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 ? |
7501 | (22*10^87863-1)/3 | 87864 | Robert Price | 12/2014 |
7502 | 2^291822-2797811 | 87848 | Enzo Doria | 04/2016 |
7503 | (2^291730+1614)/2 | 87820 | Stefano Morozzi | 01/2020 |
7504 | 2^291716-111389 | 87816 | Enzo Doria | 05/2014 |
7505 | (782*10^87809-71)/9 | 87811 | Robert Price | 06/2017 |
7506 | 3^184039+1820 | 87809 | Lelio R Paula | 12/2015 |
7507 | 12^81356+41 | 87798 | Lelio R Paula | 11/2015 |
7508 | 21307^13188+13188^21307 | 87789 | Hans Havermann | 08/2020 |
7509 | (2527^25801-1)/2526 | 87788 | Serge Batalov | 08/2020 |
7510 | (2^291596+1986)/2 | 87779 | Stefano Morozzi | 01/2020 |
7512 | 6^112792-18983 | 87770 | Enzo Doria | 03/2013 |
7511 | (3123^25117-1)/3122 | 87770 | Serge Batalov | 08/2020 |
7513 | 3*6^112787-5 | 87766 | Lelio R Paula | 10/2014 |
7514 | 7^103848+82 | 87762 | Lelio R Paula | 10/2015 |
7515 | (19*2^291534-1)/178696395 | 87754 | Lelio R Paula | 02/2014 |
7516 | (2^291506+918)/2 | 87752 | Stefano Morozzi | 01/2020 |
7517 | 24411^3932+3932^24411 | 87749 | Hans Havermann | 08/2020 |
7518 | (3^183915+5)/128 | 87748 | Lelio R Paula | 01/2013 |
7519 | 19^68619+24 | 87747 | Lelio R Paula | 12/2015 |
7521 | 2^291480+1185 | 87745 | Lelio R Paula | 10/2013 |
7520 | 2^291482+1867369 | 87745 | Enzo Doria | 03/2016 |
7522 | 2^291467-937 | 87741 | Yves Minet | 05/2013 |
7523 | (25*10^87726+161)/3 | 87727 | Robert Price | 01/2017 |
7524 | 11^84239-2850 | 87726 | Enzo Doria | 12/2012 |
7525 | 10^87720+215541 | 87721 | Patrick De Geest | 01/2011 |
7526 | 10^87720-3 | 87720 | Markus Tervooren | 07/2010 |
7527 | (2^291383+3078)/2 | 87715 | Stefano Morozzi | 01/2020 |
7529 | (5*10^87712-473)/9 | 87712 | Robert Price | 12/2017 |
7528 | (2^291373+2946)/2 | 87712 | Stefano Morozzi | 01/2020 |
7530 | 22591^7630+7630^22591 | 87711 | Hans Havermann | 08/2020 |
7531 | 22081^9378+9378^22081 | 87709 | Hans Havermann | 08/2020 |
7532 | (2^291357+5)/6847 | 87704 | Lelio R Paula | 01/2013 |
7533 | (2^291280+462)/2 | 87684 | Stefano Morozzi | 01/2020 |
7534 | 20749^16816+16816^20749 | 87680 | Hans Havermann | 08/2020 |
7536 | 93^44541-2 | 87679 | Lelio R Paula | 09/2013 |
7535 | 2^291261-109561 | 87679 | Enzo Doria | 04/2014 |
7537 | (4*10^87669+149)/9 | 87669 | Robert Price | 03/2017 |
7538 | 2^291217+4257 | 87666 | Lelio R Paula | 10/2013 |
7539 | 22748^7137+7137^22748 | 87660 | Hans Havermann | 08/2020 |
7540 | 2*12^81214-5^81214 | 87645 | Mike Oakes | 12/2010 |
7541 | (5005^23689-1)/5004 | 87632 | Serge Batalov | 08/2020 |
7542 | (5*10^87622-473)/9 | 87622 | Robert Price | 12/2017 |
7543 | 12^81188+85 | 87617 | Lelio R Paula | 11/2015 |
7544 | (2^291030+3462)/2 | 87609 | Stefano Morozzi | 01/2020 |
7545 | (21*2^290980-1)/5 | 87595 | Lelio R Paula | 02/2014 |
7546 | (2^290980+1150)/2 | 87594 | Stefano Morozzi | 01/2020 |
7547 | 2^290958+127 | 87588 | Lelio R Paula | 09/2012 |
7548 | 78531!4-8 | 87580 | Robert Price | 11/2019 |
7549 | 2^290923-67 | 87577 | Lelio R Paula | 09/2012 |
7550 | 3*10^87573-97 | 87574 | Robert Price | 08/2017 |
7551 | (13*10^87565+11)/3 | 87566 | Robert Price | 05/2015 |
7552 | (2^290881+1022)/2 | 87564 | Stefano Morozzi | 01/2020 |
7553 | (2^290857-7)/55 | 87555 | Lelio R Paula | 01/2013 |
7554 | (2^290808+1498)/2 | 87542 | Stefano Morozzi | 01/2020 |
7555 | 2^290802+82875 | 87541 | Lelio R Paula | 10/2013 |
7556 | (509203*2^290792-1)/7560987 | 87536 | Lelio R Paula | 04/2015 |
7557 | 3^183441+236 | 87524 | Lelio R Paula | 12/2015 |
7558 | 2^290618-33 | 87485 | Lelio R Paula | 09/2012 |
7559 | (184*10^87473-1)/3 | 87475 | Robert Price | 02/2017 |
7560 | 2^290534-115151 | 87460 | Enzo Doria | 06/2015 |
7561 | 2^290512-109757 | 87453 | Enzo Doria | 04/2014 |
7562 | (32*10^87448-41)/9 | 87449 | Robert Price | 05/2015 |
7563 | (2^290493+3074)/2 | 87447 | Stefano Morozzi | 01/2020 |
7564 | (2^290491+9)/3179 | 87444 | Lelio R Paula | 01/2013 |
7565 | (25*2^290489-1)/51809 | 87443 | Lelio R Paula | 02/2014 |
7566 | 2^290426+82875 | 87427 | Lelio R Paula | 11/2013 |
7567 | 3*2^290415-7 | 87425 | Lelio R Paula | 08/2014 |
7570 | 10^87408+57 | 87409 | Robert Price | 03/2011 |
7569 | 2^290366+74779 | 87409 | Enzo Doria | 04/2016 |
7568 | 22284^8365+8365^22284 | 87409 | Hans Havermann | 08/2020 |
7571 | 10^87408-123569 | 87408 | Patrick De Geest | 07/2013 |
7572 | 21517^11536+11536^21517 | 87404 | Hans Havermann | 08/2020 |
7573 | 6*10^87396+77 | 87397 | Robert Price | 06/2020 |
7574 | (2^290268+3882)/2 | 87380 | Stefano Morozzi | 01/2020 |
7575 | 2^290265+58905 | 87379 | Lelio R Paula | 10/2013 |
7576 | 7^103391+16 | 87376 | Lelio R Paula | 10/2015 |
7577 | (115*10^87368-7)/9 | 87370 | Robert Price | 11/2017 |
7578 | (92^44491-1)/91 | 87369 | Paul Bourdelais | 08/2017 |
7579 | 8*7^103379+3 | 87367 | Lelio R Paula | 10/2014 |
7580 | (2^290226+7)/71 | 87365 | Lelio R Paula | 01/2013 |
7581 | (3216^24907-1)/3215 | 87354 | Serge Batalov | 08/2020 |
7582 | (2^290160+2686)/2 | 87347 | Stefano Morozzi | 01/2020 |
7583 | 13^78407-30 | 87341 | Lelio R Paula | 11/2015 |
7584 | 4*10^87337-99 | 87338 | Robert Price | 12/2016 |
7585 | 7*10^87319+39 | 87320 | Robert Price | 07/2016 |
7586 | 3*5^124921+2 | 87317 | Lelio R Paula | 10/2014 |
7587 | 78315!4-16 | 87316 | Robert Price | 11/2019 |
7588 | 3^182984+88 | 87306 | Lelio R Paula | 12/2015 |
7589 | (10^87323-1)/(9*33909262597306826117) | 87303 | Alfred Reich | 02/2019 |
7590 | (10^87323-1)/305183363375761435053 | 87303 | Anonymous | 10/2020 |
7591 | (2^290008+958)/2 | 87301 | Stefano Morozzi | 01/2020 |
7592 | (10536^21701-1)/10535 | 87293 | Serge Batalov | 08/2020 |
7593 | 4*10^87287-99 | 87288 | Robert Price | 12/2016 |
7595 | 11^83813-1860 | 87283 | Enzo Doria | 12/2012 |
7594 | (3068^25033-1)/3067 | 87283 | Serge Batalov | 08/2020 |
7596 | 734^30453+30453^734 | 87270 | Norbert Schneider | 04/2017 |
7597 | 12^80855+29 | 87258 | Lelio R Paula | 11/2015 |
7598 | 22209^8482+8482^22209 | 87249 | Hans Havermann | 08/2020 |
7599 | 3*6^112119-5 | 87247 | Lelio R Paula | 10/2014 |
7600 | 12^80806+79 | 87205 | Lelio R Paula | 11/2015 |
7601 | 91382^9+9^91382 | 87201 | Andrey Kulsha | 11/2004 |
7602 | (21181*2^289657+1)/81 | 87198 | Lelio R Paula | 06/2014 |
7603 | 6^112054-47 | 87195 | Lelio R Paula | 12/2015 |
7604 | (55459*2^289632+1)/55 | 87191 | Lelio R Paula | 06/2014 |
7605 | 15^74131-68 | 87185 | Lelio R Paula | 11/2015 |
7606 | 2^289583-741 | 87174 | Lelio R Paula | 10/2013 |
7607 | 7^103146+12 | 87169 | Lelio R Paula | 10/2015 |
7608 | (2^289562+2182)/2 | 87167 | Stefano Morozzi | 01/2020 |
7609 | 5*2^289556-3 | 87166 | Lelio R Paula | 08/2014 |
7610 | 2^289543-112735 | 87162 | Enzo Doria | 06/2014 |
7611 | 20204^20243-20243^20204 | 87155 | Norbert Schneider | 04/2020 |
7612 | 8*3^182655-7 | 87150 | Lelio R Paula | 08/2014 |
7613 | 2*10^87144+81 | 87145 | Robert Price | 01/2017 |
7614 | 20541^17468+17468^20541 | 87140 | Hans Havermann | 07/2020 |
7616 | 7^103106+2*3^103106 | 87135 | Mike Oakes | 08/2010 |
7615 | (65*10^87134-29)/9 | 87135 | Robert Price | 01/2021 |
7617 | 8*3^182614+7 | 87130 | Lelio R Paula | 08/2014 |
7618 | 3*10^87121-73 | 87122 | Robert Price | 02/2021 |
7619 | (4982^23563-1)/4981 | 87119 | Serge Batalov | 08/2020 |
7620 | (2^289368-509203)/9 | 87108 | Lelio R Paula | 01/2013 |
7621 | 3^182566+14 | 87107 | Robert Price | 09/2013 |
7622 | 1738^26881+26881^1738 | 87096 | Norbert Schneider | 04/2020 |
7623 | 24290^3849+3849^24290 | 87089 | Hans Havermann | 08/2020 |
7624 | (3^182385-5)/2 | 87020 | Lelio R Paula | 01/2013 |
7625 | 11^83534-8 | 86992 | Lelio R Paula | 11/2015 |
7626 | 4^144479-14267 | 86986 | Mike Oakes | 03/2011 |
7627 | (145*10^86982-1)/9 | 86984 | Serge Batalov | 12/2014 |
7629 | (151*10^86963-1)/3 | 86965 | Robert Price | 04/2017 |
7628 | (2^288892+826)/2 | 86965 | Stefano Morozzi | 01/2020 |
7630 | 15^73940-58 | 86961 | Lelio R Paula | 11/2015 |
7631 | 2^288761-113143 | 86926 | Enzo Doria | 07/2014 |
7632 | 24440^3603+3603^24440 | 86925 | Hans Havermann | 08/2020 |
7634 | 2^288747+3083 | 86922 | Yves Minet | 03/2011 |
7633 | (2^288749+3102)/2 | 86922 | Stefano Morozzi | 01/2020 |
7635 | 30^58842+29 | 86917 | Norbert Schneider | 03/2017 |
7636 | (2^288705+2186)/2 | 86909 | Stefano Morozzi | 01/2020 |
7637 | (10^86906-13)/3 | 86906 | Robert Price | 02/2017 |
7638 | 2^288692-110027 | 86905 | Enzo Doria | 04/2014 |
7639 | (2^288682+7)/259895449 | 86894 | Lelio R Paula | 01/2013 |
7640 | (26*10^86885+79)/3 | 86886 | Robert Price | 09/2020 |
7641 | (2^288636+47)/1293 | 86885 | Lelio R Paula | 01/2013 |
7642 | 3^182089-34 | 86879 | Robert Price | 11/2013 |
7643 | (2^288571+1698)/2 | 86869 | Stefano Morozzi | 01/2020 |
7644 | 2^288543-109801 | 86861 | Enzo Doria | 04/2014 |
7645 | 60^48847-59 | 86858 | Norbert Schneider | 10/2014 |
7646 | (2^288484+3186)/2 | 86843 | Stefano Morozzi | 01/2020 |
7647 | 2^288470-64875 | 86839 | Lelio R Paula | 10/2013 |
7648 | 10^86816+32997 | 86817 | Patrick De Geest | 07/2013 |
7649 | 10^86816-99 | 86816 | Robert Price | 05/2011 |
7650 | (89*10^86802-539)/9 | 86803 | Robert Price | 11/2017 |
