^120985-38 | 102245 | Lelio R Paula | 10/2016 |
5916 | 10^102244+122961 | 102245 | Patrick De Geest | 12/2016 |
5917 | 10^102244-27 | 102244 | Lelio R Paula | 09/2016 |
5918 | 7^120982-62 | 102242 | Lelio R Paula | 10/2016 |
5919 | (2^339640+2670)/2 | 102242 | Stefano Morozzi | 09/2020 |
5920 | 4*6^131382-5 | 102236 | Lelio R Paula | 11/2014 |
5921 | (2^339607+3290)/2 | 102232 | Stefano Morozzi | 02/2021 |
5922 | 2^339586-1091 | 102226 | Yves Minet | 11/2012 |
5923 | 11^98161-3072000 | 102225 | Enzo Doria | 12/2012 |
5924 | 9*7^120961-4 | 102225 | Lelio R Paula | 11/2014 |
5925 | 5*3^214250+2 | 102224 | Serge Batalov | 10/2010 |
5926 | 12^94713-59 | 102213 | Lelio R Paula | 10/2016 |
5927 | 12^94705+89 | 102204 | Lelio R Paula | 10/2016 |
5928 | 2^339407-177 | 102172 | Lelio R Paula | 08/2011 |
5929 | 4*10^102163-57 | 102164 | Robert Price | 10/2018 |
5930 | 2^339355+65 | 102157 | Lelio R Paula | 07/2011 |
5931 | (28*10^102142-31)/3 | 102143 | Robert Price | 12/2019 |
5932 | (2^339323+47)/880715 | 102141 | Lelio R Paula | 12/2013 |
5933 | (2^339268+1758)/2 | 102130 | Stefano Morozzi | 09/2020 |
5934 | s(178551) | 102122 | Robert Price | 05/2020 |
5935 | (797*10^102112-23)/9 | 102114 | Robert Price | 01/2022 |
5936 | (2^339175+2790)/2 | 102102 | Stefano Morozzi | 02/2021 |
5938 | 2^339168+21 | 102100 | Lelio R Paula | 07/2011 |
5939 | 2^339167-108807 | 102100 | Enzo Doria | 03/2014 |
5937 | (2^339169+42)/2 | 102100 | Stefano Morozzi | 09/2020 |
5940 | 2^339156+163 | 102097 | Lelio R Paula | 08/2011 |
5941 | (2^339149+746)/2 | 102094 | Stefano Morozzi | 09/2020 |
5942 | (25*10^102076-67)/3 | 102077 | Robert Price | 06/2019 |
5943 | 2^339081-4543 | 102074 | Yves Minet | 03/2019 |
5944 | (3435^28867-1)/3434 | 102069 | Serge Batalov | 06/2022 |
5945 | 2*10^102062+77 | 102063 | Robert Price | 08/2020 |
5946 | 24327^15668+15668^24327 | 102053 | Hans Havermann | 02/2019 |
5947 | 10^102052-7000001 | 102052 | Robert Price | 06/2021 |
5948 | 2^339000-175635 | 102050 | Stefano Morozzi | 12/2021 |
5949 | 2^339000-151667 | 102050 | Stefano Morozzi | 12/2021 |
5950 | 2^338999+79289 | 102049 | Stefano Morozzi | 01/2022 |
5951 | 2^338994+188757 | 102048 | Stefano Morozzi | 05/2022 |
5952 | 2^338992+181591 | 102047 | Stefano Morozzi | 01/2022 |
5953 | 2^338992+136171 | 102047 | Stefano Morozzi | 03/2022 |
5954 | 2^338991-109435 | 102047 | Stefano Morozzi | 04/2022 |
5955 | 2^338988-134619 | 102046 | Stefano Morozzi | 04/2022 |
5956 | 2^338981+106391 | 102044 | Stefano Morozzi | 01/2022 |
5957 | 2^338982+34069 | 102044 | Stefano Morozzi | 03/2022 |
5958 | 2^338982-58781 | 102044 | Stefano Morozzi | 04/2022 |
