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 ? |
8751 | (8766^26227-1)/8765 | 103404 | Serge Batalov | 06/2022 |
8752 | (509203*2^343491-1)/16147 | 103403 | Lelio R Paula | 05/2015 |
8753 | 3^216685-113056 | 103386 | Enzo Doria | 07/2014 |
8754 | 24829^14580+14580^24829 | 103382 | Hans Havermann | 08/2020 |
8755 | 91355!4+1024 | 103381 | Robert Price | 09/2019 |
8756 | 44^62903-43^62903 | 103379 | Jean-Louis Charton | 08/2012 |
8757 | 11^99262-20 | 103371 | Lelio R Paula | 12/2016 |
8758 | (509203*2^343357-1)/5 | 103366 | Lelio R Paula | 05/2015 |
8759 | (2^343363+3570)/2 | 103363 | Stefano Morozzi | 09/2020 |
8760 | 2^343356-2791889 | 103361 | Enzo Doria | 03/2016 |
8761 | (25*10^103360-241)/9 | 103361 | Robert Price | 05/2018 |
8762 | Phi(104989,10)/2939693 | 103350 | Maksym Voznyy | 09/2015 |
8763 | (371*33^68040-19)/32 | 103321 | Eric Chen | 05/2023 |
8764 | 24826^14511+14511^24826 | 103319 | Hans Havermann | 09/2020 |
8765 | 2^343182-171 | 103309 | Lelio R Paula | 08/2011 |
8766 | (23619^23619+23620^23620)/7187 | 103294 | David Broadhurst | 09/2009 |
8767 | (16487^24649-1)/(16487^157-1) | 103287 | Ryan Propper | 11/2023 |
8768 | (10680^25639-1)/10679 | 103285 | Serge Batalov | 06/2022 |
8769 | (7427^26683-1)/7426 | 103282 | Serge Batalov | 06/2022 |
8770 | 2^343079-1835611 | 103278 | Enzo Doria | 05/2015 |
8771 | 2^343063-1147 | 103273 | Yves Minet | 06/2013 |
8772 | 24741^14930+14930^24741 | 103271 | Hans Havermann | 09/2020 |
8773 | (16*10^103255-73)/3 | 103256 | Robert Price | 07/2020 |
8774 | (219^44119-1)/218 | 103256 | Serge Batalov | 06/2022 |
8775 | 9*6^132689+7 | 103254 | Lelio R Paula | 11/2014 |
8776 | 2^342979-18085 | 103247 | Enzo Doria | 05/2013 |
8777 | (16*10^103244-31)/3 | 103245 | Robert Price | 03/2019 |
8778 | 2^342966-16995 | 103244 | Lelio R Paula | 11/2016 |
8779 | 23797^21774+21774^23797 | 103230 | Hans Havermann | 09/2020 |
8780 | 25388^11637+11637^25388 | 103224 | Hans Havermann | 09/2020 |
8781 | (2^342876+2086)/2 | 103216 | Stefano Morozzi | 09/2020 |
8782 | (16*10^103214-31)/3 | 103215 | Robert Price | 03/2019 |
8784 | (37*10^103211+611)/9 | 103212 | Robert Price | 11/2018 |
8783 | (238*10^103210-1)/3 | 103212 | Robert Price | 04/2020 |
8785 | 5^147650+16 | 103203 | Lelio R Paula | 11/2016 |
8786 | 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 |
8787 | Phi(852060L,10) | 103201 | Anonymous | 03/2025 |
8788 | 2^342817+2707067 | 103199 | Enzo Doria | 03/2016 |
8789 | 43^63177-42 | 103198 | Theodore Burton | 02/2007 |
8790 | 2^342805-21025 | 103195 | Enzo Doria | 06/2013 |
