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 ? |
9501 | 2^321716-2786939 | 96847 | Enzo Doria | 03/2016 |
9502 | 22496^20181+20181^22496 | 96844 | Hans Havermann | 03/2020 |
9503 | 2^321701-113053 | 96842 | Enzo Doria | 07/2014 |
9504 | 2^321700-28635 | 96842 | Lelio R Paula | 06/2016 |
9505 | (5*2^321691-1)/27 | 96838 | Lelio R Paula | 02/2014 |
9506 | 6*10^96835+13 | 96836 | Robert Price | 10/2021 |
9507 | 6^124433-124433 | 96828 | Robert Price | 01/2014 |
9508 | 2^321631+69 | 96821 | Lelio R Paula | 12/2012 |
9509 | (27109^21841-1)/27108 | 96820 | Serge Batalov | 06/2022 |
9510 | (13*2^321606+1)/119 | 96813 | Lelio R Paula | 11/2013 |
9511 | (5052^26141-1)/5051 | 96809 | Serge Batalov | 06/2022 |
9512 | (46*10^96798-1)/9 | 96799 | Serge Batalov | 10/2014 |
9513 | (59*10^96798+31)/9 | 96799 | Robert Price | 09/2015 |
9514 | (2^321536+2062)/2 | 96792 | Stefano Morozzi | 09/2020 |
9515 | 223319#+231223 | 96790 | Makoto Morimoto | 02/2016 |
9516 | 2^321519-28635 | 96787 | Lelio R Paula | 06/2016 |
9517 | 2^321487-1867237 | 96778 | Enzo Doria | 03/2016 |
9518 | 2^321461-63 | 96770 | Lelio R Paula | 12/2012 |
9519 | (16053^23011-1)/16052 | 96770 | Serge Batalov | 06/2022 |
9520 | 2^321439+123 | 96763 | Lelio R Paula | 12/2012 |
9521 | 2^321434+709 | 96762 | Lelio R Paula | 02/2013 |
9522 | 2^321419+22703 | 96757 | Enzo Doria | 09/2023 |
9523 | (2^321415+554)/2 | 96756 | Stefano Morozzi | 09/2020 |
9524 | 14^84413+89 | 96749 | Lelio R Paula | 07/2016 |
9525 | (28*10^96743+71)/9 | 96744 | Serge Batalov | 12/2008 |
9526 | 13^86847+96 | 96743 | Lelio R Paula | 06/2016 |
9527 | 9*2^321336-7 | 96733 | Lelio R Paula | 08/2014 |
9528 | (209*10^96722-17)/3 | 96724 | Robert Price | 05/2017 |
9529 | (54163^20431-1)/54162 | 96710 | Serge Batalov | 06/2022 |
9530 | 2^321237-4615 | 96702 | Yves Minet | 05/2019 |
9531 | (27273^21799-1)/27272 | 96691 | Serge Batalov | 06/2022 |
9532 | 2^321187-1159 | 96687 | Yves Minet | 11/2012 |
9533 | 2^321182-1850081 | 96686 | Enzo Doria | 08/2015 |
9535 | 23359^13768+13768^23359 | 96680 | Hans Havermann | 06/2020 |
9534 | (65630^20071-1)/65629 | 96680 | Serge Batalov | 06/2022 |
9536 | 2^321116-112829 | 96666 | Enzo Doria | 06/2014 |
9537 | 2^321108-109499 | 96664 | Enzo Doria | 04/2014 |
9538 | 2^321098+2793859 | 96661 | Enzo Doria | 03/2016 |
9539 | (2^321100+1170)/2 | 96661 | Stefano Morozzi | 09/2020 |
9540 | 2^321057+4097 | 96648 | Lelio R Paula | 06/2016 |
9541 | (4435*29^66081-4)/7 | 96640 | Curtis Bright | 05/2015 |
9542 | 22743^17750+17750^22743 | 96640 | Hans Havermann | 04/2020 |
