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 ? |
3752 | (2^(6*86654+1)-5)/3 | 156513 | Kevin Acres | 12/2013 |
3751 | (2^519925-5)/3 | 156513 | Stefano Morozzi | 08/2022 |
3753 | (4*10^156501+119)/3 | 156502 | Robert Price | 12/2021 |
3754 | (83*10^156476-101)/9 | 156477 | Robert Price | 08/2021 |
3755 | 3*10^156435-97 | 156436 | Robert Price | 07/2018 |
3756 | 10^156428+111 | 156429 | Boyan Hu | 10/2024 |
3757 | (8*10^156371-77)/3 | 156372 | Robert Price | 07/2018 |
3758 | 53^90679-52^90679 | 156356 | Robert Price | 01/2013 |
3759 | 31267^34782-34782^31267 | 156349 | Frank Doornink | 01/2024 |
3760 | (418*10^156309-31)/9 | 156311 | Robert Price | 05/2022 |
3761 | 2^519106-59141 | 156267 | Enzo Doria | 10/2015 |
3762 | 25929^35404-35404^25929 | 156266 | Frank Doornink | 03/2024 |
3763 | 2^519053+6077 | 156251 | Enzo Doria | 02/2016 |
3764 | 31199^34764-34764^31199 | 156235 | Frank Doornink | 01/2024 |
3765 | 2^518968-1860989 | 156225 | Enzo Doria | 11/2015 |
3766 | (25*10^156221-7)/9 | 156222 | Robert Price | 05/2018 |
3767 | 2^518931-2782297 | 156214 | Enzo Doria | 09/2015 |
3768 | 2^518608-2939 | 156117 | Yves Minet | 09/2014 |
3769 | 10*27^109005+697 | 156027 | Curtis Bright | 05/2015 |
3770 | (49*10^155933-319)/9 | 155934 | Robert Price | 10/2021 |
3771 | 2^517967-113485 | 155924 | Enzo Doria | 08/2014 |
3772 | (103*10^155904+17)/3 | 155906 | Robert Price | 08/2020 |
3773 | (8*10^155887-719)/9 | 155887 | Robert Price | 03/2021 |
3774 | 2^517789+29231 | 155871 | Enzo Doria | 02/2024 |
3775 | 4*10^155853-39 | 155854 | Robert Price | 09/2020 |
3776 | 2^517698+4429 | 155843 | Lelio R Paula | 01/2013 |
3777 | 32015^34584-34584^32015 | 155814 | Frank Doornink | 01/2024 |
3778 | 2^517543-113257 | 155796 | Enzo Doria | 07/2014 |
3779 | (14*10^155740-143)/3 | 155741 | Robert Price | 11/2018 |
3780 | 31959^34564-34564^31959 | 155697 | Frank Doornink | 01/2024 |
3781 | 31^104309-30^104309 | 155563 | Henri Lifchitz | 08/2013 |
3782 | 2^516742-39851 | 155555 | Enzo Doria | 09/2015 |
3783 | (22*10^155543+47)/3 | 155544 | Robert Price | 06/2021 |
3784 | (19*10^155504+131)/3 | 155505 | Robert Price | 06/2020 |
3785 | 7*10^155496+61 | 155497 | Robert Price | 08/2019 |
3786 | 2^516440+1878421 | 155464 | Enzo Doria | 08/2016 |
3787 | 26012^35173-35173^26012 | 155295 | Frank Doornink | 03/2024 |
3788 | 2^515863-1867237 | 155291 | Enzo Doria | 03/2016 |
3789 | 2^515786-461861 | 155268 | Enzo Doria | 07/2015 |
3790 | 2^515644-113247 | 155225 | Enzo Doria | 07/2014 |
3791 | (7*10^155203+107)/3 | 155204 | Robert Price | 02/2018 |
