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 ? |
4501 | (7*2^388372-1)/111 | 116911 | Lelio R Paula | 04/2014 |
4502 | (7*2^388335-1)/25 | 116900 | Lelio R Paula | 04/2014 |
4503 | 2^388180+93 | 116854 | Donovan Johnson | 07/2008 |
4504 | (2^388167+2510)/2 | 116850 | Stefano Morozzi | 09/2020 |
4505 | (85*10^116842+293)/9 | 116843 | Robert Price | 12/2019 |
4506 | 35296^2043+2043^35296 | 116840 | Hans Havermann | 02/2021 |
4507 | (2^388081+7)/1179 | 116821 | Lelio R Paula | 03/2014 |
4508 | 38633^1056+1056^38633 | 116814 | Hans Havermann | 02/2021 |
4509 | 2^387946-113261 | 116784 | Enzo Doria | 07/2014 |
4510 | 28*36^75009+305 | 116739 | Eric Chen | 12/2022 |
4511 | 28363^13056+13056^28363 | 116737 | Hans Havermann | 02/2021 |
4512 | 2^387745+75557 | 116723 | Enzo Doria | 04/2016 |
4513 | (29*10^116719+313)/9 | 116720 | Robert Price | 04/2021 |
4514 | 29622^8699+8699^29622 | 116695 | Hans Havermann | 02/2021 |
4515 | (206*10^116687-17)/9 | 116689 | Robert Price | 01/2020 |
4516 | (7*10^116684+179)/3 | 116685 | Robert Price | 09/2018 |
4517 | 2^387554+71455 | 116666 | Enzo Doria | 04/2016 |
4518 | (602*10^116647-71)/9 | 116649 | Robert Price | 12/2021 |
4519 | 3^244461-10 | 116638 | Tyler NeSmith | 09/2022 |
4520 | (2^387447+78557)/365 | 116631 | Lelio R Paula | 12/2014 |
4521 | (2*10^116620+43)/9 | 116620 | Serge Batalov | 11/2022 |
4522 | (19*10^116617+287)/9/12893 | 116614 | Alfred Reich | 11/2022 |
4523 | (145*10^116610-1)/9 | 116612 | Serge Batalov | 12/2014 |
4524 | (25*10^116601-331)/9 | 116602 | Robert Price | 08/2018 |
4525 | (17*2^387284-1)/59 | 116584 | Lelio R Paula | 04/2014 |
4526 | (2^387284+3726)/2 | 116584 | Stefano Morozzi | 09/2020 |
4527 | 2^387211+111546435 | 116563 | Paul Bourdelais | 10/2021 |
4528 | 10^116557+987654321 | 116558 | Robert Price | 12/2019 |
4529 | 10^116557+9141 | 116558 | Patrick De Geest | 02/2020 |
4530 | 10^116557-609431 | 116557 | Patrick De Geest | 02/2020 |
4531 | (28*10^116546+41)/3 | 116547 | Robert Price | 07/2021 |
4532 | (2^387123+1814)/2 | 116536 | Stefano Morozzi | 02/2021 |
4533 | (2^387076+2046)/2 | 116522 | Stefano Morozzi | 09/2020 |
4534 | (2^387073+2570)/2 | 116521 | Stefano Morozzi | 09/2020 |
4535 | (7*10^116514-1)/3 | 116515 | Robert Price | 08/2022 |
4536 | 28846^10931+10931^28846 | 116500 | Hans Havermann | 02/2021 |
4537 | 10^116496+3593037 | 116497 | Paul Bourdelais | 10/2021 |
4538 | (31*2^386968+1)/7 | 116490 | Lelio R Paula | 03/2014 |
4539 | 2^386947+1869143 | 116483 | Enzo Doria | 05/2016 |
4540 | (30^78857-1)/29 | 116480 | Robert Price | 12/2011 |
4541 | (2^386903-3)/5 | 116469 | Lelio R Paula | 03/2014 |
4542 | 28107^13892+13892^28107 | 116441 | Hans Havermann | 02/2021 |
4543 | 29110^9957+9957^29110 | 116386 | Hans Havermann | 02/2021 |
4544 | 26469^24838+24838^26469 | 116335 | Hans Havermann | 01/2021 |
4545 | 2^386389-1864783 | 116315 | Enzo Doria | 03/2016 |
4546 | (31*2^386231-1)/1711 | 116266 | Lelio R Paula | 03/2014 |
4547 | 297^47012+47012^297 | 116250 | Norbert Schneider | 07/2018 |
4548 | (35*10^116223-341)/9 | 116224 | Robert Price | 08/2021 |
