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 | (55*10^106222+377)/9 | 106223 | Robert Price | 02/2019 |
4502 | 2^352814-131 | 106208 | Lelio R Paula | 10/2011 |
4503 | 2^352752+73951 | 106189 | Enzo Doria | 04/2016 |
4504 | 6^136434+5 | 106167 | Lelio R Paula | 11/2014 |
4505 | 25296^15737+15737^25296 | 106166 | Hans Havermann | 09/2020 |
4506 | 8*10^106157-13 | 106158 | Robert Price | 08/2019 |
4507 | 7*5^151872-8 | 106155 | Lelio R Paula | 11/2014 |
4508 | 105^52520-2 | 106153 | Lelio R Paula | 10/2017 |
4509 | 2^352601-1604809 | 106144 | Enzo Doria | 05/2015 |
4510 | 12^98353-43 | 106141 | Lelio R Paula | 02/2017 |
4511 | 2^352585-18901 | 106139 | Enzo Doria | 05/2013 |
4512 | 26363^10614+10614^26363 | 106135 | Hans Havermann | 09/2020 |
4513 | (2^352554-3)/193 | 106128 | Lelio R Paula | 12/2013 |
4514 | 2^352540+15015 | 106126 | Lelio R Paula | 01/2017 |
4515 | 25068^17119+17119^25068 | 106125 | Hans Havermann | 09/2020 |
4516 | 2^352524-3149 | 106121 | Yves Minet | 10/2014 |
4517 | 2^352486-1858781 | 106109 | Enzo Doria | 04/2016 |
4518 | 2^352444+12705 | 106097 | Lelio R Paula | 01/2017 |
4519 | 28561^5182+5182^28561 | 106090 | Hans Havermann | 10/2020 |
4520 | (182*10^106083+43)/9 | 106085 | Robert Price | 02/2020 |
4521 | (266*10^106066+1)/3 | 106068 | Robert Price | 04/2020 |
4522 | 2^352228-18953 | 106032 | Enzo Doria | 05/2013 |
4523 | (2^352223+1118)/2 | 106030 | Stefano Morozzi | 09/2020 |
4524 | (2^352212+2250)/2 | 106027 | Stefano Morozzi | 01/2021 |
4525 | (2^352136+2638)/2 | 106004 | Stefano Morozzi | 09/2020 |
4526 | 2^352107+1872533 | 105995 | Enzo Doria | 05/2016 |
4527 | (2^352096+2086)/2 | 105992 | Stefano Morozzi | 09/2020 |
4528 | 2^352066-2591 | 105983 | Yves Minet | 07/2014 |
4529 | 2^352030+715 | 105972 | Lelio R Paula | 10/2011 |
4530 | 9*6^136176-5 | 105967 | Lelio R Paula | 11/2014 |
4531 | 17099*2^351991+58329 | 105965 | Paul Vanderveen | 12/2017 |
4532 | 28582^5097+5097^28582 | 105963 | Hans Havermann | 09/2020 |
4533 | 155^48377+2 | 105962 | Lelio R Paula | 08/2013 |
4534 | 2^351865+1869707 | 105922 | Enzo Doria | 05/2016 |
4535 | 24974^17427+17427^24974 | 105921 | Hans Havermann | 09/2020 |
4536 | 25163^16194+16194^25163 | 105920 | Hans Havermann | 09/2020 |
4537 | 2^351831+23 | 105912 | Lelio R Paula | 10/2011 |
4538 | (2^351818+3798)/2 | 105908 | Stefano Morozzi | 02/2021 |
4539 | 9*2^351784-5 | 105899 | Lelio R Paula | 09/2014 |
4540 | (2^351776+2766)/2 | 105895 | Stefano Morozzi | 01/2021 |
4541 | (85*10^105881+203)/9 | 105882 | Robert Price | 09/2020 |