3792 | 10^155163+206871 | 155164 | Patrick De Geest | 05/2021 |
3793 | 10^155163-40001 | 155163 | Robert Price | 01/2021 |
3794 | (17*10^155133-89)/9 | 155134 | Robert Price | 07/2018 |
3795 | 26024^35131-35131^26024 | 155117 | Frank Doornink | 03/2024 |
3796 | 2^515249+214457 | 155106 | Enzo Doria | 10/2023 |
3797 | 2^515227-114997 | 155099 | Enzo Doria | 06/2015 |
3798 | 26510^35061-35061^26510 | 155090 | Frank Doornink | 03/2024 |
3800 | EPrime(155025) | 155025 | Eric W. Weisstein | 10/2010 |
3799 | (14*10^155024-221)/9 | 155025 | Robert Price | 04/2018 |
3801 | (4*10^155022+173)/3 | 155023 | Robert Price | 02/2018 |
3802 | 2^514652+6561 | 154926 | Henri Lifchitz | 11/2024 |
3803 | (266*10^154897+1)/3 | 154899 | Robert Price | 04/2020 |
3804 | (115*10^154869-1)/3 | 154871 | Serge Batalov | 12/2014 |
3805 | (86*10^154853-221)/9 | 154854 | Robert Price | 09/2019 |
3806 | 2^514361+78167 | 154839 | Enzo Doria | 05/2016 |
3807 | 2^514281-115033 | 154815 | Enzo Doria | 06/2015 |
3808 | 25701^35104-35104^25701 | 154807 | Frank Doornink | 02/2024 |
3809 | 2^514194-3311 | 154788 | Yves Minet | 12/2014 |
3810 | 25930^35067-35067^25930 | 154779 | Frank Doornink | 03/2024 |
3811 | (14*10^154773+37)/3 | 154774 | Robert Price | 12/2018 |
3812 | (16*10^154748+131)/3 | 154749 | Robert Price | 03/2022 |
3813 | 6*10^154742-31 | 154743 | Robert Price | 07/2020 |
3814 | (14*10^154645+139)/9 | 154646 | Robert Price | 11/2021 |
3815 | 2^513690+216949 | 154637 | Enzo Doria | 12/2023 |
3816 | 5*10^154618-79 | 154619 | Robert Price | 03/2021 |
3817 | (167*10^154595+1)/3 | 154597 | Robert Price | 04/2021 |
3818 | 2^513437+1879337 | 154560 | Enzo Doria | 09/2016 |
3819 | 2^513306-113201 | 154521 | Enzo Doria | 07/2014 |
3820 | 26415^34942-34942^26415 | 154509 | Frank Doornink | 03/2024 |
3821 | 18^120818+19^120818-2 | 154497 | Harsh Aggarwal | 05/2024 |
3822 | (10^154494-13)/3 | 154494 | Robert Price | 12/2017 |
3823 | 23462^35343+35343^23462 | 154462 | Norbert Schneider | 04/2023 |
3824 | 2^513042-113787 | 154442 | Enzo Doria | 09/2014 |
3825 | (151*10^154434-7)/9 | 154436 | Robert Price | 01/2020 |
3826 | (118*10^154428+11)/3 | 154430 | Robert Price | 03/2022 |
3827 | 2^512987+29153 | 154425 | Enzo Doria | 02/2024 |
3828 | (14*10^154413+139)/9 | 154414 | Robert Price | 11/2021 |
3829 | (77*10^154389-149)/9 | 154390 | Robert Price | 03/2022 |
3830 | 32184^34247-34247^32184 | 154374 | Frank Doornink | 02/2024 |
3831 | (27393*35^99972-499)/34 | 154367 | Eric Chen | 05/2025 |
3832 | 10^154359+49 | 154360 | Robert Price | 02/2023 |
3833 | 2^512628-114053 | 154317 | Enzo Doria | 11/2014 |
