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 ? |
| 3501 | (29*10^165796+67)/3 | 165797 | Robert Price | 12/2019 |
| 3502 | 81^86837-80^86837 | 165728 | Robert Price | 01/2015 |
| 3503 | (71*10^165716-17)/9 | 165717 | Serge Batalov | 01/2023 |
| 3504 | fibonacci(792909)+3 | 165708 | Jon Grantham | 08/2023 |
| 3505 | (71*10^165663-539)/9 | 165664 | Robert Price | 07/2019 |
| 3506 | 25682^37567-37567^25682 | 165657 | Frank Doornink | 02/2024 |
| 3507 | 2^550270-69551 | 165648 | Enzo Doria | 12/2015 |
| 3508 | 2^550250+29329 | 165642 | Enzo Doria | 03/2024 |
| 3509 | (202*10^165569-13)/9 | 165571 | Robert Price | 01/2020 |
| 3510 | 2^550000-23625 | 165567 | Enzo Doria | 08/2013 |
| 3511 | 31688^36783-36783^31688 | 165557 | Frank Doornink | 01/2024 |
| 3512 | 2^549675-1852057 | 165469 | Enzo Doria | 08/2015 |
| 3513 | 2^549663+212543 | 165466 | Enzo Doria | 08/2023 |
| 3514 | (3^549633)%(2^549633) | 165456 | Henri Lifchitz | 06/2025 |
| 3515 | (529*10^165394-7)/9 | 165396 | Robert Price | 08/2021 |
| 3516 | 26253^37424-37424^26253 | 165384 | Frank Doornink | 03/2024 |
| 3517 | -tau(103^14938) | 165374 | Nik Lygeros / Olivier Rozier | 10/2013 |
| 3518 | (28*10^165358-1)/9 | 165359 | Serge Batalov | 10/2014 |
| 3519 | 2^549278+1879669 | 165350 | Enzo Doria | 09/2016 |
| 3520 | (7*10^165350-439)/9 | 165350 | Robert Price | 02/2018 |
| 3521 | 2^549252-351269 | 165342 | Enzo Doria | 06/2015 |
| 3522 | 2^549221+335931 | 165332 | Enzo Doria | 09/2024 |
| 3523 | 67*10^165327-9 | 165329 | Robert Price | 03/2021 |
| 3524 | (14*10^165282+73)/3 | 165283 | Robert Price | 12/2018 |
| 3525 | (8*10^165283-629)/9 | 165283 | Robert Price | 01/2022 |
| 3526 | 2^549049+2797607 | 165281 | Enzo Doria | 04/2016 |
| 3527 | (218*10^165258+7)/9 | 165260 | Robert Price | 06/2022 |
| 3528 | (7*10^165239+107)/3 | 165240 | Robert Price | 02/2018 |
| 3529 | 2^548880+23961 | 165230 | Enzo Doria | 10/2023 |
| 3530 | 5*2^548778+3 | 165200 | Jon Grantham | 07/2023 |
| 3531 | (44^100511-1)/43 | 165184 | Paul Bourdelais | 11/2017 |
| 3532 | (7*10^165134+107)/3 | 165135 | Robert Price | 02/2018 |
| 3533 | (301*10^165082+23)/9 | 165084 | Robert Price | 06/2021 |
| 3534 | (23*10^165060-83)/3 | 165061 | Robert Price | 08/2019 |
| 3535 | 7*10^165023-27 | 165024 | Robert Price | 11/2022 |
| 3536 | (76*10^164984+77)/9 | 164985 | Robert Price | 10/2019 |
| 3537 | 31058^36715-36715^31058 | 164931 | Frank Doornink | 01/2024 |
| 3538 | (4*10^164929-211)/9 | 164929 | Robert Price | 03/2022 |
| 3539 | 10^164878-500001 | 164878 | Robert Price | 03/2022 |
| 3540 | 9*10^164791-89 | 164792 | Robert Price | 04/2020 |
| 3541 | (2^788414)%F(788414) | 164769 | Henri Lifchitz | 05/2025 |