5959 | 2^338977+83007 | 102043 | Stefano Morozzi | 02/2022 |
5960 | 2^338979-83601 | 102043 | Stefano Morozzi | 04/2022 |
5961 | 2^338975+28355 | 102042 | Stefano Morozzi | 01/2022 |
5962 | 2^338973+134619 | 102042 | Stefano Morozzi | 03/2022 |
5963 | 2^338974+183777 | 102042 | Stefano Morozzi | 03/2022 |
5968 | 25080^11711+11711^25080 | 102041 | Hans Havermann | 02/2019 |
5964 | 2^338972+152343 | 102041 | Stefano Morozzi | 12/2021 |
5965 | 2^338972-180623 | 102041 | Stefano Morozzi | 02/2022 |
5966 | 2^338972+84441 | 102041 | Stefano Morozzi | 03/2022 |
5967 | 2^338971+64875 | 102041 | Stefano Morozzi | 05/2022 |
5969 | 2^338968+53461 | 102040 | Stefano Morozzi | 01/2022 |
5970 | 2^338969+160175 | 102040 | Stefano Morozzi | 03/2022 |
5971 | 2^338967+13409 | 102040 | Stefano Morozzi | 04/2022 |
5972 | 2^338963-127455 | 102039 | Stefano Morozzi | 12/2021 |
5973 | 2^338963+132195 | 102039 | Stefano Morozzi | 02/2022 |
5974 | 2^338966+137919 | 102039 | Stefano Morozzi | 04/2022 |
5979 | (2^338982-7)/1754517 | 102038 | Lelio R Paula | 12/2013 |
5975 | 2^338960+53935 | 102038 | Stefano Morozzi | 01/2022 |
5976 | 2^338962+83157 | 102038 | Stefano Morozzi | 02/2022 |
5977 | 2^338961+12341 | 102038 | Stefano Morozzi | 04/2022 |
5978 | 2^338960+63933 | 102038 | Stefano Morozzi | 05/2022 |
5980 | 2^338959-30919 | 102037 | Stefano Morozzi | 02/2022 |
5981 | 2^338959+164769 | 102037 | Stefano Morozzi | 05/2022 |
5982 | 2^338954-177003 | 102036 | Stefano Morozzi | 12/2021 |
5983 | 2^338954+154207 | 102036 | Stefano Morozzi | 01/2022 |
5984 | 2^338956+182493 | 102036 | Stefano Morozzi | 02/2022 |
5985 | 2^338954+162715 | 102036 | Stefano Morozzi | 04/2022 |
5986 | 2^338955-158241 | 102036 | Stefano Morozzi | 04/2022 |
5987 | 2^338956-33395 | 102036 | Stefano Morozzi | 04/2022 |
5988 | 2^338950+175077 | 102035 | Stefano Morozzi | 12/2021 |
5989 | 2^338951-51687 | 102035 | Stefano Morozzi | 12/2021 |
5990 | 2^338949+85449 | 102034 | Stefano Morozzi | 03/2022 |
5991 | 2^338949-184585 | 102034 | Stefano Morozzi | 04/2022 |
5992 | 2^338943+127103 | 102033 | Stefano Morozzi | 12/2021 |
5993 | 2^338944-84003 | 102033 | Stefano Morozzi | 04/2022 |
5994 | 2^338945+138605 | 102033 | Stefano Morozzi | 05/2022 |
5995 | 2^338940+78607 | 102032 | Stefano Morozzi | 01/2022 |
5996 | 2^338939+151569 | 102031 | Stefano Morozzi | 12/2021 |
5997 | 2^338938+183975 | 102031 | Stefano Morozzi | 03/2022 |
5998 | 2^338937-33429 | 102031 | Stefano Morozzi | 04/2022 |
5999 | 2^338939-183259 | 102031 | Stefano Morozzi | 04/2022 |
6000 | 2^338935+179999 | 102030 | Stefano Morozzi | 01/2022 |