3834 | 2^512529-49783 | 154287 | Enzo Doria | 10/2015 |
3835 | (355573#+6991)/(6991*2031599) | 154285 | Makoto Morimoto | 06/2019 |
3836 | 2^512315-115297 | 154223 | Enzo Doria | 06/2015 |
3837 | (47*10^154221+241)/9 | 154222 | Robert Price | 06/2022 |
3838 | (7*10^154211+173)/9 | 154211 | Robert Price | 01/2022 |
3839 | 2^512243+22703 | 154201 | Enzo Doria | 09/2023 |
3840 | 2^512168+79561 | 154178 | Enzo Doria | 06/2016 |
3841 | (77*10^154175+67)/9 | 154176 | Robert Price | 12/2021 |
3842 | (59*19^120562-5)/18 | 154170 | Anonymous | 12/2023 |
3843 | (5*10^154158-311)/9 | 154158 | Robert Price | 12/2021 |
3844 | 27169^34758-34758^27169 | 154120 | Frank Doornink | 01/2024 |
3845 | (89*10^154103-737)/9 | 154104 | Robert Price | 01/2022 |
3846 | 2^511857+23327 | 154085 | Enzo Doria | 10/2023 |
3847 | 4*10^154078-57 | 154079 | Robert Price | 10/2018 |
3848 | 26242^34863-34863^26242 | 154060 | Frank Doornink | 03/2024 |
3849 | (355049#+1859)/(11*13*10148891*62587751) | 154051 | Makoto Morimoto | 06/2019 |
3850 | 7*10^154038-57 | 154039 | Robert Price | 06/2020 |
3851 | 2^511658-114371 | 154025 | Enzo Doria | 02/2015 |
3852 | (57^87719+1)/58 | 154022 | Paul Bourdelais | 11/2017 |
3853 | 2^511580-1846679 | 154001 | Enzo Doria | 07/2015 |
3854 | (259*10^153998-1)/3 | 154000 | Robert Price | 12/2020 |
3855 | 2^511541-113713 | 153990 | Enzo Doria | 09/2014 |
3856 | 49958^32768+49957^32768 | 153965 | Kellen Shenton | 11/2020 |
3857 | 10^153935+970801 | 153936 | Patrick De Geest | 05/2021 |
3858 | 10^153935-1000001 | 153935 | Robert Price | 08/2020 |
3859 | (16*10^153868-73)/3 | 153869 | Robert Price | 07/2020 |
3860 | 9*10^153851-67 | 153852 | Robert Price | 11/2020 |
3861 | 2^511077-112723 | 153850 | Enzo Doria | 06/2014 |
3862 | 69^83653-68^83653 | 153826 | Robert Price | 12/2013 |
3863 | 2^510963-113707 | 153816 | Enzo Doria | 09/2014 |
3864 | (154*10^153810-1)/9 | 153812 | Serge Batalov | 12/2014 |
3865 | (181*25^109991-75077)/8 | 153763 | Curtis Bright | 05/2015 |
3866 | 2*10^153746-3 | 153747 | Robert Price | 07/2015 |
3867 | 2^510626+1179379 | 153714 | Enzo Doria | 05/2016 |
3868 | (7*10^153664+113)/3 | 153665 | Robert Price | 02/2018 |
3869 | 2^510448+24991 | 153661 | Enzo Doria | 11/2023 |
3870 | 2^510420+81121 | 153652 | Enzo Doria | 07/2016 |
3871 | (197*7^181761-1)/2 | 153608 | Karsten Bonath | 05/2017 |
3872 | 2^509894-459161 | 153494 | Enzo Doria | 05/2015 |
3873 | (74*10^153489+673)/9 | 153490 | Robert Price | 02/2022 |
3874 | (49*10^153484-211)/9 | 153485 | Robert Price | 06/2022 |
3875 | 2^509841+71657 | 153478 | Enzo Doria | 04/2016 |