| 3542 | 2^547178-1856501 | 164717 | Enzo Doria | 10/2015 |
| 3543 | 11^158160-60 | 164707 | Enzo Doria | 07/2012 |
| 3544 | (26*10^164643-167)/3 | 164644 | Robert Price | 10/2019 |
| 3545 | (10^164643+83)/3 | 164643 | Robert Price | 08/2021 |
| 3546 | 2^546398-59201 | 164483 | Enzo Doria | 10/2015 |
| 3547 | 2^546332-114893 | 164463 | Enzo Doria | 05/2015 |
| 3548 | 36087^36087-36080^36080 | 164462 | Henri Lifchitz | 08/2024 |
| 3549 | 32003^36504-36504^32003 | 164458 | Frank Doornink | 01/2024 |
| 3550 | (65*10^164427+277)/9 | 164428 | Robert Price | 08/2021 |
| 3551 | 2^545958+1876189 | 164350 | Enzo Doria | 07/2016 |
| 3552 | 2^545878-49841 | 164326 | Enzo Doria | 10/2015 |
| 3553 | (10^164284+77)/3 | 164284 | Robert Price | 01/2018 |
| 3554 | 2^545668-4019 | 164263 | Yves Minet | 04/2017 |
| 3555 | 10^164216+780639 | 164217 | Patrick De Geest | 08/2021 |
| 3557 | 10^164216-50000000001 | 164216 | Robert Price | 12/2020 |
| 3556 | 10^164216-891051 | 164216 | Patrick De Geest | 08/2021 |
| 3558 | (13*53^238539*617^9)/10^247183 | 164151 | Hans Havermann | 07/2017 |
| 3559 | (16*10^164122+197)/3 | 164123 | Robert Price | 04/2019 |
| 3560 | 2^545185+76757 | 164118 | Enzo Doria | 05/2016 |
| 3561 | 2^545076-114329 | 164085 | Enzo Doria | 01/2015 |
| 3562 | (2*10^164043-143)/3 | 164043 | Robert Price | 03/2018 |
| 3563 | (33085^36481-1)/(33085^191-1) | 164018 | Ryan Propper | 11/2023 |
| 3564 | (29^112153+1)/30 | 164011 | Lelio R Paula | 03/2017 |
| 3565 | 2^544757+1874777 | 163989 | Enzo Doria | 06/2016 |
| 3566 | 2^544723+211193 | 163978 | Enzo Doria | 08/2023 |
| 3567 | 2^544444-457199 | 163894 | Enzo Doria | 01/2015 |
| 3568 | (32804^36481-1)/(32804^191-1) | 163883 | Ryan Propper | 11/2023 |
| 3569 | 2^544264-343529 | 163840 | Enzo Doria | 07/2015 |
| 3570 | 7*10^163780+9 | 163781 | Robert Price | 09/2015 |
| 3571 | (2^783598)%L(783598) | 163761 | Henri Lifchitz | 05/2025 |
| 3572 | 2^543989+2793047 | 163758 | Enzo Doria | 02/2016 |
| 3573 | (28*10^163709+113)/3 | 163710 | Robert Price | 03/2020 |
| 3574 | (2^543398)/(543398^2) | 163568 | Henri Lifchitz | 05/2025 |
| 3575 | (2013^49729-1)/(2013^223-1) | 163561 | Ryan Propper | 11/2023 |
| 3576 | (23*10^163523+301)/9 | 163524 | Robert Price | 06/2021 |
| 3577 | (68*10^163521-329)/9 | 163522 | Robert Price | 03/2021 |
| 3578 | 35820^35899+35899^35820 | 163489 | Norbert Schneider | 11/2021 |
| 3579 | 2^542870-69551 | 163421 | Enzo Doria | 12/2015 |
| 3580 | 2^542502+24799 | 163310 | Enzo Doria | 10/2023 |
| 3581 | 2^542395+72533 | 163278 | Enzo Doria | 04/2016 |
| 3582 | 2^542258+216949 | 163236 | Enzo Doria | 12/2023 |