8791 | 24129^18890+18890^24129 | 103182 | Hans Havermann | 09/2020 |
8792 | (622*10^103171-1)/9 | 103173 | Robert Price | 11/2021 |
8793 | 2^342714+12285 | 103168 | Lelio R Paula | 11/2016 |
8794 | (2*10^103161-17)/3 | 103161 | Serge Batalov | 11/2022 |
8795 | 18^82178-4481 | 103156 | Enzo Doria | 03/2013 |
8796 | 2^342665-17349 | 103153 | Enzo Doria | 05/2013 |
8797 | 30408^2467+2467^30408 | 103150 | Hans Havermann | 09/2020 |
8798 | (5^147571-3^147571)/2 | 103148 | Kellen Shenton | 05/2022 |
8799 | (2^342615+3090)/2 | 103138 | Stefano Morozzi | 09/2020 |
8800 | 2^342612+175 | 103137 | Lelio R Paula | 08/2011 |
8801 | 5*2^342585-3 | 103130 | Lelio R Paula | 09/2014 |
8802 | (16*10^103127+77)/3 | 103128 | Robert Price | 03/2019 |
8803 | (2^342527+3774)/2 | 103111 | Stefano Morozzi | 02/2021 |
8804 | 2^342503-19209 | 103104 | Enzo Doria | 05/2013 |
8805 | (2^342484+2586)/2 | 103098 | Stefano Morozzi | 09/2020 |
8806 | 2^342476+51 | 103096 | Lelio R Paula | 08/2011 |
8807 | 2^342460-18885 | 103091 | Enzo Doria | 05/2013 |
8808 | (2^342442+862)/2 | 103086 | Stefano Morozzi | 02/2021 |
8809 | (194*10^103074-17)/3 | 103076 | Robert Price | 09/2020 |
8810 | 7^121968+82 | 103075 | Lelio R Paula | 10/2016 |
8811 | 2^342392-108863 | 103071 | Enzo Doria | 03/2014 |
8812 | (2^342389+806)/2 | 103070 | Stefano Morozzi | 01/2021 |
8813 | 3^216014+28 | 103065 | Lelio R Paula | 11/2016 |
8814 | 2^342353-108879 | 103059 | Enzo Doria | 03/2014 |
8815 | 10^103055+276081 | 103056 | Patrick De Geest | 12/2016 |
8816 | 10^103055-53 | 103055 | Lelio R Paula | 11/2016 |
8817 | (58*10^103046+419)/9 | 103047 | Robert Price | 05/2019 |
8818 | 7^121933+4 | 103046 | Lelio R Paula | 11/2014 |
8819 | (71*10^103028-143)/9 | 103029 | Robert Price | 10/2021 |
8820 | (2^342235+3618)/2 | 103023 | Stefano Morozzi | 09/2020 |
8821 | 19^80560-68 | 103017 | Lelio R Paula | 10/2016 |
8822 | 2^342209+76607 | 103016 | Enzo Doria | 05/2016 |
8824 | 328^40945+40945^328 | 103013 | Norbert Schneider | 12/2014 |
8823 | 12^95454-91 | 103013 | Lelio R Paula | 10/2016 |
8825 | 25833^9716+9716^25833 | 103009 | Hans Havermann | 01/2019 |
8826 | (893*10^103004+7)/9 | 103006 | Robert Price | 06/2021 |
8827 | (1475^32503-1)/1474 | 102993 | Serge Batalov | 06/2022 |
8828 | 65^56809-64^56809 | 102990 | Robert Price | 08/2013 |
8829 | 5^147338-6 | 102985 | Lelio R Paula | 11/2014 |
8830 | 2^342084-112739 | 102978 | Enzo Doria | 06/2014 |
8831 | 9*6^132330+7 | 102974 | Lelio R Paula | 11/2014 |
8832 | 2^342034-16737 | 102963 | Enzo Doria | 05/2013 |