3876 | 9*10^153446+41 | 153447 | Robert Price | 08/2020 |
3877 | 2^509533-1846723 | 153385 | Enzo Doria | 07/2015 |
3878 | 2^509468-39989 | 153366 | Enzo Doria | 09/2015 |
3879 | 70^83117-69^83117 | 153360 | Robert Price | 01/2014 |
3880 | (209246*34^99997-81)/11 | 153148 | Eric Chen | 08/2023 |
3881 | (7*10^153116-1)/3 | 153117 | Robert Price | 08/2022 |
3882 | (112*10^153025-1)/3 | 153027 | Serge Batalov | 12/2014 |
3883 | 31912^33975-33975^31912 | 153022 | Frank Doornink | 01/2024 |
3884 | (19*10^153010+131)/3 | 153011 | Robert Price | 06/2020 |
3885 | 2^508174-1855151 | 152976 | Enzo Doria | 09/2015 |
3886 | 181^67757-2 | 152974 | Lelio R Paula | 12/2017 |
3887 | 2^508160-1864829 | 152972 | Enzo Doria | 01/2016 |
3888 | 10^152960+412563 | 152961 | Patrick De Geest | 05/2021 |
3889 | 10^152960-50000000001 | 152960 | Robert Price | 12/2020 |
3890 | 10^152960-423749 | 152960 | Patrick De Geest | 05/2021 |
3891 | (71*10^152933-143)/9 | 152934 | Robert Price | 10/2021 |
3892 | 32109^33932-33932^32109 | 152919 | Frank Doornink | 02/2024 |
3893 | 2^507857-1848199 | 152881 | Enzo Doria | 07/2015 |
3894 | 2^507638-115361 | 152815 | Enzo Doria | 07/2015 |
3895 | 2^507551-114025 | 152789 | Enzo Doria | 11/2014 |
3896 | (6^196337+1)/7 | 152780 | Paul Bourdelais | 02/2010 |
3897 | (22*10^152670-13)/9 | 152671 | Robert Price | 06/2018 |
3898 | 2^507085+28517 | 152648 | Enzo Doria | 01/2024 |
3899 | 2^506917+28847 | 152598 | Enzo Doria | 02/2024 |
3900 | 2^506451-22339 | 152457 | Enzo Doria | 07/2013 |
3901 | 26463^34460-34460^26463 | 152405 | Frank Doornink | 03/2024 |
3902 | 31898^33837-33837^31898 | 152394 | Frank Doornink | 01/2024 |
3903 | 2^505995-113185 | 152320 | Enzo Doria | 07/2014 |
3904 | (8*10^152260-17)/9 | 152260 | Serge Batalov | 11/2022 |
3905 | (4*10^152253+149)/9 | 152253 | Robert Price | 04/2018 |
3906 | 33530^33643-33643^33530 | 152250 | Norbert Schneider | 04/2024 |
3907 | 578^55117-577^55117 | 152230 | Jean-Louis Charton | 08/2012 |
3908 | 2^505431-505431 | 152150 | Henri Lifchitz | 04/2005 |
3909 | (203*10^152143+1)/3 | 152145 | Robert Price | 03/2020 |
3910 | 31730^33787-33787^31730 | 152092 | Frank Doornink | 01/2024 |
3911 | 2^505089+217787 | 152047 | Enzo Doria | 01/2024 |
3912 | 33565^33594-33594^33565 | 152043 | Norbert Schneider | 04/2024 |
3913 | 2^505053+73097 | 152037 | Enzo Doria | 04/2016 |
3914 | 22*10^151998+3 | 152000 | Robert Price | 01/2020 |
3915 | (238*10^151979-1)/3 | 151981 | Robert Price | 04/2020 |
3916 | (2^504793-7)/46765 | 151954 | Lelio R Paula | 12/2017 |
3917 | 3^318465+4 | 151947 | Aurel Gibier | 09/2024 |