| 3583 | 2^542072-3683 | 163180 | Yves Minet | 03/2017 |
| 3584 | fibonacci(780566)-4 | 163129 | Jon Grantham | 08/2023 |
| 3585 | (682*10^163106-7)/9 | 163108 | Robert Price | 08/2021 |
| 3586 | 2^541779-1867357 | 163092 | Enzo Doria | 04/2016 |
| 3587 | (29*10^163056+313)/9 | 163057 | Robert Price | 04/2021 |
| 3588 | 2^541476+214743 | 163001 | Enzo Doria | 11/2023 |
| 3589 | 13^142159+14^142159-2 | 162933 | Harsh Aggarwal | 05/2024 |
| 3590 | 8*10^162924+9 | 162925 | Robert Price | 12/2015 |
| 3591 | 2^540905+81557 | 162829 | Enzo Doria | 08/2016 |
| 3592 | 2^540628-2939 | 162746 | Yves Minet | 09/2014 |
| 3593 | 919^54919-918^54919 | 162743 | Jean-Louis Charton | 01/2013 |
| 3594 | (203*10^162689+1)/3 | 162691 | Robert Price | 03/2020 |
| 3595 | 3*10^162620-59 | 162621 | Robert Price | 09/2018 |
| 3596 | 2^540180+26791 | 162611 | Enzo Doria | 12/2023 |
| 3597 | 3^340694-112960 | 162553 | Enzo Doria | 07/2014 |
| 3598 | 2^539925+81 | 162534 | Henri Lifchitz | 11/2024 |
| 3599 | 2^539807-114081 | 162499 | Enzo Doria | 12/2014 |
| 3600 | 414^62093+62093^414 | 162498 | Norbert Schneider | 03/2022 |
| 3601 | 2^539654-114893 | 162453 | Enzo Doria | 05/2015 |
| 3602 | 9*10^162449-71 | 162450 | Robert Price | 11/2020 |
| 3603 | 2^539617-1851583 | 162441 | Enzo Doria | 08/2015 |
| 3604 | (509*10^162422+31)/9 | 162424 | Robert Price | 08/2022 |
| 3605 | (91*10^162410+11)/3 | 162412 | Robert Price | 02/2020 |
| 3606 | 2^539346+81049 | 162360 | Enzo Doria | 07/2016 |
| 3607 | 2^539220-115499 | 162322 | Enzo Doria | 07/2015 |
| 3608 | 2^539217-113713 | 162321 | Enzo Doria | 09/2014 |
| 3609 | 9^170099-8^170099 | 162316 | Jean-Louis Charton | 05/2005 |
| 3610 | 4*10^162263-39 | 162264 | Robert Price | 09/2020 |
| 3611 | 31243^36098-36098^31243 | 162252 | Frank Doornink | 01/2024 |
| 3612 | 31987^36000-36000^31987 | 162180 | Frank Doornink | 01/2024 |
| 3613 | (115*10^162153-1)/3 | 162155 | Serge Batalov | 12/2014 |
| 3614 | (10^162138+11)/3 | 162138 | Serge Batalov | 07/2015 |
| 3615 | (61*10^162100-403)/9 | 162101 | Robert Price | 10/2021 |
| 3616 | 2^538473-112723 | 162097 | Enzo Doria | 06/2014 |
| 3617 | 2^538157+79337 | 162002 | Enzo Doria | 06/2016 |
| 3618 | 2^538136-114749 | 161996 | Enzo Doria | 05/2015 |
| 3619 | 2^538104+1531 | 161986 | Lelio R Paula | 01/2013 |
| 3620 | (9*29^110727-1199)/14 | 161927 | Curtis Bright | 05/2015 |
| 3621 | ((2^179294-1)^3+2)/11 | 161918 | Serge Batalov | 03/2023 |
| 3622 | 31198^36027-36027^31198 | 161910 | Frank Doornink | 01/2024 |
| 3623 | 2^537815-22737 | 161899 | Enzo Doria | 07/2013 |
| 3624 | 2^537804-39839 | 161896 | Enzo Doria | 09/2015 |
| 3625 | (754*10^161816-7)/9 | 161818 | Robert Price | 05/2020 |