8833 | (9787*2^342011-1)/25 | 102959 | Lelio R Paula | 05/2015 |
8834 | (3^215802-5)/124343756 | 102956 | Lelio R Paula | 12/2013 |
8835 | 2^341991-17421 | 102950 | Enzo Doria | 05/2013 |
8836 | 2^341969-112225 | 102943 | Enzo Doria | 06/2014 |
8837 | (9787*2^341864-1)/3 | 102915 | Lelio R Paula | 05/2015 |
8838 | 2^341859+1869413 | 102910 | Enzo Doria | 05/2016 |
8839 | 10^102909+61 | 102910 | Lelio R Paula | 11/2016 |
8840 | 10^102909-67049 | 102909 | Patrick De Geest | 12/2016 |
8841 | 23814^20935+20935^23814 | 102898 | Hans Havermann | 02/2019 |
8842 | 28345^4264+4264^28345 | 102888 | Hans Havermann | 02/2019 |
8843 | 25665^10204+10204^25665 | 102886 | Hans Havermann | 02/2019 |
8844 | 26288^8181+8181^26288 | 102860 | Hans Havermann | 02/2019 |
8845 | 2^341659+1836773 | 102850 | Enzo Doria | 03/2016 |
8846 | 2^341608-110639 | 102835 | Enzo Doria | 05/2014 |
8847 | 10^102824+49 | 102825 | Lelio R Paula | 11/2016 |
8848 | 10^102824-168011 | 102824 | Patrick De Geest | 12/2016 |
8849 | (156^46877-1)/155 | 102805 | Anonymous | 06/2024 |
8850 | 11^98708-8 | 102794 | Lelio R Paula | 10/2016 |
8851 | 2^341466+4429 | 102792 | Lelio R Paula | 09/2011 |
8852 | 2^341460-107 | 102790 | Lelio R Paula | 08/2011 |
8853 | (2^341388+47)/21014943 | 102761 | Lelio R Paula | 12/2013 |
8854 | 2^341355+21 | 102759 | Lelio R Paula | 08/2011 |
8855 | 2^341319+224885 | 102748 | Enzo Doria | 07/2024 |
8856 | (419^39181-1)/418 | 102739 | Tom Wu | 07/2024 |
8857 | (6496^26947-1)/6495 | 102736 | Serge Batalov | 06/2022 |
8858 | 2^341246+40755 | 102726 | Lelio R Paula | 11/2016 |
8859 | (2^341256-5)/991699 | 102723 | Lelio R Paula | 12/2013 |
8860 | 7^121547+78 | 102720 | Lelio R Paula | 10/2016 |
8861 | 2^341223-461197 | 102719 | Enzo Doria | 06/2015 |
8862 | 27838^4893+4893^27838 | 102711 | Hans Havermann | 03/2019 |
8864 | 2^341190-20577 | 102709 | Enzo Doria | 06/2013 |
8863 | (27*2^341212+1)/84674449 | 102709 | Lelio R Paula | 12/2013 |
8865 | (25*10^102698+77)/3 | 102699 | Robert Price | 09/2022 |
8866 | 6^131976-65 | 102698 | Lelio R Paula | 10/2016 |
8867 | 2^341134+2533 | 102692 | Henri Lifchitz | 09/2005 |
8868 | 13^92184-54 | 102688 | Lelio R Paula | 10/2016 |
8869 | 2^341116-20873 | 102687 | Enzo Doria | 06/2013 |
8870 | 2^341094-19953 | 102680 | Enzo Doria | 06/2013 |
8871 | 23761^20950+20950^23761 | 102676 | Hans Havermann | 03/2019 |
8873 | 2^341053+129 | 102668 | Lelio R Paula | 08/2011 |
8872 | (4724^27943-1)/4723 | 102668 | Serge Batalov | 06/2022 |
8875 | (31*2^341006-1)/3 | 102655 | Lelio R Paula | 03/2014 |
8874 | 7^121471+94 | 102655 | Lelio R Paula | 10/2016 |