4549 | (2^386087-5)/40959 | 116220 | Lelio R Paula | 03/2014 |
4550 | (2^386054+714)/2 | 116214 | Stefano Morozzi | 09/2020 |
4551 | (10^116201+7)/39623 | 116197 | Alfred Reich | 06/2021 |
4552 | 2^385969-823 | 116189 | Yves Minet | 05/2012 |
4553 | 57^66162+2 | 116173 | Lelio R Paula | 08/2013 |
4554 | 2^385854+75049 | 116154 | Enzo Doria | 04/2016 |
4555 | (2^385832+2590)/2 | 116147 | Stefano Morozzi | 09/2020 |
4556 | 2^385775-109587 | 116130 | Enzo Doria | 04/2014 |
4557 | (14*10^116112-101)/3 | 116113 | Robert Price | 02/2019 |
4558 | Phi(236354,10)/24817171 | 116109 | Makoto Kamada | 08/2019 |
4559 | 3*10^116107+23 | 116108 | Robert Price | 03/2022 |
4560 | 28729^10990+10990^28729 | 116094 | Hans Havermann | 02/2021 |
4561 | 2^385622+71839 | 116084 | Enzo Doria | 04/2016 |
4562 | 2^385564+781 | 116067 | Lelio R Paula | 01/2012 |
4563 | (2^385565+1562)/2 | 116067 | Stefano Morozzi | 09/2020 |
4564 | (2^385563+378)/2 | 116066 | Stefano Morozzi | 02/2021 |
4565 | (2^385557+2214)/2 | 116064 | Stefano Morozzi | 09/2020 |
4566 | 29431^8776+8776^29431 | 116056 | Hans Havermann | 01/2021 |
4567 | (266*10^116024-17)/3 | 116026 | Robert Price | 07/2020 |
4568 | 30263^6820+6820^30263 | 116022 | Hans Havermann | 02/2021 |
4569 | 2^385410+71893 | 116020 | Enzo Doria | 04/2016 |
4570 | 38225^1084+1084^38225 | 116014 | Hans Havermann | 02/2021 |
4571 | 26548^23433+23433^26548 | 116011 | Hans Havermann | 02/2021 |
4572 | (25*2^385091+1)/27 | 115924 | Lelio R Paula | 04/2014 |
4573 | (11*10^115910-131)/3 | 115911 | Robert Price | 09/2018 |
4574 | 2^384953-111943 | 115883 | Paul Bourdelais | 02/2012 |
4575 | 43^70942-2 | 115882 | Lelio R Paula | 04/2014 |
4576 | (94*10^115879-7)/3 | 115881 | Robert Price | 03/2020 |
4577 | 7*10^115863+3 | 115864 | Robert Price | 08/2015 |
4578 | 2^384834+93 | 115847 | Donovan Johnson | 07/2008 |
4579 | 2^384809+71297 | 115840 | Enzo Doria | 04/2016 |
4580 | (55*10^115829+323)/9 | 115830 | Robert Price | 10/2021 |
4581 | (7*2^384759-1)/5 | 115825 | Lelio R Paula | 04/2014 |
4582 | 31348^4951+4951^31348 | 115822 | Hans Havermann | 02/2021 |
4583 | (793*10^115810+53)/9 | 115812 | Robert Price | 01/2022 |
4584 | (71*10^115810-17)/9 | 115811 | Ray Chandler | 08/2011 |
4585 | (97*10^115794-1)/3 | 115796 | Serge Batalov | 12/2014 |
4586 | 31115^5266+5266^31115 | 115794 | Hans Havermann | 02/2021 |
4587 | 2*10^115784-27 | 115785 | Robert Price | 05/2018 |
4588 | 10^115773+126739 | 115774 | Patrick De Geest | 04/2018 |
4590 | 10^115773-1000000001 | 115773 | Robert Price | 03/2018 |
4589 | 10^115773-396221 | 115773 | Patrick De Geest | 04/2018 |
4591 | (2^384582+778)/2 | 115771 | Stefano Morozzi | 09/2020 |
4592 | 30537^6178+6178^30537 | 115762 | Hans Havermann | 01/2021 |
4593 | 2^384530+1874305 | 115756 | Enzo Doria | 06/2016 |
4594 | (3*2^384524-1)/7631 | 115750 | Lelio R Paula | 03/2014 |
4595 | 28275^12406+12406^28275 | 115748 | Hans Havermann | 01/2021 |
4596 | (44*10^115697-413)/9 | 115698 | Robert Price | 03/2019 |
4597 | 31408^4825+4825^31408 | 115692 | Hans Havermann | 02/2021 |
4598 | (2^384317+1026)/2 | 115691 | Stefano Morozzi | 09/2020 |