| 3626 | 2^537461+29867 | 161792 | Enzo Doria | 03/2024 |
| 3627 | 2^537458-59201 | 161791 | Enzo Doria | 10/2015 |
| 3628 | 2*10^161766+51 | 161767 | Robert Price | 05/2018 |
| 3629 | (7*10^161760+131)/3 | 161761 | Robert Price | 06/2020 |
| 3630 | 2^537209-49849 | 161717 | Enzo Doria | 10/2015 |
| 3631 | (58*10^161691+329)/9 | 161692 | Robert Price | 05/2019 |
| 3632 | 23*10^161668-3 | 161670 | Robert Price | 10/2021 |
| 3633 | 2^536937+1879397 | 161635 | Enzo Doria | 09/2016 |
| 3634 | 3*2^536868-5 | 161614 | Jon Grantham | 07/2023 |
| 3635 | 34*10^161593-9 | 161595 | Robert Price | 06/2020 |
| 3636 | (31*10^161582-1)/3 | 161584 | Robert Price | 02/2020 |
| 3637 | 24494^36813+36813^24494 | 161575 | Norbert Schneider | 11/2023 |
| 3638 | 2^536569-22881 | 161524 | Enzo Doria | 07/2013 |
| 3639 | (46*10^161505+593)/9 | 161506 | Robert Price | 04/2022 |
| 3640 | 10^161253+573493 | 161254 | Patrick De Geest | 07/2018 |
| 3641 | 10^161253-8001 | 161253 | Robert Price | 03/2018 |
| 3642 | 97*10^161247-3 | 161249 | Robert Price | 05/2020 |
| 3643 | 7*10^161192-9 | 161193 | Robert Price | 10/2015 |
| 3644 | 2^535415-19 | 161176 | Stefano Morozzi | 07/2022 |
| 3645 | 2^535269-113803 | 161133 | Enzo Doria | 09/2014 |
| 3646 | 2^535205-23295 | 161113 | Enzo Doria | 08/2013 |
| 3647 | (49*10^161105+437)/9 | 161106 | Robert Price | 09/2021 |
| 3648 | 2^534961+6561 | 161040 | Henri Lifchitz | 11/2024 |
| 3649 | (89^82609+1)/90 | 161036 | Paul Bourdelais | 10/2017 |
| 3650 | 35385^35385-17692^17692 | 160960 | Henri Lifchitz | 10/2024 |
| 3651 | 2^534429-19 | 160880 | Stefano Morozzi | 07/2022 |
| 3652 | 2^534359-23229 | 160859 | Enzo Doria | 08/2013 |
| 3653 | 10^160836-1001 | 160836 | Robert Price | 10/2022 |
| 3654 | 457^60454+60454^457 | 160803 | Norbert Schneider | 07/2021 |
| 3655 | 8*10^160782+53 | 160783 | Robert Price | 08/2020 |
| 3656 | 67^88037-66^88037 | 160763 | Robert Price | 10/2013 |
| 3657 | (88*10^160761-637)/9 | 160762 | Robert Price | 11/2020 |
| 3658 | 2^533981+81 | 160745 | Henri Lifchitz | 11/2024 |
| 3659 | 2^533948-1852559 | 160735 | Enzo Doria | 08/2015 |
| 3660 | (2^769082)%F(769082) | 160728 | Henri Lifchitz | 05/2025 |
| 3661 | 2^533665-23335 | 160650 | Enzo Doria | 08/2013 |
| 3662 | 2^533600+217981 | 160630 | Enzo Doria | 01/2024 |
| 3663 | (19*10^160619+77)/3 | 160620 | Robert Price | 05/2019 |
| 3664 | 2^533350-5 | 160555 | Stefano Morozzi | 07/2022 |
| 3665 | (38*10^160508+151)/9 | 160509 | Robert Price | 03/2022 |
| 3666 | 3^336353-2^336353 | 160482 | Mike Oakes | 10/2007 |
| 3667 | 26134^36315-36315^26134 | 160411 | Frank Doornink | 03/2024 |