8877 | 2^341004-173 | 102653 | Lelio R Paula | 08/2011 |
8876 | 20^78901+3 | 102653 | Lelio R Paula | 10/2016 |
8878 | (2^341001+1674)/2 | 102652 | Stefano Morozzi | 09/2020 |
8879 | 5^146853-52 | 102646 | Lelio R Paula | 10/2016 |
8880 | 2^340969-216931 | 102642 | Enzo Doria | 05/2014 |
8881 | 122^49196+121 | 102641 | Norbert Schneider | 10/2018 |
8882 | 2^340952+1870021 | 102637 | Enzo Doria | 05/2016 |
8883 | 5*2^340940-3 | 102634 | Lelio R Paula | 08/2014 |
8884 | (6^131891+1)/7 | 102631 | Paul Bourdelais | 01/2010 |
8885 | (11*10^102629-23)/3 | 102630 | Robert Price | 09/2021 |
8886 | 3^215094+5*2^215094 | 102626 | Jon Grantham | 08/2023 |
8887 | 25876^9239+9239^25876 | 102615 | Hans Havermann | 02/2019 |
8888 | (4*10^102612+83)/3 | 102613 | Robert Price | 02/2018 |
8889 | (234^43313-1)/20184091 | 102611 | Aurel Gibier | 03/2025 |
8891 | 75^54717+74 | 102598 | Norbert Schneider | 01/2015 |
8890 | (17*10^102597+13)/3 | 102598 | Robert Price | 01/2019 |
8892 | 2^340806-20393 | 102593 | Enzo Doria | 06/2013 |
8893 | (2^340790+5)/138267 | 102583 | Lelio R Paula | 12/2013 |
8894 | 6^131821+77 | 102577 | Lelio R Paula | 10/2016 |
8895 | (203*10^102561-11)/3 | 102563 | Robert Price | 12/2020 |
8896 | (67^56171-1)/2053280351166 | 102561 | Paul Vanderveen | 05/2025 |
8897 | 2^340688-29 | 102558 | Henri Lifchitz | 05/2005 |
8898 | 2^340685-109101 | 102557 | Enzo Doria | 04/2014 |
8899 | 65^56567-64 | 102552 | Norbert Schneider | 11/2014 |
8900 | 26230^8121+8121^26230 | 102550 | Hans Havermann | 03/2019 |
8901 | (64*10^102547+161)/9 | 102548 | Robert Price | 08/2022 |
8902 | (8*10^102541+49)/3 | 102542 | Robert Price | 05/2020 |
8903 | (536*10^102533-23)/9 | 102535 | Robert Price | 02/2020 |
8904 | 2^340547-108897 | 102515 | Enzo Doria | 03/2014 |
8905 | (2^340536+778)/2 | 102512 | Stefano Morozzi | 09/2020 |
8906 | 25592^10119+10119^25592 | 102500 | Hans Havermann | 02/2019 |
8907 | (5*10^102492-23)/9 | 102492 | Serge Batalov | 11/2022 |
8908 | 7^121268+82 | 102484 | Lelio R Paula | 10/2016 |
8909 | (6883*2^340443-1)/162401 | 102483 | Lelio R Paula | 05/2015 |
8910 | 2^340434-5955 | 102481 | Lelio R Paula | 11/2016 |
8911 | (7*10^102480-19)/3 | 102481 | Robert Price | 10/2020 |
8912 | (2^340381+47)/7 | 102465 | Lelio R Paula | 12/2013 |
8913 | 111^50095+110 | 102461 | Norbert Schneider | 04/2017 |
8914 | 180^45427-45427^180 | 102451 | Norman Luhn | 09/2024 |
8915 | 2^340291-20301 | 102438 | Enzo Doria | 06/2013 |
8916 | 12^94915+89 | 102431 | Lelio R Paula | 10/2016 |
8917 | (106*10^102422+17)/3 | 102424 | Robert Price | 04/2021 |