9543 | 22882^16679+16679^22882 | 96612 | Hans Havermann | 02/2020 |
9544 | 23452^13163+13163^23452 | 96608 | Hans Havermann | 02/2020 |
9545 | (1042^32009-1)/1041 | 96596 | Serge Batalov | 06/2022 |
9546 | (85*10^96594-373)/9 | 96595 | Robert Price | 05/2021 |
9547 | (2^320863+666)/2 | 96590 | Stefano Morozzi | 09/2020 |
9548 | p(19849)#/2-64 | 96581 | Robert Price | 04/2017 |
9549 | 2^320817+189 | 96576 | Lelio R Paula | 12/2012 |
9550 | 2^320816-4685 | 96576 | Yves Minet | 05/2019 |
9552 | (361*10^96567-163)/9 | 96569 | Robert Price | 10/2017 |
9551 | (19*10^96568+137)/3 | 96569 | Robert Price | 03/2022 |
9553 | 2^320788-110327 | 96567 | Enzo Doria | 05/2014 |
9554 | (3065^27697-1)/3064 | 96561 | Serge Batalov | 06/2022 |
9555 | 2^320764+793 | 96560 | Lelio R Paula | 02/2013 |
9556 | (2^320752+1842)/2 | 96556 | Stefano Morozzi | 02/2021 |
9557 | (48331^20611-1)/48330 | 96542 | Serge Batalov | 06/2022 |
9558 | (107746^19321-1)/(107746^139-1) | 96532 | Ryan Propper | 11/2023 |
9559 | (50595^20521-1)/50594 | 96529 | Serge Batalov | 06/2022 |
9560 | (9179*2^320652-1)/221 | 96528 | Lelio R Paula | 05/2015 |
9561 | 23288^13947+13947^23288 | 96517 | Hans Havermann | 07/2020 |
9563 | 2^320616-78795 | 96516 | Lelio R Paula | 03/2016 |
9562 | (19*10^96520+287)/9/109/409 | 96516 | Alfred Reich | 11/2022 |
9564 | 2^320612-461939 | 96514 | Enzo Doria | 07/2015 |
9565 | (21245^22303-1)/21244 | 96507 | Serge Batalov | 06/2022 |
9566 | 10^96504+987654321 | 96505 | Robert Price | 12/2019 |
9567 | 10^96504+71887 | 96505 | Patrick De Geest | 02/2020 |
9568 | 22654^18195+18195^22654 | 96505 | Hans Havermann | 07/2020 |
9569 | 10^96504-139277 | 96504 | Patrick De Geest | 02/2020 |
9570 | (218^41263-1)/217 | 96490 | Serge Batalov | 06/2022 |
9571 | (47220^20641-1)/47219 | 96474 | Serge Batalov | 06/2022 |
9572 | 5*2^320431+7 | 96461 | Lelio R Paula | 08/2014 |
9573 | 19^75431-90 | 96458 | Lelio R Paula | 06/2016 |
9574 | (11853^23677-1)/11852 | 96452 | Serge Batalov | 06/2022 |
9575 | (2^320386+2814)/2 | 96446 | Stefano Morozzi | 02/2021 |
9576 | (2^320370+3)/67 | 96440 | Lelio R Paula | 10/2013 |
9577 | 3*10^96431-17 | 96432 | Robert Price | 03/2021 |
9578 | 2^320309+1871477 | 96423 | Enzo Doria | 05/2016 |
9579 | (10412^24001-1)/10411 | 96421 | Serge Batalov | 06/2022 |
9580 | tau(809^6028) | 96410 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
9581 | (29*10^96407-23)/3 | 96408 | Robert Price | 03/2022 |
9582 | (21*2^320252-1)/1895 | 96404 | Lelio R Paula | 02/2014 |
9583 | 2^320232-173 | 96400 | Lelio R Paula | 12/2012 |
9584 | 175^42975-2 | 96395 | Lelio R Paula | 10/2017 |