| 3668 | 25821^36358-36358^25821 | 160411 | Frank Doornink | 03/2024 |
| 3669 | 2*10^160339-63 | 160340 | Robert Price | 01/2022 |
| 3670 | 24419^36532+36532^24419 | 160293 | Norbert Schneider | 10/2023 |
| 3671 | (91*10^160249+11)/3 | 160251 | Robert Price | 02/2020 |
| 3672 | (772*10^160239-7)/9 | 160241 | Robert Price | 06/2022 |
| 3673 | (2*10^160226+187)/9 | 160226 | Robert Price | 06/2022 |
| 3674 | 2^532253+1868417 | 160225 | Enzo Doria | 04/2016 |
| 3675 | 2^532108-291989 | 160181 | Enzo Doria | 07/2015 |
| 3676 | (28*10^160171-79)/3 | 160172 | Robert Price | 02/2021 |
| 3677 | (26*10^160166-131)/3 | 160167 | Robert Price | 10/2019 |
| 3678 | 55*10^160145+9 | 160147 | Robert Price | 03/2022 |
| 3679 | (17*10^160143+67)/3 | 160144 | Robert Price | 01/2019 |
| 3680 | (394^61637-1)/393 | 159977 | Jinyuan Wang | 04/2023 |
| 3681 | (19*10^159921-49)/3 | 159922 | Robert Price | 05/2019 |
| 3682 | 2^531110-1853771 | 159881 | Enzo Doria | 09/2015 |
| 3683 | (275^65536+53^65536)/2 | 159864 | Kellen Shenton | 11/2020 |
| 3684 | 2^530992-1845299 | 159845 | Enzo Doria | 07/2015 |
| 3685 | 2^530810+25 | 159790 | Stefano Morozzi | 04/2021 |
| 3686 | 25590^36241-36241^25590 | 159753 | Frank Doornink | 02/2024 |
| 3687 | (136*10^159736-1)/9 | 159738 | Serge Batalov | 12/2014 |
| 3688 | 2^530417-113833 | 159672 | Enzo Doria | 09/2014 |
| 3689 | 2^530406-69701 | 159669 | Enzo Doria | 02/2016 |
| 3690 | 5*10^159656+41 | 159657 | Robert Price | 03/2019 |
| 3691 | 8*10^159614+29 | 159615 | Robert Price | 11/2021 |
| 3692 | 7^188858-188858^7 | 159604 | Norbert Schneider | 05/2018 |
| 3693 | 28^110278+27 | 159590 | Alfred Reich | 02/2024 |
| 3694 | (45^96517-2^96517)/43 | 159562 | Robert Price | 11/2024 |
| 3695 | 7*10^159553-27 | 159554 | Robert Price | 11/2022 |
| 3696 | 3^334409-5*2^334409 | 159554 | Jon Grantham | 08/2023 |
| 3697 | (24^115597-1)/23 | 159547 | Paul Bourdelais | 09/2017 |
| 3698 | 25472^36209-36209^25472 | 159540 | Frank Doornink | 01/2024 |
| 3699 | (17*10^159438-377)/9 | 159439 | Robert Price | 03/2018 |
| 3700 | (49^94321+2^94321)/51 | 159420 | Robert Price | 06/2025 |
| 3701 | (203*10^159381+1)/3 | 159383 | Robert Price | 03/2020 |
| 3702 | 32282^35343-35343^32282 | 159361 | Frank Doornink | 02/2024 |
| 3703 | 2^529215+219383 | 159310 | Enzo Doria | 02/2024 |
| 3704 | 89*10^159258-9 | 159260 | Robert Price | 05/2020 |
| 3705 | 2^528944-461315 | 159229 | Enzo Doria | 07/2015 |
| 3706 | (13*10^159147+161)/3 | 159148 | Robert Price | 11/2018 |
| 3707 | 2^528620-113879 | 159131 | Enzo Doria | 10/2014 |
| 3708 | 32125^35306-35306^32125 | 159119 | Frank Doornink | 02/2024 |
| 3709 | (47*10^159083+259)/9 | 159084 | Robert Price | 03/2022 |