8918 | 24466^15349+15349^24466 | 102417 | Hans Havermann | 03/2019 |
8919 | 2^340212+111546435 | 102415 | Paul Bourdelais | 10/2021 |
8920 | 2^340208-20373 | 102413 | Enzo Doria | 06/2013 |
8921 | 6^131598-17273 | 102404 | Enzo Doria | 02/2013 |
8922 | (2^340132+1182)/2 | 102390 | Stefano Morozzi | 09/2020 |
8923 | (2^340133+1166)/2 | 102390 | Stefano Morozzi | 09/2020 |
8924 | 2^340115-21181 | 102385 | Enzo Doria | 06/2013 |
8925 | (55459*2^340090+1)/1591 | 102379 | Lelio R Paula | 07/2014 |
8926 | (2^340064+3630)/2 | 102370 | Stefano Morozzi | 09/2020 |
8927 | 9348*31^68638+13 | 102369 | Eric Chen | 04/2024 |
8928 | 89^52511-88^52511 | 102365 | Robert Price | 06/2015 |
8929 | 3^214539-112756 | 102362 | Enzo Doria | 07/2014 |
8930 | Phi(562221,2)/53723397926257372614541581386770268796893300042857 | 102361 | Anonymous | 06/2025 |
8931 | 23159^23450-23450^23159 | 102353 | Norbert Schneider | 12/2019 |
8932 | 2^340000-19877 | 102351 | Enzo Doria | 06/2013 |
8934 | 2^339990-114845 | 102348 | Enzo Doria | 05/2015 |
8933 | 311^41058-41058^311 | 102348 | Norman Luhn | 09/2024 |
8935 | (3^214516-2)/51383 | 102346 | Lelio R Paula | 12/2013 |
8936 | 20^78661-99 | 102341 | Lelio R Paula | 09/2016 |
8937 | 2^339951-19585 | 102336 | Enzo Doria | 05/2013 |
8938 | (17*10^102335-167)/3 | 102336 | Robert Price | 05/2022 |
8939 | (13*2^339943+1)/45 | 102333 | Lelio R Paula | 12/2013 |
8940 | (236282#+121441)/(121441*25499429) | 102332 | Makoto Morimoto | 08/2020 |
8941 | 5*10^102323-91 | 102324 | Robert Price | 03/2021 |
8943 | 2^339889+29327 | 102317 | Yves Minet | 06/2010 |
8942 | (167*10^102315-17)/3 | 102317 | Robert Price | 03/2020 |
8944 | 26348^7641+7641^26348 | 102314 | Hans Havermann | 03/2019 |
8945 | Phi(239414,10)/6333218543 | 102303 | Makoto Kamada | 08/2019 |
8946 | 2^339834-109493 | 102301 | Enzo Doria | 04/2014 |
8947 | (16*10^102299-19)/3 | 102300 | Robert Price | 02/2019 |
8948 | (11^98221-5^98221)/6 | 102286 | Robert Price | 01/2013 |
8949 | 3^214365+1820 | 102279 | Lelio R Paula | 10/2016 |
8950 | 31978^1579+1579^31978 | 102278 | Hans Havermann | 03/2019 |
8951 | (89*10^102262-467)/9 | 102263 | Robert Price | 04/2022 |
8952 | (11*2^339705-1)/183 | 102261 | Lelio R Paula | 03/2014 |
8953 | 2^339694-16995 | 102259 | Lelio R Paula | 09/2016 |
8954 | (56*10^102255+691)/9 | 102256 | Robert Price | 05/2019 |
8955 | 122^49008+121 | 102249 | Norbert Schneider | 10/2018 |
8956 | (2^339652+3178)/2 | 102246 | Stefano Morozzi | 09/2020 |
8957 | 7^120985-38 | 102245 | Lelio R Paula | 10/2016 |