4599 | 2^384312+81091 | 115690 | Yves Minet | 02/2010 |
4600 | (2^384265+2150)/2 | 115675 | Stefano Morozzi | 09/2020 |
4601 | (2^384246+2182)/2 | 115670 | Stefano Morozzi | 09/2020 |
4602 | (4*10^115668-79)/3 | 115669 | Robert Price | 02/2018 |
4603 | (2^384193+242)/2 | 115654 | Stefano Morozzi | 09/2020 |
4604 | (266947#+1267)/(7^2*181) | 115647 | Makoto Morimoto | 06/2019 |
4605 | (266947#+273808)/(2*109*157) | 115646 | Makoto Morimoto | 10/2017 |
4606 | 18^92125-92125^18 | 115642 | Dylan Delgado | 08/2019 |
4607 | 3^242337-4042 | 115625 | Enzo Doria | 05/2014 |
4608 | (2^383931+674)/2 | 115575 | Stefano Morozzi | 09/2020 |
4609 | 7*10^115561+79 | 115562 | Robert Price | 07/2019 |
4610 | (2^383872+3046)/2 | 115557 | Stefano Morozzi | 09/2020 |
4611 | (11*2^383859-13)/15 | 115553 | Renaud Lifchitz | 03/2011 |
4612 | 2^383780-3149 | 115530 | Yves Minet | 10/2014 |
4613 | (41*10^115523-383)/9 | 115524 | Robert Price | 11/2018 |
4614 | 27316^16923+16923^27316 | 115506 | Hans Havermann | 02/2021 |
4615 | (2^383618+618)/2 | 115481 | Stefano Morozzi | 09/2020 |
4616 | 2^383560-455 | 115464 | Yves Minet | 03/2012 |
4617 | (2^383544+5)/3 | 115458 | Lelio R Paula | 03/2014 |
4618 | (22*10^115454+119)/3 | 115455 | Robert Price | 04/2021 |
4619 | 89^59221-88^59221 | 115445 | Robert Price | 06/2015 |
4620 | 2^383347+1869413 | 115399 | Enzo Doria | 05/2016 |
4621 | (103*10^115366+17)/3 | 115368 | Robert Price | 08/2020 |
4622 | (2^383232+750)/2 | 115365 | Stefano Morozzi | 09/2020 |
4623 | (2^383130-7)/131793 | 115329 | Lelio R Paula | 03/2014 |
4624 | (12^106859+1)/13 | 115320 | Paul Bourdelais | 02/2010 |
4625 | 29961^7060+7060^29961 | 115315 | Hans Havermann | 01/2021 |
4627 | 2^383039+1869509 | 115307 | Enzo Doria | 05/2016 |
4626 | 32918^3183+3183^32918 | 115307 | Hans Havermann | 01/2021 |
4628 | 36964^1315+1315^36964 | 115288 | Hans Havermann | 02/2021 |
4629 | (13*2^382968+1)/7 | 115286 | Lelio R Paula | 04/2014 |
4630 | (22*10^115282+29)/3 | 115283 | Robert Price | 02/2021 |
4631 | (2^382945+2046)/2 | 115278 | Stefano Morozzi | 09/2020 |
4632 | (2^382789+3182)/2 | 115231 | Stefano Morozzi | 09/2020 |
4633 | 30416^6141+6141^30416 | 115224 | Hans Havermann | 02/2021 |
4634 | 6*10^115211+79 | 115212 | Robert Price | 09/2020 |
4635 | (2^382722+3598)/2 | 115211 | Stefano Morozzi | 09/2020 |
4636 | 27987^13078+13078^27987 | 115210 | Hans Havermann | 02/2021 |
4637 | (2^382699+2090)/2 | 115204 | Stefano Morozzi | 09/2020 |
4638 | (22*10^115188-193)/9 | 115189 | Robert Price | 07/2022 |
4639 | 2^382633+73697 | 115185 | Enzo Doria | 04/2016 |
4640 | 2^382509-113593 | 115147 | Enzo Doria | 08/2014 |
4641 | 2^382502-2791721 | 115145 | Enzo Doria | 03/2016 |
4642 | (2^382495+1274)/2 | 115143 | Stefano Morozzi | 01/2021 |
4643 | 26752^20065+20065^26752 | 115099 | Hans Havermann | 02/2021 |
4644 | 26383^23008+23008^26383 | 115080 | Hans Havermann | 02/2021 |
4645 | 9^120590-120590 | 115073 | Robert Price | 03/2014 |
4646 | 28203^12016+12016^28203 | 115062 | Hans Havermann | 02/2021 |
4647 | 2^382198+9 | 115054 | Lelio R Paula | 03/2013 |
4648 | 30101^6630+6630^30101 | 115032 | Hans Havermann | 01/2021 |