3918 | 350377#+797581 | 151937 | Nicolas PALAU | 08/2017 |
3919 | (79*10^151906-529)/9 | 151907 | Robert Price | 01/2022 |
3920 | 2^504618-343991 | 151906 | Enzo Doria | 07/2015 |
3921 | 2^504259+211193 | 151798 | Enzo Doria | 08/2023 |
3922 | 2^504212-461939 | 151783 | Enzo Doria | 07/2015 |
3923 | 7*10^151772-41 | 151773 | Robert Price | 08/2021 |
3924 | 2^504054-2231 | 151736 | Yves Minet | 10/2011 |
3925 | 25237^34458-34458^25237 | 151686 | Frank Doornink | 11/2023 |
3926 | (28*10^151682+191)/3 | 151683 | Robert Price | 02/2020 |
3927 | 2^503817+23961 | 151665 | Enzo Doria | 10/2023 |
3928 | 2^503475-933075 | 151562 | Paul Bourdelais | 06/2021 |
3929 | 2^503371+26289 | 151530 | Enzo Doria | 12/2023 |
3930 | 2^503361-39883 | 151527 | Enzo Doria | 09/2015 |
3931 | 2^503225+382667 | 151486 | Enzo Doria | 10/2016 |
3932 | (206*10^151483-11)/3 | 151485 | Robert Price | 02/2022 |
3933 | (844*10^151396-7)/9 | 151398 | Robert Price | 05/2021 |
3934 | (26*10^151391-503)/9 | 151392 | Robert Price | 07/2018 |
3935 | 2^502769+1877567 | 151349 | Enzo Doria | 08/2016 |
3936 | 2^502576-4019 | 151291 | Yves Minet | 10/2011 |
3937 | 5*10^151257+41 | 151258 | Robert Price | 03/2019 |
3938 | 2^502290-2792501 | 151205 | Enzo Doria | 02/2016 |
3939 | 2^502181+1868417 | 151172 | Enzo Doria | 04/2016 |
3940 | 2^502055+22583 | 151134 | Enzo Doria | 09/2023 |
3941 | (3*23^110967+41)/22 | 151106 | Anonymous | 12/2023 |
3942 | 6*10^151063-77 | 151064 | Robert Price | 09/2021 |
3943 | 2^501781-1850503 | 151052 | Enzo Doria | 08/2015 |
3944 | (2^82589933-1)%(10^151020) | 151020 | Anonymous | 05/2025 |
3945 | (5*10^151007+37)/3 | 151008 | Robert Price | 03/2018 |
3946 | 3^316494+5*2^316494 | 151007 | Jon Grantham | 08/2023 |
3947 | 2^501627-991 | 151005 | Yves Minet | 02/2014 |
3948 | 2^501577-115333 | 150990 | Enzo Doria | 07/2015 |
3949 | 25239^34298-34298^25239 | 150983 | Frank Doornink | 12/2023 |
3950 | (19*10^150977-7)/3 | 150978 | Robert Price | 12/2023 |
3951 | 8*10^150975+3 | 150976 | Robert Price | 08/2015 |
3952 | 2^501512-113669 | 150971 | Enzo Doria | 09/2014 |
3953 | 34180^26101+26101^34180 | 150962 | Norman Luhn | 04/2024 |
3954 | U(1083477)/2166953 | 150949 | Serge Batalov | 07/2024 |
3955 | (83*10^150908+493)/9 | 150909 | Robert Price | 12/2021 |
3956 | 36267^14482+14482^36267 | 150901 | Norman Luhn | 04/2024 |
3957 | 2*10^150818+83 | 150819 | Robert Price | 12/2021 |
3958 | 48753^1240+1240^48753 | 150814 | Norman Luhn | 04/2024 |
3959 | (12^139739+1)/13 | 150803 | Paul Bourdelais | 09/2011 |
3960 | (31^101111-1)/30 | 150792 | Alexander Zhirkov | 11/2010 |