4542 | 2^351671-112387 | 105864 | Enzo Doria | 06/2014 |
4543 | 2^351669-109501 | 105863 | Enzo Doria | 04/2014 |
4544 | 2^351644+74731 | 105856 | Enzo Doria | 04/2016 |
4545 | 13^95025-42 | 105853 | Lelio R Paula | 12/2016 |
4546 | 2^351599+15015 | 105842 | Lelio R Paula | 01/2017 |
4547 | 19^82763+90 | 105834 | Lelio R Paula | 01/2017 |
4548 | 26050^11551+11551^26050 | 105832 | Hans Havermann | 09/2020 |
4549 | (9179*2^351540-1)/221 | 105826 | Lelio R Paula | 05/2015 |
4550 | 20^81329-51 | 105812 | Lelio R Paula | 01/2017 |
4551 | (5*10^105811+211)/9 | 105811 | Robert Price | 02/2018 |
4552 | 2^351483+28425 | 105807 | Lelio R Paula | 01/2017 |
4553 | 25231^15610+15610^25231 | 105804 | Hans Havermann | 09/2020 |
4554 | 15^89960+86 | 105802 | Lelio R Paula | 02/2017 |
4555 | 2^351460-17289 | 105801 | Enzo Doria | 05/2013 |
4556 | 19^82735+22 | 105798 | Lelio R Paula | 01/2017 |
4557 | 2^351435-109077 | 105793 | Enzo Doria | 04/2014 |
4558 | 2^351393+1089 | 105780 | Lelio R Paula | 01/2017 |
4559 | 2^351364+72631 | 105772 | Enzo Doria | 04/2016 |
4560 | (2^351333+3626)/2 | 105762 | Stefano Morozzi | 09/2020 |
4561 | 26818^8783+8783^26818 | 105761 | Hans Havermann | 09/2020 |
4564 | 2^351305+1872467 | 105754 | Enzo Doria | 05/2016 |
4563 | 3^221649+16 | 105754 | Lelio R Paula | 11/2016 |
4562 | 32511^1790+1790^32511 | 105754 | Hans Havermann | 09/2020 |
4565 | 8*3^221637-5 | 105749 | Lelio R Paula | 09/2014 |
4566 | 2^351279-2780449 | 105746 | Enzo Doria | 03/2016 |
4567 | (2^351263+3198)/2 | 105741 | Stefano Morozzi | 09/2020 |
4568 | 15^89903-26 | 105735 | Lelio R Paula | 02/2017 |
4570 | 2^351213-73 | 105726 | Lelio R Paula | 10/2011 |
4569 | (2^351214+1210)/2 | 105726 | Stefano Morozzi | 09/2020 |
4571 | 2^351192-4589 | 105720 | Yves Minet | 07/2011 |
4572 | 20^81250-33 | 105709 | Lelio R Paula | 12/2016 |
4573 | (2^351133+3530)/2 | 105702 | Stefano Morozzi | 09/2020 |
4574 | (2^351103+534)/2 | 105693 | Stefano Morozzi | 09/2020 |
4575 | 2^351076-113159 | 105685 | Enzo Doria | 07/2014 |
4576 | 93^53688-2 | 105684 | Lelio R Paula | 09/2013 |
4577 | 2^351060-4559 | 105680 | Yves Minet | 07/2011 |
4578 | ((10^52836-1)*10000/9+5927)*10^52836+(10^52836-1)/9 | 105676 | Hans Havermann | 08/2019 |
4579 | 11^101473-750 | 105674 | Enzo Doria | 11/2012 |
4580 | 3*5^151181-8 | 105672 | Lelio R Paula | 11/2014 |
4581 | 15^89831-34 | 105650 | Lelio R Paula | 02/2017 |
4583 | 11^101448-360 | 105648 | Enzo Doria | 08/2012 |
4582 | 3*7^125011+4 | 105648 | Lelio R Paula | 11/2014 |
4584 | 2^350943-113269 | 105645 | Enzo Doria | 07/2014 |
4585 | 2*10^105636-91 | 105637 | Robert Price | 04/2020 |
4586 | 24231^22880+22880^24231 | 105634 | Hans Havermann | 09/2020 |
4587 | 35296^983+983^35296 | 105626 | Hans Havermann | 09/2020 |
4588 | 2^350868-19725 | 105622 | Enzo Doria | 06/2013 |
4589 | 2^350858-59291 | 105619 | Enzo Doria | 10/2015 |
4590 | 2^350851+135 | 105617 | Lelio R Paula | 10/2011 |
4591 | 2^350841-741 | 105614 | Lelio R Paula | 01/2017 |
4592 | 3*10^105582+77 | 105583 | Robert Price | 09/2018 |
4593 | 24915^17284+17284^24915 | 105581 | Hans Havermann | 09/2020 |
4594 | (5*10^105576-53)/3 | 105577 | Robert Price | 09/2020 |
4595 | 8*10^105571+3 | 105572 | Robert Price | 08/2015 |
4596 | 2^350674-78557 | 105564 | Jorge Menendez | 03/2018 |
4597 | 2*5^151026-3 | 105563 | Lelio R Paula | 11/2014 |
4598 | (2^350702+5)/34264674063 | 105562 | Lelio R Paula | 12/2013 |
4599 | 6^135647-5^135647 | 105554 | Jean-Louis Charton | 02/2009 |
4600 | 3^221211+236 | 105545 | Lelio R Paula | 11/2016 |
4601 | (32*10^105530-77)/9 | 105531 | Robert Price | 06/2018 |
4603 | 10^105515+33 | 105516 | Lelio R Paula | 11/2016 |
4602 | 24644^19125+19125^24644 | 105516 | Hans Havermann | 10/2020 |
4604 | 10^105515-41033 | 105515 | Patrick De Geest | 12/2016 |
4605 | (29*2^350473-1)/267927 | 105499 | Lelio R Paula | 03/2014 |
4606 | 17^85732-72 | 105489 | Lelio R Paula | 01/2017 |
4607 | (2^350399+3618)/2 | 105481 | Stefano Morozzi | 09/2020 |
4608 | (2^350379+1010)/2 | 105475 | Stefano Morozzi | 09/2020 |
4609 | 25233^15118+15118^25233 | 105462 | Hans Havermann | 09/2020 |
4610 | 26261^10366+10366^26261 | 105454 | Hans Havermann | 09/2020 |
4611 | 2^350307-110329 | 105453 | Enzo Doria | 05/2014 |
4612 | 2^350253-20365 | 105437 | Enzo Doria | 06/2013 |
4613 | (3*29^72089-1075)/4 | 105423 | Curtis Bright | 05/2015 |
4614 | (2^350187+2658)/2 | 105417 | Stefano Morozzi | 09/2020 |
4615 | 2^350139-109009 | 105403 | Enzo Doria | 04/2014 |
4616 | (2^350129+3026)/2 | 105400 | Stefano Morozzi | 09/2020 |
4617 | (2^350097+2070)/2 | 105390 | Stefano Morozzi | 09/2020 |
4618 | 3*5^150763+4 | 105380 | Lelio R Paula | 11/2014 |
4619 | 12^97643+91 | 105375 | Lelio R Paula | 02/2017 |
4620 | 6*5^150744-7 | 105367 | Lelio R Paula | 11/2014 |
4621 | (2^350018+2074)/2 | 105366 | Stefano Morozzi | 09/2020 |
4622 | 4*10^105364-39 | 105365 | Robert Price | 09/2020 |
4623 | 2^349996-112169 | 105360 | Enzo Doria | 06/2014 |
4624 | (2^349997+3702)/2 | 105360 | Stefano Morozzi | 09/2020 |
4625 | (418*10^105343+23)/9 | 105345 | Robert Price | 02/2020 |
4626 | 25490^13563+13563^25490 | 105334 | Hans Havermann | 10/2020 |
4627 | 11^101142-1080 | 105329 | Enzo Doria | 12/2012 |
4628 | (68*10^105320-41)/9 | 105321 | Robert Price | 10/2013 |
4629 | 24800^17649+17649^24800 | 105319 | Hans Havermann | 09/2020 |
4630 | 3^220712-9934 | 105307 | Enzo Doria | 05/2014 |
4631 | 7*10^105296+9 | 105297 | Robert Price | 09/2015 |
4632 | 89*10^105284-7 | 105286 | Robert Price | 04/2020 |
4633 | 2^349739+1873373 | 105282 | Enzo Doria | 06/2016 |
4634 | 7^124574+24 | 105278 | Lelio R Paula | 11/2016 |
4635 | 2^349708-20957 | 105273 | Enzo Doria | 06/2013 |
4636 | 34115^1218+1218^34115 | 105267 | Hans Havermann | 09/2020 |
4637 | 2^349649-20469 | 105255 | Enzo Doria | 06/2013 |
4638 | 2^349607-20109 | 105243 | Enzo Doria | 06/2013 |
4639 | (85*10^105242+473)/9 | 105243 | Robert Price | 01/2020 |
4640 | 2^349594-19085 | 105239 | Enzo Doria | 05/2013 |
4641 | 2^349530+1874479 | 105220 | Enzo Doria | 06/2016 |
4642 | (3^220528+8)/8081 | 105215 | Lelio R Paula | 12/2013 |
4643 | (2^349452+3570)/2 | 105196 | Stefano Morozzi | 09/2020 |
4644 | 3^220464-112852 | 105189 | Enzo Doria | 07/2014 |
4645 | 2^349419-17349 | 105186 | Enzo Doria | 05/2013 |
4646 | 2^349400-2579 | 105180 | Yves Minet | 07/2014 |
4647 | 8*5^150471+3 | 105176 | Lelio R Paula | 11/2014 |
4648 | (2^349359+2298)/2 | 105168 | Stefano Morozzi | 02/2021 |
4649 | 655^37342+37342^655 | 105165 | Norbert Schneider | 08/2018 |
4650 | 13^94403+94 | 105160 | Lelio R Paula | 12/2016 |
4651 | 2^349301-112309 | 105151 | Enzo Doria | 06/2014 |
4652 | 30291^2960+2960^30291 | 105149 | Hans Havermann | 09/2020 |
4653 | 27350^6981+6981^27350 | 105132 | Hans Havermann | 09/2020 |
4655 | 9*2^349220+5 | 105127 | Lelio R Paula | 09/2014 |
4654 | (2^349225+3122)/2 | 105127 | Stefano Morozzi | 09/2020 |
4656 | 7*3^220317+8 | 105119 | Lelio R Paula | 09/2014 |
4658 | 7^124386+88 | 105119 | Lelio R Paula | 11/2016 |
4657 | (2^349197+570)/2 | 105119 | Stefano Morozzi | 09/2020 |
4659 | 2^349171-19657 | 105111 | Enzo Doria | 06/2013 |
4660 | (2^349115+590)/2 | 105094 | Stefano Morozzi | 09/2020 |
4661 | (2^(6*58185+1)-5)/3 | 105093 | Kevin Acres | 11/2013 |
4662 | (2^349111-5)/3 | 105093 | Lelio R Paula | 12/2013 |
4663 | 675^37142+37142^675 | 105087 | Norbert Schneider | 08/2018 |
4664 | 2^348981-2797573 | 105054 | Enzo Doria | 04/2016 |
4665 | 2^348869-6555 | 105021 | Lelio R Paula | 01/2017 |
4666 | 5^150217+78 | 104998 | Lelio R Paula | 11/2016 |
4667 | (88*10^104989-1)/3 | 104991 | Robert Price | 04/2020 |
4668 | (28*10^104972+149)/3 | 104973 | Robert Price | 12/2019 |
4669 | 24927^16262+16262^24927 | 104972 | Hans Havermann | 10/2020 |
4670 | 2^348703-113005 | 104971 | Enzo Doria | 06/2014 |
4672 | 2^348612-809 | 104943 | Yves Minet | 05/2013 |
4671 | (73*10^104942-1)/9 | 104943 | Robert Price | 01/2015 |
4673 | 2^348583-350407 | 104934 | Enzo Doria | 06/2015 |
4674 | 2^348531-949 | 104919 | Yves Minet | 05/2012 |
4675 | 20^80636-59 | 104910 | Lelio R Paula | 12/2016 |
4676 | 8*10^104893-13 | 104894 | Robert Price | 08/2019 |
4678 | 2^348437+35 | 104890 | Lelio R Paula | 09/2011 |
4677 | 25208^14487+14487^25208 | 104890 | Hans Havermann | 09/2020 |
4679 | 2^348348+135 | 104864 | Lelio R Paula | 09/2011 |
4680 | 6*10^104852-31 | 104853 | Robert Price | 07/2020 |
4682 | (2^348284+9)/25 | 104843 | Lelio R Paula | 12/2013 |
4681 | 75^55914-74 | 104843 | Norbert Schneider | 01/2015 |
4683 | 2^348268+255255 | 104840 | Lelio R Paula | 01/2017 |
4684 | 35540^891+891^35540 | 104839 | Hans Havermann | 09/2020 |
4685 | 5^149985-52 | 104836 | Lelio R Paula | 11/2016 |
4686 | 2^348224-20249 | 104826 | Enzo Doria | 06/2013 |
4687 | (2^348217+2522)/2 | 104824 | Stefano Morozzi | 09/2020 |
4688 | 2^348212-20105 | 104823 | Enzo Doria | 06/2013 |
4689 | 24647^17898+17898^24647 | 104819 | Hans Havermann | 10/2020 |
4690 | 20^80562+19 | 104814 | Lelio R Paula | 12/2016 |
4691 | 2*10^104811-87 | 104812 | Robert Price | 04/2018 |
4692 | 2^348149-20053 | 104804 | Enzo Doria | 06/2013 |
4693 | 2^348095-171 | 104788 | Lelio R Paula | 09/2011 |
4694 | 25383^13412+13412^25383 | 104769 | Hans Havermann | 09/2020 |
4695 | 2^348003-20377 | 104760 | Enzo Doria | 06/2013 |
4697 | 2^347994-741 | 104757 | Lelio R Paula | 12/2016 |
4696 | (2^347993+2234)/2 | 104757 | Stefano Morozzi | 01/2021 |
4698 | (2^347990+790)/2 | 104756 | Stefano Morozzi | 09/2020 |
4699 | 25137^14702+14702^25137 | 104756 | Hans Havermann | 09/2020 |
4700 | (13*2^347972-1)/282309 | 104746 | Lelio R Paula | 03/2014 |
4701 | 2^347910-109937 | 104732 | Enzo Doria | 04/2014 |
4702 | (88*10^104709-637)/9 | 104710 | Robert Price | 11/2020 |
4703 | 5^149794+28 | 104702 | Lelio R Paula | 11/2016 |
4704 | 105^51800-2 | 104698 | Lelio R Paula | 10/2017 |
4705 | (16^86939+3^86939)/19 | 104684 | Robert Price | 06/2013 |
4706 | (15^89009-1)/14 | 104682 | Paul Bourdelais | 03/2010 |
4707 | 2^347731-18609 | 104678 | Enzo Doria | 05/2013 |
4708 | 4*6^134519+7 | 104677 | Lelio R Paula | 11/2014 |