4649 | (89*10^115030-179)/9 | 115031 | Robert Price | 12/2019 |
4650 | (2^381935+2774)/2 | 114974 | Stefano Morozzi | 02/2021 |
4651 | 27300^16271+16271^27300 | 114972 | Hans Havermann | 02/2021 |
4652 | 2^381883-59197 | 114959 | Enzo Doria | 10/2015 |
4653 | 27423^15542+15542^27423 | 114944 | Hans Havermann | 02/2021 |
4654 | (2^381820+1626)/2 | 114939 | Stefano Morozzi | 09/2020 |
4656 | 2^381806-4481 | 114936 | Yves Minet | 07/2011 |
4655 | 2^381808-1851507 | 114936 | Enzo Doria | 08/2015 |
4657 | (2^381792+2502)/2 | 114931 | Stefano Morozzi | 09/2020 |
4658 | 2^381774-112373 | 114926 | Enzo Doria | 06/2014 |
4659 | 26341^22980+22980^26341 | 114883 | Hans Havermann | 01/2021 |
4660 | 2^381555-109051 | 114860 | Enzo Doria | 04/2014 |
4661 | 2^381546-113501 | 114857 | Enzo Doria | 08/2014 |
4662 | (7*10^114845-19)/3 | 114846 | Robert Price | 10/2020 |
4663 | 26675^20178+20178^26675 | 114833 | Hans Havermann | 02/2021 |
4664 | (2^381436+1182)/2 | 114824 | Stefano Morozzi | 09/2020 |
4665 | 3^240644+16 | 114817 | Robert Price | 04/2020 |
4666 | 2^381360-110429 | 114801 | Enzo Doria | 05/2014 |
4667 | (17*10^114795+79)/3 | 114796 | Robert Price | 04/2019 |
4668 | (66^63079+1)/67 | 114774 | Paul Bourdelais | 09/2017 |
4669 | 2^381252-115079 | 114769 | Enzo Doria | 06/2015 |
4670 | (2^381251+3186)/2 | 114768 | Stefano Morozzi | 09/2020 |
4671 | 2^381209-29323 | 114756 | Enzo Doria | 07/2015 |
4672 | 2^381207-111546435 | 114755 | Paul Bourdelais | 10/2021 |
4673 | 569^41651-568^41651 | 114754 | Jean-Louis Charton | 11/2009 |
4674 | 2^381201+1873205 | 114753 | Enzo Doria | 06/2016 |
4675 | (85*10^114737+293)/9 | 114738 | Robert Price | 12/2019 |
4676 | (265*10^114736+17)/3 | 114738 | Robert Price | 03/2020 |
4677 | (2^381062+2170)/2 | 114711 | Stefano Morozzi | 09/2020 |
4678 | 2^380974-112991 | 114685 | Enzo Doria | 07/2014 |
4679 | 37097^1234+1234^37097 | 114679 | Hans Havermann | 02/2021 |
4680 | 88^58949-87 | 114626 | Norbert Schneider | 11/2015 |
4681 | (2^380773+662)/2 | 114624 | Stefano Morozzi | 09/2020 |
4682 | 29913^6782+6782^29913 | 114608 | Hans Havermann | 02/2021 |
4683 | (331*10^114605-7)/9 | 114607 | Robert Price | 04/2022 |
4684 | 2^380700+91 | 114603 | Lelio R Paula | 03/2013 |
4685 | (2^380632+2578)/2 | 114582 | Stefano Morozzi | 09/2020 |
4686 | 7*10^114577+27 | 114578 | Robert Price | 11/2020 |
4688 | 10^114575+3^114576 | 114576 | Mike Oakes | 09/2010 |
4687 | 10^114575+66561 | 114576 | Patrick De Geest | 02/2017 |
4689 | 10^114575-581081 | 114575 | Patrick De Geest | 02/2017 |
4690 | 2^380587-115315 | 114569 | Enzo Doria | 06/2015 |
4691 | 31874^3927+3927^31874 | 114558 | Hans Havermann | 02/2021 |
4692 | 2^380513-216925 | 114546 | Enzo Doria | 05/2014 |
4693 | 2^380451-1840087 | 114528 | Enzo Doria | 06/2015 |
4694 | (2^380426+1510)/2 | 114520 | Stefano Morozzi | 09/2020 |
4695 | (15*2^380388-1)/539 | 114507 | Lelio R Paula | 04/2014 |
4696 | 2^380279-461347 | 114476 | Enzo Doria | 07/2015 |
4697 | (331*10^114454-7)/9 | 114456 | Robert Price | 04/2022 |
4698 | 27121^16590+16590^27121 | 114447 | Hans Havermann | 01/2021 |