3961 | 2^500899-500899 | 150786 | Henri Lifchitz | 04/2005 |
3962 | 2^500832+1876801 | 150766 | Enzo Doria | 07/2016 |
3963 | (23*10^150763-59)/9 | 150764 | Robert Price | 05/2018 |
3964 | 46002^1891+1891^46002 | 150735 | Norman Luhn | 04/2024 |
3965 | 32001^33458-33458^32001 | 150734 | Frank Doornink | 01/2024 |
3966 | 2^500699+1884953 | 150726 | Enzo Doria | 02/2017 |
3967 | (13*10^150675-37)/3 | 150676 | Robert Price | 01/2019 |
3968 | (2^500527+lucasV(1,2,500527)+1)/4 | 150674 | Mike Oakes | 11/2016 |
3969 | 35365^18204+18204^35365 | 150661 | Norman Luhn | 05/2024 |
3971 | 2^500337-71101 | 150617 | Enzo Doria | 04/2016 |
3970 | (74*10^150616+133)/9 | 150617 | Robert Price | 06/2019 |
3972 | (2*10^150612-17)/3 | 150612 | Serge Batalov | 11/2022 |
3973 | 4*10^150577+91 | 150578 | Robert Price | 08/2018 |
3974 | 5649^40132+40132^5649 | 150575 | Norbert Schneider | 04/2024 |
3975 | 2^500160+80071 | 150564 | Enzo Doria | 06/2016 |
3976 | (2^500120+5)/9 | 150551 | Lelio R Paula | 12/2017 |
3978 | (43*10^150515-421)/9 | 150516 | Robert Price | 01/2019 |
3977 | 4695^40994+40994^4695 | 150516 | Norbert Schneider | 11/2023 |
3979 | 2^499999+15981 | 150515 | Norman Luhn | 08/2023 |
3980 | (10^150551-1)/9/385207015049/57498437921/7808738631274081 | 150512 | Maksym Voznyy | 08/2014 |
3981 | (10^150551-1)/1556587827131683185788921941743157225041 | 150512 | Robert Price | 07/2016 |
3982 | 26614^34005-34005^26614 | 150476 | Frank Doornink | 03/2024 |
3983 | 2^499861-29503 | 150474 | Enzo Doria | 08/2015 |
3984 | 2^499653-113533 | 150411 | Enzo Doria | 08/2014 |
3985 | (14*10^150391-191)/3 | 150392 | Robert Price | 10/2020 |
3986 | 29033^33696+33696^29033 | 150382 | Norbert Schneider | 10/2023 |
3987 | 2^499534-221 | 150375 | Yves Minet | 06/2012 |
3988 | 30110^33567+33567^30110 | 150337 | Norbert Schneider | 10/2023 |
3989 | 26212^34023-34023^26212 | 150331 | Frank Doornink | 03/2024 |
3990 | (28*10^150294+41)/3 | 150295 | Robert Price | 07/2021 |
3991 | 2^499225+49727 | 150282 | Enzo Doria | 09/2016 |
3992 | (695*10^150274+7)/9 | 150276 | Robert Price | 12/2021 |
3993 | 2^499054-29261 | 150231 | Enzo Doria | 07/2015 |
3994 | 18580^35187+35187^18580 | 150215 | Norbert Schneider | 10/2023 |
3995 | (844*10^150196-7)/9 | 150198 | Robert Price | 05/2021 |
3996 | 2^498907-933075 | 150186 | Paul Bourdelais | 06/2021 |
3997 | (2^498898-7)/9 | 150183 | Lelio R Paula | 10/2015 |
3998 | 3*10^150182+17 | 150183 | Robert Price | 07/2018 |
3999 | 5*10^150175+63 | 150176 | Robert Price | 09/2020 |
4000 | 29273^33618-33618^29273 | 150154 | Frank Doornink | 06/2024 |