9585 | (2^320170+2718)/2 | 96381 | Stefano Morozzi | 09/2020 |
9586 | 10^96377+199911 | 96378 | Patrick De Geest | 12/2016 |
9587 | 10^96377-20001 | 96377 | Robert Price | 11/2016 |
9588 | (18079^22639-1)/18078 | 96374 | Serge Batalov | 06/2022 |
9590 | 2^320124-2790449 | 96367 | Enzo Doria | 03/2016 |
9589 | 10^96366+19197 | 96367 | Patrick De Geest | 08/2021 |
9592 | 10^96366-10000000001 | 96366 | Robert Price | 04/2021 |
9591 | 10^96366-287367 | 96366 | Patrick De Geest | 08/2021 |
9593 | (26351^21799-1)/26350 | 96365 | Serge Batalov | 06/2022 |
9594 | 22488^19255+19255^22488 | 96351 | Hans Havermann | 11/2019 |
9595 | 23248^13939+13939^23248 | 96346 | Hans Havermann | 11/2019 |
9596 | (5928^25537-1)/5927 | 96345 | Serge Batalov | 06/2022 |
9598 | 2^320043-108775 | 96343 | Enzo Doria | 04/2014 |
9597 | 24989^7168+7168^24989 | 96343 | Hans Havermann | 11/2019 |
9599 | 10^96338+424939 | 96339 | Patrick De Geest | 03/2016 |
9600 | 10^96338-9281 | 96338 | Robert Price | 09/2015 |
9601 | (8527^24509-1)/8526 | 96336 | Serge Batalov | 06/2022 |
9602 | (3*2^320008+1)/7 | 96332 | Lelio R Paula | 09/2013 |
9603 | (2444^28433-1)/2443 | 96331 | Serge Batalov | 06/2022 |
9604 | (48690^20551-1)/48689 | 96327 | Serge Batalov | 06/2022 |
9605 | (61935^20101-1)/61934 | 96318 | Serge Batalov | 06/2022 |
9606 | 25699^5596+5596^25699 | 96317 | Hans Havermann | 07/2020 |
9607 | (7*10^96312+83)/9 | 96312 | Tyler Busby | 01/2023 |
9608 | (11*10^96298+529)/9 | 96299 | Robert Price | 09/2021 |
9610 | 26502^4297+4297^26502 | 96287 | Hans Havermann | 07/2020 |
9609 | (54180^20341-1)/54179 | 96287 | Serge Batalov | 06/2022 |
9611 | 2^319838+4257 | 96281 | Lelio R Paula | 03/2016 |
9612 | 2^319830-16995 | 96279 | Lelio R Paula | 03/2016 |
9613 | 6^123713+143 | 96268 | Lelio R Paula | 05/2016 |
9614 | (28*10^96265-97)/3 | 96266 | Robert Price | 07/2016 |
9615 | (3799^26893-1)/3798 | 96265 | Serge Batalov | 06/2022 |
9617 | 103^47825-102 | 96264 | Norbert Schneider | 10/2016 |
9616 | (54041^20341-1)/54040 | 96264 | Serge Batalov | 06/2022 |
9618 | (2^319777+2082)/2 | 96263 | Stefano Morozzi | 09/2020 |
9619 | (2^319771+3686)/2 | 96261 | Stefano Morozzi | 09/2020 |
9620 | 2^319713-81 | 96244 | Lelio R Paula | 11/2012 |
9621 | 2^319711+213413 | 96243 | Enzo Doria | 09/2023 |
9622 | (55459*2^319706+1)/20167 | 96242 | Lelio R Paula | 07/2014 |
9623 | 22834^16393+16393^22834 | 96238 | Hans Havermann | 05/2020 |
9624 | 8*10^96236+27 | 96237 | Robert Price | 09/2021 |
9625 | 12^89153-43 | 96213 | Lelio R Paula | 06/2016 |
9626 | (25048^21871-1)/25047 | 96202 | Serge Batalov | 06/2022 |
9628 | (2*10^96195-59)/3 | 96195 | Robert Price | 05/2017 |
9627 | 9*10^96194-77 | 96195 | Robert Price | 08/2020 |
9629 | 2^319545+147 | 96193 | Lelio R Paula | 11/2012 |
9630 | 23357^13128+13128^23357 | 96189 | Hans Havermann | 02/2020 |
9631 | (19838^22381-1)/19837 | 96179 | Serge Batalov | 06/2022 |
9632 | (221^41023-1)/220 | 96172 | Serge Batalov | 06/2022 |
9633 | 25952^5073+5073^25952 | 96160 | Hans Havermann | 12/2019 |
9634 | (50891^20431-1)/50890 | 96157 | Serge Batalov | 06/2022 |
9635 | (50^56597+1)/51 | 96155 | Robert Price | 03/2014 |
9636 | 22313^20374+20374^22313 | 96149 | Hans Havermann | 12/2019 |
9637 | 22279^20668+20668^22279 | 96141 | Hans Havermann | 11/2019 |
9638 | (41*10^96137-59)/9 | 96138 | Robert Price | 10/2017 |
9639 | 2^319353-110343 | 96135 | Enzo Doria | 05/2014 |
9640 | 11^92309-42 | 96130 | Lelio R Paula | 05/2016 |
9641 | (21675^22171-1)/21674 | 96129 | Serge Batalov | 06/2022 |
9642 | 14^83869-13^83869 | 96125 | Jean-Louis Charton | 01/2009 |
9643 | (7*10^96112+143)/3 | 96113 | Robert Price | 04/2016 |
9644 | 26731^3940+3940^26731 | 96112 | Hans Havermann | 07/2020 |
9645 | 22640^17583+17583^22640 | 96109 | Hans Havermann | 07/2020 |
9646 | (19178^22441-1)/19177 | 96107 | Serge Batalov | 06/2022 |
9647 | 2^319237+65835 | 96100 | Lelio R Paula | 03/2016 |
9648 | 2^319194-1841 | 96087 | Yves Minet | 03/2013 |
9649 | 23949^10280+10280^23949 | 96084 | Hans Havermann | 07/2020 |
9650 | (1927^29251-1)/1926 | 96083 | Serge Batalov | 06/2022 |
9651 | 20^73847-31 | 96078 | Lelio R Paula | 05/2016 |
9652 | (60229^20101-1)/60228 | 96075 | Serge Batalov | 06/2022 |
9653 | (28044^21601-1)/28043 | 96074 | Serge Batalov | 06/2022 |
9654 | 50^56543-49 | 96065 | Norbert Schneider | 09/2018 |
9655 | 2^319107-109555 | 96061 | Enzo Doria | 04/2014 |
9656 | (3*2^319114-1)/2627 | 96060 | Lelio R Paula | 02/2014 |
9657 | 2^319098-112157 | 96059 | Enzo Doria | 06/2014 |
9658 | 15^81673+94 | 96055 | Lelio R Paula | 07/2016 |
9659 | (85761^19471-1)/85760 | 96052 | Serge Batalov | 06/2022 |
9660 | 2^319074+1871749 | 96051 | Enzo Doria | 05/2016 |
9661 | 25670^5517+5517^25670 | 96050 | Hans Havermann | 06/2020 |
9662 | 6*10^96037+19 | 96038 | Robert Price | 11/2021 |
9663 | 2^319007-112111 | 96031 | Enzo Doria | 06/2014 |
9664 | (57987^20161-1)/57986 | 96029 | Serge Batalov | 06/2022 |
9666 | 2^318946-1421 | 96013 | Yves Minet | 01/2013 |
9665 | (68310^19861-1)/68309 | 96013 | Serge Batalov | 06/2022 |
9667 | (20294^22291-1)/20293 | 96012 | Serge Batalov | 06/2022 |
9668 | (304^38669-1)/303 | 96008 | Serge Batalov | 06/2022 |
9669 | (1840^29401-1)/1839 | 95986 | Serge Batalov | 06/2022 |
9670 | 22830^16003+16003^22830 | 95982 | Hans Havermann | 02/2020 |
9671 | 2^318834-110411 | 95979 | Enzo Doria | 05/2014 |
9672 | 2^318831-1965 | 95978 | Lelio R Paula | 03/2016 |
9673 | (5*10^95976+7)/3 | 95977 | Erik Branger | 03/2014 |
9674 | (12797^23369-1)/12796 | 95975 | Serge Batalov | 06/2022 |
9675 | (10534^23857-1)/10533 | 95963 | Serge Batalov | 06/2022 |
9676 | 22871^15696+15696^22871 | 95962 | Hans Havermann | 01/2020 |
9677 | (2^318760+858)/2 | 95957 | Stefano Morozzi | 02/2021 |
9678 | 23167^13860+13860^23167 | 95953 | Hans Havermann | 01/2020 |
9679 | (823^32911-1)/822 | 95946 | Serge Batalov | 06/2022 |
9681 | 5*3^201089-8 | 95945 | Lelio R Paula | 08/2014 |
9680 | (81783^19531-1)/81782 | 95945 | Serge Batalov | 06/2022 |
9682 | 2^318711-139 | 95942 | Lelio R Paula | 11/2012 |
9683 | 2^318635-1167 | 95919 | Lelio R Paula | 03/2016 |
9684 | 15^81555+16 | 95917 | Lelio R Paula | 07/2016 |
9685 | (37*10^95895+377)/9 | 95896 | Robert Price | 10/2017 |
9686 | (22699*2^318527+1)/21 | 95890 | Lelio R Paula | 07/2014 |
9687 | 45^57997+196072 | 95882 | Enzo Doria | 10/2023 |
9688 | 2^318496-97305 | 95877 | Lelio R Paula | 03/2016 |
9689 | 22129^21496+21496^22129 | 95871 | Hans Havermann | 07/2020 |
9691 | 2^318466+15015 | 95868 | Lelio R Paula | 03/2016 |
9690 | 24847^7216+7216^24847 | 95868 | Hans Havermann | 07/2020 |
9692 | 23706^11065+11065^23706 | 95866 | Hans Havermann | 07/2020 |
9693 | (2863^27733-1)/2862 | 95865 | Serge Batalov | 06/2022 |
9694 | 6^123187-173129 | 95859 | Mike Oakes | 04/2011 |
9695 | 2^318423-487 | 95855 | Yves Minet | 03/2012 |
9696 | 10^95848+443709 | 95849 | Patrick De Geest | 10/2022 |
9697 | 10^95848-5000000001 | 95848 | Robert Price | 09/2022 |
9698 | 10^95848-61971 | 95848 | Patrick De Geest | 10/2022 |
9699 | (2^318359+678)/2 | 95836 | Stefano Morozzi | 09/2020 |
9700 | (10223*2^318350+1)/1563 | 95834 | Lelio R Paula | 07/2014 |
9701 | (8060^24533-1)/8059 | 95831 | Serge Batalov | 06/2022 |
9702 | (9853^23993-1)/9852 | 95814 | Serge Batalov | 06/2022 |
9704 | (2^318267+2550)/2 | 95808 | Stefano Morozzi | 09/2020 |
9703 | (5048^25873-1)/5047 | 95808 | Serge Batalov | 06/2022 |
9706 | 4*10^95799+13 | 95800 | Robert Price | 02/2021 |
9705 | (65533^19891-1)/65532 | 95800 | Serge Batalov | 06/2022 |
9707 | 2^318203+211343 | 95789 | Enzo Doria | 08/2023 |
9708 | 45369!2+1024 | 95787 | Robert Price | 02/2015 |
9709 | 3^200758-86 | 95786 | Lelio R Paula | 05/2016 |
9710 | (2^318271+1)/(3*46031404238891*654090604244843) | 95781 | Gord Palameta | 12/2018 |
9711 | (2^318171+2030)/2 | 95779 | Stefano Morozzi | 09/2020 |
9712 | 3*10^95776-59 | 95777 | Robert Price | 10/2017 |
9713 | (82970^19471-1)/82969 | 95772 | Serge Batalov | 06/2022 |
9714 | 193^41901-2 | 95768 | Lelio R Paula | 10/2017 |
9715 | (2^318124+2502)/2 | 95765 | Stefano Morozzi | 09/2020 |
9716 | (12900^23297-1)/12899 | 95761 | Serge Batalov | 06/2022 |
9717 | 4*10^95752-71 | 95753 | Robert Price | 11/2017 |
9718 | (16955^22639-1)/16954 | 95743 | Serge Batalov | 06/2022 |
9719 | (81795^19489-1)/81794 | 95740 | Serge Batalov | 06/2022 |
9720 | (2742^27847-1)/2741 | 95737 | Serge Batalov | 06/2022 |
9721 | (6072^25303-1)/6071 | 95726 | Serge Batalov | 06/2022 |
9722 | 23640^11197+11197^23640 | 95721 | Hans Havermann | 12/2019 |
9723 | 10^95712+89329 | 95713 | Patrick De Geest | 09/2022 |
9725 | 10^95712-4000001 | 95712 | Robert Price | 07/2021 |
9724 | 10^95712-989313 | 95712 | Patrick De Geest | 09/2022 |
9726 | (60657^20011-1)/60656 | 95706 | Serge Batalov | 06/2022 |
9727 | ((10^47850-1)*10000/9+6421)*10^47850+(10^47850-1)/9 | 95704 | Hans Havermann | 05/2019 |
9728 | (509203*2^317881-1)/52805 | 95693 | Lelio R Paula | 04/2015 |
9729 | (26161^21661-1)/26160 | 95687 | Serge Batalov | 06/2022 |
9730 | (2^317853+1086)/2 | 95683 | Stefano Morozzi | 09/2020 |
9731 | 4*10^95672+91 | 95673 | Robert Price | 11/2017 |
9732 | 6*10^95665+43 | 95666 | Robert Price | 07/2020 |
9733 | (20819^22153-1)/20818 | 95663 | Serge Batalov | 06/2022 |
9734 | (64464^19891-1)/64463 | 95658 | Serge Batalov | 06/2022 |
9735 | (934^32203-1)/933 | 95652 | Serge Batalov | 06/2022 |
9736 | (11*10^95641-113)/3 | 95642 | Robert Price | 01/2017 |
9737 | (72330^19681-1)/72329 | 95632 | Serge Batalov | 06/2022 |
9738 | (13224^23203-1)/13223 | 95624 | Serge Batalov | 06/2022 |
9739 | (18211^22441-1)/18210 | 95602 | Serge Batalov | 06/2022 |
9740 | (71975^19681-1)/71974 | 95590 | Serge Batalov | 06/2022 |
9741 | (13*10^95588+11)/3 | 95589 | Robert Price | 05/2015 |
9742 | Phi(191188,10)/191189 | 95587 | Maksym Voznyy | 11/2015 |
9743 | (13368^23167-1)/13367 | 95585 | Serge Batalov | 06/2022 |
9744 | (25874^21661-1)/25873 | 95583 | Serge Batalov | 06/2022 |
9745 | 3^200325+70 | 95580 | Lelio R Paula | 05/2016 |
9746 | (25*10^95574-7)/9 | 95575 | Robert Price | 12/2014 |
9747 | (2^317485-3)/21313 | 95569 | Lelio R Paula | 06/2013 |
9748 | 2^317464+31 | 95567 | Lelio R Paula | 11/2012 |
9749 | (2^317451+2114)/2 | 95562 | Stefano Morozzi | 09/2020 |
9750 | 7^113073+82 | 95558 | Lelio R Paula | 04/2016 |