4699 | 25966^25555+25555^25966 | 114445 | Hans Havermann | 01/2021 |
4700 | (11*10^114400+529)/9 | 114401 | Robert Price | 09/2021 |
4701 | 2^380021+1871147 | 114398 | Enzo Doria | 05/2016 |
4702 | 2^380005+1869707 | 114393 | Enzo Doria | 05/2016 |
4703 | 2^379958-113297 | 114379 | Enzo Doria | 07/2014 |
4704 | (2^379897+1274)/2 | 114361 | Stefano Morozzi | 01/2021 |
4705 | 2^379881-109105 | 114356 | Enzo Doria | 04/2014 |
4706 | (21*2^379857-1)/13 | 114349 | Lelio R Paula | 03/2014 |
4707 | (4*10^114340+83)/3 | 114341 | Robert Price | 02/2018 |
4708 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
4709 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
4710 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
4711 | 27764^13065+13065^27764 | 114280 | Hans Havermann | 01/2021 |
4712 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
4713 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
4714 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
4715 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
4716 | (22*10^114178+47)/3 | 114179 | Robert Price | 06/2021 |
4717 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
4718 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
4719 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
4720 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
4721 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
4722 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
4723 | 13643^27572-27572^13643 | 114008 | Norbert Schneider | 12/2022 |
4725 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
4724 | (83*10^113969+133)/9 | 113970 | Robert Price | 08/2021 |
4726 | 32226^3439+3439^32226 | 113966 | Hans Havermann | 02/2021 |
4727 | (77*10^113963-149)/9 | 113964 | Robert Price | 03/2022 |
4728 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
4729 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
4730 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
4733 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
4731 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
4732 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
4734 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
4735 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
4736 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
4738 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
4737 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
4739 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
4740 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
4741 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
4742 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
4743 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
4744 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
4745 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
4746 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
4747 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
4748 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
4749 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
4750 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |