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) !!!
Submit your PRPs
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
3751 | 38237^8214+8214^38237 | 149681 | Hans Havermann | 09/2020 |
3752 | (23*10^149655+19)/3 | 149656 | Robert Price | 07/2019 |
3753 | 2^497068-113579 | 149633 | Enzo Doria | 08/2014 |
3754 | 2^497037+73727 | 149624 | Enzo Doria | 04/2016 |
3755 | 11*10^149618+3 | 149620 | Robert Price | 07/2021 |
3756 | 34942^19067+19067^34942 | 149562 | Hans Havermann | 09/2020 |
3757 | 33696^27449+27449^33696 | 149561 | Hans Havermann | 09/2020 |
3758 | 2^496773+77747 | 149544 | Enzo Doria | 05/2016 |
3759 | 29937^33400-33400^29937 | 149506 | Frank Doornink | 06/2024 |
3760 | (7*10^149483-493)/9 | 149483 | Robert Price | 04/2021 |
3761 | (206*10^149459+7)/3 | 149461 | Robert Price | 07/2020 |
3762 | 34589^20940+20940^34589 | 149459 | Hans Havermann | 09/2020 |
3763 | 2^496477-113485 | 149455 | Enzo Doria | 08/2014 |
3764 | 3*2^496411-5 | 149436 | Jon Grantham | 07/2023 |
3765 | ((10^74703-1)*10000/9+7481)*10^74703+(10^74703-1)/9 | 149410 | Hans Havermann | 09/2019 |
3766 | 2^496270-49451 | 149393 | Enzo Doria | 10/2015 |
3767 | 2^496238-1865081 | 149383 | Enzo Doria | 01/2016 |
3768 | 33831^26030+26030^33831 | 149380 | Hans Havermann | 09/2020 |
3769 | 34131^23780+23780^34131 | 149365 | Hans Havermann | 09/2020 |
3770 | (5*10^149335-53)/3 | 149336 | Robert Price | 09/2020 |
3771 | 26043^33818-33818^26043 | 149330 | Frank Doornink | 03/2024 |
3772 | 2^495968-1853699 | 149302 | Enzo Doria | 09/2015 |
3773 | 2^495935-4117 | 149292 | Yves Minet | 02/2018 |
3774 | 5*2^495930+3 | 149291 | Jon Grantham | 07/2023 |
3775 | 35816^14709+14709^35816 | 149267 | Hans Havermann | 09/2020 |
3776 | (10^149264+7)/191/2683/80557 | 149254 | Alfred Reich | 07/2021 |
3777 | 10^149240-40000001 | 149240 | Robert Price | 07/2022 |
3778 | 10^149240-10809 | 149240 | Alfred Reich | 08/2024 |
3779 | 2^495712-396765 | 149225 | Enzo Doria | 07/2013 |
3780 | (21*2^495676-1)/4835 | 149211 | Lelio R Paula | 10/2015 |
3781 | 2^495420+1880701 | 149137 | Enzo Doria | 10/2016 |
3782 | 25646^33823-33823^25646 | 149127 | Frank Doornink | 02/2024 |
3783 | 2^495337-1847293 | 149112 | Enzo Doria | 07/2015 |
3784 | (2^495193-7)/5 | 149068 | Lelio R Paula | 10/2015 |
3785 | 2^495163+111546435 | 149059 | Paul Bourdelais | 10/2021 |
3786 | 33282^30097+30097^33282 | 149055 | Hans Havermann | 09/2020 |
3787 | 37384^9705+9705^37384 | 149050 | Hans Havermann | 09/2020 |
3788 | 34808^19143+19143^34808 | 149049 | Hans Havermann | 09/2020 |
3789 | 413^56963-412^56963 | 149013 | Jean-Louis Charton | 08/2012 |
3790 | 30657^33212-33212^30657 | 149007 | Frank Doornink | 06/2024 |