8958 | 10^102244+122961 | 102245 | Patrick De Geest | 12/2016 |
8959 | 10^102244-27 | 102244 | Lelio R Paula | 09/2016 |
8960 | 7^120982-62 | 102242 | Lelio R Paula | 10/2016 |
8961 | (2^339640+2670)/2 | 102242 | Stefano Morozzi | 09/2020 |
8962 | 4*6^131382-5 | 102236 | Lelio R Paula | 11/2014 |
8963 | (2^339607+3290)/2 | 102232 | Stefano Morozzi | 02/2021 |
8964 | 2^339586-1091 | 102226 | Yves Minet | 11/2012 |
8965 | 11^98161-3072000 | 102225 | Enzo Doria | 12/2012 |
8966 | 9*7^120961-4 | 102225 | Lelio R Paula | 11/2014 |
8967 | 5*3^214250+2 | 102224 | Serge Batalov | 10/2010 |
8968 | 12^94713-59 | 102213 | Lelio R Paula | 10/2016 |
8969 | 12^94705+89 | 102204 | Lelio R Paula | 10/2016 |
8970 | L(2*339475)%(2^339475) | 102193 | Henri Lifchitz | 06/2025 |
8971 | 2^339407-177 | 102172 | Lelio R Paula | 08/2011 |
8972 | 4*10^102163-57 | 102164 | Robert Price | 10/2018 |
8973 | 2^339355+65 | 102157 | Lelio R Paula | 07/2011 |
8974 | (28*10^102142-31)/3 | 102143 | Robert Price | 12/2019 |
8975 | (2^339323+47)/880715 | 102141 | Lelio R Paula | 12/2013 |
8976 | (2^339268+1758)/2 | 102130 | Stefano Morozzi | 09/2020 |
8977 | U(610865) | 102130 | Serge Batalov | 07/2024 |
8978 | s(178551) | 102122 | Robert Price | 05/2020 |
8979 | (797*10^102112-23)/9 | 102114 | Robert Price | 01/2022 |
8980 | (5*10^102107+7)/3 | 102108 | Tyler Busby | 02/2023 |
8981 | (2^339175+2790)/2 | 102102 | Stefano Morozzi | 02/2021 |
8984 | 2^339168+21 | 102100 | Lelio R Paula | 07/2011 |
8983 | 2^339167-108807 | 102100 | Enzo Doria | 03/2014 |
8982 | (2^339169+42)/2 | 102100 | Stefano Morozzi | 09/2020 |
8985 | 2^339156+163 | 102097 | Lelio R Paula | 08/2011 |
8986 | (2^339149+746)/2 | 102094 | Stefano Morozzi | 09/2020 |
8987 | (25*10^102076-67)/3 | 102077 | Robert Price | 06/2019 |
8988 | 2^339081-4543 | 102074 | Yves Minet | 03/2019 |
8989 | (3435^28867-1)/3434 | 102069 | Serge Batalov | 06/2022 |
8990 | 2*10^102062+77 | 102063 | Robert Price | 08/2020 |
8991 | 24327^15668+15668^24327 | 102053 | Hans Havermann | 02/2019 |
8992 | 10^102052+66793 | 102053 | Patrick De Geest | 09/2022 |
8994 | 10^102052-7000001 | 102052 | Robert Price | 06/2021 |
8993 | 10^102052-618623 | 102052 | Patrick De Geest | 09/2022 |
8995 | 2^339000-175635 | 102050 | Stefano Morozzi | 12/2021 |
8996 | 2^339000-151667 | 102050 | Stefano Morozzi | 12/2021 |
8997 | 2^338999+79289 | 102049 | Stefano Morozzi | 01/2022 |
8998 | 2^338998-63897 | 102049 | Stefano Morozzi | 12/2022 |
8999 | 2^338997+99659 | 102049 | Stefano Morozzi | 02/2023 |
9000 | 2^338994+188757 | 102048 | Stefano Morozzi | 05/2022 |