| 3710 | (861^54181-1)/860 | 159019 | Jinyuan Wang | 11/2023 |
| 3711 | 2^528000-23307 | 158944 | Enzo Doria | 08/2013 |
| 3712 | (29*10^158909+61)/9 | 158910 | Robert Price | 08/2018 |
| 3713 | 2^527817+1876817 | 158889 | Enzo Doria | 07/2016 |
| 3714 | 2^527774-49451 | 158876 | Enzo Doria | 10/2015 |
| 3715 | 2^527745-291553 | 158868 | Enzo Doria | 07/2015 |
| 3716 | (91*10^158862+11)/3 | 158864 | Robert Price | 02/2020 |
| 3717 | 2^527644-1856999 | 158837 | Enzo Doria | 10/2015 |
| 3718 | (37200^37199-37198^37199)/(2^37199) | 158822 | Henri Lifchitz | 06/2013 |
| 3719 | (17*10^158819+7)/3 | 158820 | Robert Price | 06/2023 |
| 3720 | 2^527573-1863253 | 158816 | Enzo Doria | 12/2015 |
| 3721 | 2^527553-23401 | 158810 | Enzo Doria | 08/2013 |
| 3722 | (49*10^158783+437)/9 | 158784 | Robert Price | 09/2021 |
| 3723 | 10^158755+310207 | 158756 | Patrick De Geest | 08/2021 |
| 3724 | 10^158755-50001 | 158755 | Robert Price | 08/2020 |
| 3725 | (61*10^158748-619)/9 | 158749 | Robert Price | 03/2022 |
| 3726 | 11^152420-210 | 158730 | Enzo Doria | 09/2012 |
| 3727 | (13*10^158694+401)/9 | 158695 | Robert Price | 01/2018 |
| 3728 | 2^527052-69059 | 158659 | Enzo Doria | 11/2015 |
| 3729 | 2^526849-113233 | 158598 | Enzo Doria | 07/2014 |
| 3730 | 24286^36161+36161^24286 | 158579 | Norbert Schneider | 08/2023 |
| 3731 | (2^526763-lucasV(1,2,526763)+1)/2 | 158572 | Mike Oakes | 11/2016 |
| 3732 | (299*10^158544-17)/3 | 158546 | Robert Price | 04/2020 |
| 3733 | 2^526658-3551 | 158540 | Yves Minet | 03/2017 |
| 3734 | 7*10^158465-11 | 158466 | Robert Price | 06/2019 |
| 3735 | (7^187507-3^187507)/4 | 158462 | Jon Grantham | 08/2023 |
| 3736 | 2^526375-23115 | 158455 | Enzo Doria | 08/2013 |
| 3737 | (77*10^158407-293)/9 | 158408 | Robert Price | 09/2019 |
| 3738 | 31209^35242-35242^31209 | 158388 | Frank Doornink | 01/2024 |
| 3739 | 32071^35144-35144^32071 | 158363 | Frank Doornink | 01/2024 |
| 3740 | 2^526028-112793 | 158351 | Enzo Doria | 07/2014 |
| 3741 | (229*10^158339-1)/3 | 158341 | Robert Price | 07/2020 |
| 3742 | 32042^35123-35123^32042 | 158255 | Frank Doornink | 01/2024 |
| 3743 | 2^525690-9221 | 158249 | Lelio R Paula | 01/2013 |
| 3744 | 2*10^158242-7 | 158243 | Robert Price | 10/2015 |
| 3745 | 2^525667-525667^2 | 158242 | Henri Lifchitz | 12/2017 |
| 3746 | (56*10^158230+187)/9 | 158231 | Robert Price | 10/2021 |
| 3747 | 10^158209-7000000001 | 158209 | Robert Price | 12/2021 |
| 3748 | 31050^35207-35207^31050 | 158153 | Frank Doornink | 01/2024 |
| 3749 | 2^525186-1857431 | 158097 | Enzo Doria | 10/2015 |
| 3750 | 31527^35108-35108^31527 | 157940 | Frank Doornink | 01/2024 |