3791 | 47^89113-46^89113 | 149006 | Robert Price | 10/2012 |
3792 | V(891190) | 148997 | Serge Batalov | 07/2024 |
3793 | 2^494916+87855 | 148985 | Lelio R Paula | 01/2016 |
3794 | 6*10^148982+73 | 148983 | Robert Price | 04/2019 |
3795 | 3^312155-4 | 148936 | Anonymous | 02/2023 |
3796 | 10^148934+1530019 | 148935 | Patrick De Geest | 02/2021 |
3797 | 10^148934-1000000001 | 148934 | Robert Price | 11/2020 |
3798 | 10^148934-512057 | 148934 | Patrick De Geest | 02/2021 |
3799 | 39000^6571+6571^39000 | 148888 | Hans Havermann | 10/2020 |
3800 | 2^494586-2791991 | 148886 | Enzo Doria | 02/2016 |
3801 | 8*10^148860-3 | 148861 | Ray Chandler | 04/2011 |
3802 | 38262^7769+7769^38262 | 148854 | Hans Havermann | 10/2020 |
3803 | 10^148816-500001 | 148816 | Robert Price | 03/2022 |
3804 | (53*10^148811-287)/9 | 148812 | Robert Price | 02/2022 |
3805 | 2^494333-286183 | 148810 | Enzo Doria | 06/2015 |
3806 | 2^494144-264119 | 148753 | Enzo Doria | 05/2014 |
3807 | 6*10^148734+47 | 148735 | Robert Price | 06/2022 |
3808 | 28724^33353-33353^28724 | 148696 | Frank Doornink | 06/2024 |
3809 | 36502^11823+11823^36502 | 148663 | Hans Havermann | 11/2020 |
3810 | (5*2^493838-1)/3721 | 148658 | Lelio R Paula | 10/2015 |
3811 | 64239^206+206^64239 | 148641 | Hans Havermann | 10/2020 |
3812 | (23*2^493761-1)/555 | 148636 | Lelio R Paula | 10/2015 |
3813 | (23*10^148627-137)/3 | 148628 | Robert Price | 08/2021 |
3814 | (151*10^148601-1)/3 | 148603 | Robert Price | 03/2020 |
3815 | 2^493636+82081 | 148600 | Enzo Doria | 09/2016 |
3816 | ((((((((((((((34251^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2 | 148593 | Serge Batalov | 07/2016 |
3817 | 27134^33509-33509^27134 | 148563 | Frank Doornink | 01/2024 |
3818 | 177487#+(519643#)/(177487#) | 148558 | Makoto Morimoto | 11/2017 |
3819 | (2*10^148460-23)/3 | 148460 | Robert Price | 04/2018 |
3820 | (28*10^148445+101)/3 | 148446 | Robert Price | 09/2020 |
3821 | 2^493065+225 | 148428 | Henri Lifchitz | 11/2024 |
3822 | 36203^12556+12556^36203 | 148391 | Hans Havermann | 10/2020 |
3823 | 33912^23749+23749^33912 | 148387 | Hans Havermann | 10/2020 |
3824 | (8*10^148328+1)/9 | 148328 | Serge Batalov | 05/2010 |
3825 | 34294^21117+21117^34294 | 148309 | Hans Havermann | 10/2020 |
3826 | (71*10^148306-539)/9 | 148307 | Robert Price | 07/2019 |
3827 | 30814^33039-33039^30814 | 148304 | Frank Doornink | 06/2024 |
3828 | 38042^7875+7875^38042 | 148222 | Hans Havermann | 10/2020 |
3829 | 5^211959-12874 | 148153 | Enzo Doria | 05/2014 |
3830 | 34142^21831+21831^34142 | 148145 | Hans Havermann | 10/2020 |
3831 | 10^148112+34059 | 148113 | Patrick De Geest | 05/2021 |
3832 | 10^148112-70000001 | 148112 | Robert Price | 03/2021 |
3833 | 10^148112-106593 | 148112 | Patrick De Geest | 05/2021 |
3834 | 36303^12016+12016^36303 | 148108 | Hans Havermann | 10/2020 |
3835 | (77*10^148089-149)/9 | 148090 | Robert Price | 03/2022 |
3836 | (16*10^148071+41)/3 | 148072 | Robert Price | 03/2019 |
3837 | 187^65175-2 | 148068 | Lelio R Paula | 11/2017 |
3838 | 2^491851+1871453 | 148062 | Enzo Doria | 05/2016 |
3839 | (68*10^148060-257)/9 | 148061 | Robert Price | 07/2019 |
3840 | (151*10^148020-1)/3 | 148022 | Robert Price | 03/2020 |
3841 | 39153^6032+6032^39153 | 148017 | Hans Havermann | 10/2020 |
3842 | 36798^10507+10507^36798 | 147983 | Hans Havermann | 12/2020 |
3843 | (26*10^147973-23)/3 | 147974 | Robert Price | 12/2019 |
3844 | 2^491521-1865323 | 147963 | Enzo Doria | 02/2016 |
3845 | (2^491476+7)/71 | 147948 | Lelio R Paula | 10/2015 |
3846 | 2*10^147928+47 | 147929 | Robert Price | 12/2021 |
3847 | 2^491391-7807 | 147924 | Lelio R Paula | 01/2013 |
3848 | 39736^5279+5279^39736 | 147920 | Hans Havermann | 11/2020 |
3849 | 2^491370-113297 | 147918 | Enzo Doria | 07/2014 |
3850 | (32760^32760+32759^32759)/95546586737133567708319 | 147900 | Robert Price | 07/2016 |
3851 | 44368^2155+2155^44368 | 147899 | Hans Havermann | 12/2020 |
3852 | (17*10^147890+79)/3 | 147891 | Robert Price | 04/2019 |
3853 | 94*10^147871-3 | 147873 | Robert Price | 05/2020 |
3854 | 10^147843+620967 | 147844 | Patrick De Geest | 05/2021 |
3855 | 10^147843-50000000001 | 147843 | Robert Price | 12/2020 |
3856 | 10^147843-182891 | 147843 | Patrick De Geest | 05/2021 |
3857 | 35805^13346+13346^35805 | 147709 | Hans Havermann | 11/2020 |
3858 | 2^490672+255255 | 147707 | Paul Bourdelais | 10/2021 |
3859 | 2^490660-17 | 147704 | Lelio R Paula | 12/2017 |
3860 | (229*10^147699-1)/3 | 147701 | Robert Price | 07/2020 |
3861 | 40656^4289+4289^40656 | 147678 | Hans Havermann | 11/2020 |
3862 | (85*10^147676-31)/9 | 147677 | Rytis Slatkevicius | 12/2024 |
3863 | 2^490550+1179439 | 147671 | Enzo Doria | 06/2016 |
3864 | 27249^33292-33292^27249 | 147662 | Frank Doornink | 06/2024 |
3865 | (23*10^147649-59)/9 | 147650 | Robert Price | 05/2018 |
3866 | 2^490303+2793653 | 147596 | Enzo Doria | 03/2016 |
3867 | 2^490266+255255 | 147585 | Paul Bourdelais | 10/2021 |
3868 | 199^64181-2 | 147543 | Lelio R Paula | 11/2017 |
3869 | 32849^30958+30958^32849 | 147518 | Hans Havermann | 11/2020 |
3870 | 2^489871-286987 | 147466 | Enzo Doria | 07/2015 |
3871 | 2^489864-351899 | 147464 | Enzo Doria | 07/2015 |
3872 | 175^65724-2 | 147422 | Lelio R Paula | 11/2017 |
3873 | 2^489712+1874851 | 147419 | Enzo Doria | 06/2016 |
3874 | 34473^18830+18830^34473 | 147367 | Hans Havermann | 11/2020 |
3875 | 63^81896-2 | 147359 | Lelio R Paula | 10/2017 |
3876 | 34580^18237+18237^34580 | 147344 | Hans Havermann | 11/2020 |
3877 | (11*10^147313-107)/3 | 147314 | Robert Price | 09/2018 |
3878 | (397*10^147310+17)/9 | 147312 | Robert Price | 03/2022 |
3879 | 2^489305-59815 | 147296 | Enzo Doria | 10/2015 |
3880 | (2^489280-7)/6381 | 147285 | Lelio R Paula | 09/2015 |
3881 | 2*10^147264-61 | 147265 | Robert Price | 10/2021 |
3882 | 2*10^147260+29 | 147261 | Robert Price | 02/2021 |
3883 | 33556^24457+24457^33556 | 147258 | Hans Havermann | 11/2020 |
3884 | 12^136450-11 | 147255 | Riley Fisher | 12/2024 |
3885 | 2^489125+80357 | 147242 | Enzo Doria | 06/2016 |
3886 | 10^147240-40000001 | 147240 | Robert Price | 07/2022 |
3887 | 10^147240-371331 | 147240 | Alfred Reich | 09/2024 |
3888 | 10^147240-464727 | 147240 | Alfred Reich | 09/2024 |
3889 | 32835^30452+30452^32835 | 147220 | Hans Havermann | 12/2020 |
3890 | (19*10^147213-97)/3 | 147214 | Robert Price | 10/2021 |
3891 | 27123^33190-33190^27123 | 147143 | Frank Doornink | 01/2024 |
3892 | (148*10^147139-1)/3 | 147141 | Robert Price | 03/2020 |
3893 | 34338^19255+19255^34338 | 147123 | Hans Havermann | 11/2020 |
3894 | 2^488648-112799 | 147098 | Enzo Doria | 07/2014 |
3895 | 2^488627-350317 | 147092 | Enzo Doria | 06/2015 |
3896 | 33432^25075+25075^33432 | 147076 | Hans Havermann | 12/2020 |
3897 | (2^488441-1)/(61543567*30051203516986199) | 147012 | Dario Alpern | 12/2014 |
3898 | 2^488280-2165 | 146987 | Yves Minet | 07/2014 |
3899 | 2^488278+45 | 146987 | Lelio R Paula | 12/2017 |
3900 | 32568^32568+32561^32561 | 146973 | Henri Lifchitz | 08/2024 |
3901 | 2^487927-39 | 146881 | Lelio R Paula | 12/2017 |
3902 | (7*10^146860-13)/3 | 146861 | Robert Price | 07/2018 |
3903 | (26*10^146856-77)/3 | 146857 | Robert Price | 11/2019 |
3904 | 32948^28611+28611^32948 | 146834 | Hans Havermann | 11/2020 |
3905 | 42596^2799+2799^42596 | 146829 | Hans Havermann | 12/2020 |
3906 | 2^487726-1863341 | 146821 | Enzo Doria | 12/2015 |
3907 | 43838^2229+2229^43838 | 146775 | Hans Havermann | 11/2020 |
3908 | 35005^15594+15594^35005 | 146775 | Hans Havermann | 11/2020 |
3909 | (167*10^146764+7)/3 | 146766 | Robert Price | 04/2021 |
3910 | 53^85114+85114 | 146761 | Kellen Shenton | 05/2022 |
3911 | (59*10^146738+157)/9 | 146739 | Robert Price | 12/2021 |
3912 | ((10^73362-1)*10000/9+6701)*10^73362+(10^73362-1)/9 | 146728 | Hans Havermann | 09/2019 |
3913 | 32963^28234+28234^32963 | 146711 | Hans Havermann | 12/2020 |
3914 | (22*10^146661-19)/3 | 146662 | Robert Price | 04/2019 |
3915 | 33958^20825+20825^33958 | 146651 | Hans Havermann | 12/2020 |
3916 | 26289^33178-33178^26289 | 146640 | Frank Doornink | 03/2024 |
3917 | 2^487121-39523 | 146639 | Enzo Doria | 09/2015 |
3918 | 61063^252+252^61063 | 146637 | Hans Havermann | 12/2020 |
3919 | (7*2^486967+1)/39 | 146591 | Lelio R Paula | 09/2015 |
3920 | 34099^19900+19900^34099 | 146587 | Hans Havermann | 11/2020 |
3921 | 2^486893-79 | 146570 | Yves Minet | 09/2017 |
3922 | 36818^9567+9567^36818 | 146565 | Hans Havermann | 12/2020 |
3923 | 2^486840+1879141 | 146554 | Enzo Doria | 08/2016 |
3924 | 2^486774-87 | 146534 | Lelio R Paula | 12/2017 |
3925 | 38655^6176+6176^38655 | 146530 | Hans Havermann | 11/2020 |
3926 | 8*10^146522-81 | 146523 | Robert Price | 02/2021 |
3927 | 32820^29059+29059^32820 | 146485 | Hans Havermann | 12/2020 |
3928 | 37261^8530+8530^37261 | 146472 | Hans Havermann | 12/2020 |
3929 | 66953^154+154^66953 | 146462 | Hans Havermann | 12/2020 |
3930 | 96^73877-95^73877 | 146445 | Robert Price | 10/2015 |
3931 | 8*10^146421+81 | 146422 | Robert Price | 08/2019 |
3932 | 33153^26090+26090^33153 | 146420 | Hans Havermann | 11/2020 |
3933 | 2^486275+23961 | 146384 | Enzo Doria | 10/2023 |
3934 | (52^85259+1)/53 | 146304 | Paul Bourdelais | 09/2017 |
3935 | 179^64935+2 | 146290 | Lelio R Paula | 10/2017 |
3936 | 32595^30758+30758^32595 | 146285 | Hans Havermann | 11/2020 |
3937 | 34972^15157+15157^34972 | 146205 | Hans Havermann | 12/2020 |
3938 | (16*10^146200+131)/3 | 146201 | Robert Price | 03/2022 |
3939 | (19*10^146181-49)/3 | 146182 | Robert Price | 05/2019 |
3940 | 2^485531-114621 | 146160 | Enzo Doria | 05/2015 |
3941 | 2^485473-933075 | 146142 | Paul Bourdelais | 06/2021 |
3942 | (2^485461-9)/4249 | 146135 | Lelio R Paula | 09/2015 |
3943 | (22*10^146130-193)/9 | 146131 | Robert Price | 07/2022 |
3944 | 2^485389+81887 | 146117 | Enzo Doria | 08/2016 |
3945 | 32554^30773+30773^32554 | 146108 | Hans Havermann | 11/2020 |
3946 | 28075^32844-32844^28075 | 146101 | Frank Doornink | 06/2024 |
3947 | (10^146063+71)/9 | 146063 | Serge Batalov | 07/2015 |
3948 | 2*10^146051-63 | 146052 | Robert Price | 01/2022 |
3949 | (2^485119+7)/2745 | 146032 | Lelio R Paula | 09/2015 |
3950 | (22*10^146029+257)/9 | 146030 | Robert Price | 06/2018 |
3951 | 7*10^145982-53 | 145983 | Robert Price | 06/2020 |
3952 | 2^484743+1872923 | 145923 | Enzo Doria | 05/2016 |
3953 | 33140^25287+25287^33140 | 145913 | Hans Havermann | 01/2021 |
3954 | 37875^7112+7112^37875 | 145895 | Hans Havermann | 01/2021 |
3955 | 30542^32529-32529^30542 | 145890 | Frank Doornink | 06/2024 |
3956 | 32862^27473+27473^32862 | 145872 | Hans Havermann | 01/2021 |
3957 | V(1046652)/2093303 | 145817 | Serge Batalov | 07/2024 |
3958 | 32836^27587+27587^32836 | 145815 | Hans Havermann | 01/2021 |
3959 | 2^484341+1879157 | 145802 | Enzo Doria | 08/2016 |
3960 | 2*10^145784-91 | 145785 | Robert Price | 04/2020 |
3961 | 2^484271-51 | 145781 | Lelio R Paula | 11/2017 |
3962 | (188*10^145750-11)/3 | 145752 | Robert Price | 02/2022 |
3963 | 70^78989-69^78989 | 145743 | Robert Price | 01/2014 |
3964 | 51^85334+2 | 145714 | Robert Price | 07/2015 |
3965 | 2^483886-1857101 | 145665 | Enzo Doria | 10/2015 |
3966 | 29312^32607-32607^29312 | 145657 | Frank Doornink | 06/2024 |
3967 | 33350^23307+23307^33350 | 145656 | Hans Havermann | 01/2021 |
3968 | (19*10^145585+131)/3 | 145586 | Robert Price | 06/2020 |
3969 | 31815^32332-32332^31815 | 145580 | Frank Doornink | 01/2024 |
3970 | 2^483563-1863457 | 145567 | Enzo Doria | 12/2015 |
3971 | 4*10^145563-69 | 145564 | Robert Price | 01/2022 |
3972 | 2^483516-1861409 | 145553 | Enzo Doria | 11/2015 |
3973 | 2^483384-115019 | 145514 | Enzo Doria | 06/2015 |
3974 | (259*10^145501-1)/3 | 145503 | Robert Price | 12/2020 |
3975 | (19*10^145499-61)/3 | 145500 | Robert Price | 05/2019 |
3976 | 2^483302-3221 | 145489 | Lelio R Paula | 12/2012 |
3977 | (23*10^145463+61)/3 | 145464 | Robert Price | 01/2022 |
3978 | 35523^12434+12434^35523 | 145453 | Hans Havermann | 01/2021 |
3979 | (29121^32761-1)/(29121^181-1) | 145444 | Ryan Propper | 11/2023 |
3980 | 2^483046+1874029 | 145412 | Enzo Doria | 06/2016 |
3981 | 2^482991-2789317 | 145395 | Enzo Doria | 12/2015 |
3982 | 2^482978-2789711 | 145391 | Enzo Doria | 12/2015 |
3983 | 2^482966-2789711 | 145388 | Enzo Doria | 12/2015 |
3984 | 2^482817-1858843 | 145343 | Enzo Doria | 10/2015 |
3985 | 2^482751-113635 | 145323 | Enzo Doria | 08/2014 |
3986 | 2^482705-459763 | 145309 | Enzo Doria | 05/2015 |
3987 | (4^241306+11)/3 | 145281 | Robert Price | 03/2017 |
3988 | (238*10^145276-1)/3 | 145278 | Robert Price | 04/2020 |
3989 | (19*2^482534+1)/7 | 145258 | Lelio R Paula | 09/2015 |
3990 | 53^84239-52^84239 | 145252 | Robert Price | 01/2013 |
3991 | U(1042497) | 145243 | Serge Batalov | 07/2024 |
3992 | 2^482386+212419 | 145213 | Enzo Doria | 08/2023 |
3993 | 47952^1067+1067^47952 | 145207 | Hans Havermann | 01/2021 |
3994 | (2^482333+5)/9287 | 145193 | Lelio R Paula | 09/2015 |
3995 | 2^482087+72623 | 145123 | Enzo Doria | 04/2016 |
3996 | 2^482016-286565 | 145102 | Enzo Doria | 07/2015 |
3997 | (19*10^145093+101)/3 | 145094 | Robert Price | 09/2019 |
3998 | 2^481956+121 | 145084 | Stefano Morozzi | 04/2021 |
3999 | 27535^32676-32676^27535 | 145078 | Frank Doornink | 06/2024 |
4000 | 2^481840+15 | 145049 | Robert Price | 09/2015 |