4709 | 25321^13588+13588^25321 | 104656 | Hans Havermann | 09/2020 |
4710 | 2^347612-18545 | 104642 | Enzo Doria | 05/2013 |
4713 | 6^134465+95 | 104635 | Lelio R Paula | 11/2016 |
4711 | (2^347588+3718)/2 | 104635 | Stefano Morozzi | 09/2020 |
4712 | (166*10^104633-7)/3 | 104635 | Robert Price | 11/2020 |
4714 | 2^347517-17301 | 104614 | Enzo Doria | 05/2013 |
4715 | (2^347493+3186)/2 | 104606 | Stefano Morozzi | 09/2020 |
4716 | (2^347490+2574)/2 | 104605 | Stefano Morozzi | 09/2020 |
4717 | 3^219222-310 | 104596 | Lelio R Paula | 11/2016 |
4718 | 25993^10554+10554^25993 | 104581 | Hans Havermann | 09/2020 |
4719 | (15*2^347399-1)/13 | 104578 | Lelio R Paula | 03/2014 |
4720 | 2^347372-17853 | 104570 | Enzo Doria | 05/2013 |
4721 | 2^347314+99 | 104552 | Lelio R Paula | 09/2011 |
4722 | 2^347300-22233 | 104548 | Enzo Doria | 06/2013 |
4723 | 2^347287-19845 | 104544 | Enzo Doria | 06/2013 |
4724 | (31*2^347254-1)/891 | 104533 | Lelio R Paula | 03/2014 |
4725 | 5*F(250091)*(F(250091)-1)+1 | 104532 | Serge Batalov | 04/2015 |
4726 | 7*2^347239-3 | 104531 | Lelio R Paula | 09/2014 |
4727 | (3^219098-8)/514147439969 | 104525 | Lelio R Paula | 12/2013 |
4728 | 5*3^219047+8 | 104513 | Lelio R Paula | 09/2014 |
4729 | 2^347148-109073 | 104502 | Enzo Doria | 04/2014 |
4730 | 2^347129-20145 | 104497 | Enzo Doria | 06/2013 |
4731 | 4*10^104461-57 | 104462 | Robert Price | 10/2018 |
4732 | (2^346989+590)/2 | 104454 | Stefano Morozzi | 09/2020 |
4733 | 2^346923-19585 | 104435 | Enzo Doria | 05/2013 |
4734 | (11*2^346864-1)/175 | 104416 | Lelio R Paula | 03/2014 |
4735 | 2^346854+346854^2+1 | 104414 | Robert Price | 06/2014 |
4736 | 2^346834+11715 | 104408 | Lelio R Paula | 04/2015 |
4737 | 24276^19961+19961^24276 | 104392 | Hans Havermann | 09/2020 |
4738 | 79^55009+78 | 104387 | Norbert Schneider | 02/2015 |
4739 | 3^218744+58 | 104368 | Lelio R Paula | 11/2016 |
4740 | 9*10^104361+89 | 104362 | Robert Price | 11/2019 |
4741 | 25048^14665+14665^25048 | 104358 | Hans Havermann | 09/2020 |
4742 | 2^346649+65 | 104352 | Lelio R Paula | 09/2011 |
4743 | 2^346638-17265 | 104349 | Enzo Doria | 05/2013 |
4744 | (2^346606+2082)/2 | 104339 | Stefano Morozzi | 09/2020 |
4745 | (2^346607+2070)/2 | 104339 | Stefano Morozzi | 09/2020 |
4746 | (2^346582+3202)/2 | 104332 | Stefano Morozzi | 09/2020 |
4747 | 26092^9967+9967^26092 | 104331 | Hans Havermann | 09/2020 |
4748 | 5*10^104326+77 | 104327 | Robert Price | 12/2018 |
4749 | (2^346564+2026)/2 | 104326 | Stefano Morozzi | 09/2020 |
4750 | 6^134055+101 | 104316 | Lelio R Paula | 11/2016 |