7651 | 20272^19123+19123^20272 | 86796 | Hans Havermann | 07/2020 |
7652 | (2^288316+1530)/2 | 86792 | Stefano Morozzi | 01/2020 |
7653 | (2^288316+2218)/2 | 86792 | Stefano Morozzi | 01/2020 |
7654 | 41487!2+256 | 86786 | Robert Price | 01/2015 |
7655 | (5*2^288301-1)/999 | 86785 | Lelio R Paula | 02/2014 |
7656 | (2^288278+1750)/2 | 86781 | Stefano Morozzi | 01/2020 |
7657 | 2^288243+76043 | 86770 | Enzo Doria | 04/2016 |
7658 | (29*2^288232+1)/15 | 86767 | Lelio R Paula | 01/2013 |
7659 | 9*6^111495+5 | 86761 | Lelio R Paula | 10/2014 |
7660 | (5*2^288202+1)/81 | 86757 | Lelio R Paula | 01/2013 |
7661 | 9*7^102654-2 | 86754 | Lelio R Paula | 10/2014 |
7662 | (2^288155+1154)/2 | 86743 | Stefano Morozzi | 01/2020 |
7663 | 2^288122-112841 | 86734 | Enzo Doria | 06/2014 |
7664 | 5^124062-68 | 86716 | Lelio R Paula | 10/2015 |
7665 | (2^288007-3)/125 | 86697 | Lelio R Paula | 01/2013 |
7666 | 379*10^86691+9 | 86694 | Arkadiusz Wesolowski | 10/2011 |
7667 | 7^102555+48 | 86670 | Lelio R Paula | 10/2015 |
7668 | 2^287902-4961 | 86668 | Yves Minet | 05/2019 |
7669 | 77779!4+2 | 86660 | Robert Price | 08/2017 |
7670 | (2^287842+3250)/2 | 86649 | Stefano Morozzi | 01/2020 |
7671 | 2^287809-111355 | 86640 | Enzo Doria | 05/2014 |
7672 | (47*2^287795+1)/7 | 86636 | Lelio R Paula | 04/2015 |
7673 | 9*7^102504+8 | 86627 | Lelio R Paula | 10/2014 |
7674 | 3^181551+32 | 86622 | Robert Price | 11/2013 |
7675 | 2^287735-521347 | 86617 | Paul Bourdelais | 05/2013 |
7676 | 21912^8953+8953^21912 | 86596 | Hans Havermann | 08/2020 |
7677 | (1017*2^287600-1)/91 | 86578 | Lelio R Paula | 04/2015 |
7678 | (2^287552+1302)/2 | 86562 | Stefano Morozzi | 01/2020 |
7679 | 2^287540-112967 | 86559 | Enzo Doria | 06/2014 |
7680 | 3^181407+20 | 86554 | Robert Price | 10/2013 |
7681 | (28*10^86550+101)/3 | 86551 | Robert Price | 09/2020 |
7683 | (13*10^86546+83)/3 | 86547 | Robert Price | 09/2016 |
7682 | (2^287502+2238)/2 | 86547 | Stefano Morozzi | 01/2020 |
7684 | 2^287482-4715 | 86541 | Yves Minet | 05/2019 |
7685 | 3^181378-2^(181378-1) | 86540 | Robert Price | 01/2016 |
7686 | 24588^3305+3305^24588 | 86530 | Hans Havermann | 08/2020 |
7687 | 83^45080-2 | 86513 | Lelio R Paula | 08/2013 |
7688 | (2^287367+2570)/2 | 86506 | Stefano Morozzi | 01/2020 |
7690 | (78557*2^287351+1)/611 | 86504 | Lelio R Paula | 06/2014 |
7689 | (2^287360+1006)/2 | 86504 | Stefano Morozzi | 01/2020 |
7691 | 25*29^59147+20936 | 86498 | Curtis Bright | 05/2015 |
7692 | 2^287298+1871359 | 86486 | Enzo Doria | 05/2016 |
7693 | 118^41740-117 | 86481 | Norbert Schneider | 08/2018 |
7694 | 3*10^86478+89 | 86479 | Robert Price | 03/2017 |
7695 | 2^287272-79049 | 86478 | Paul Bourdelais | 02/2012 |
7696 | (2^287259+2846)/2 | 86474 | Stefano Morozzi | 01/2020 |
7697 | 906705*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7698 | 811824*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7699 | 779064*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7700 | 768385*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7701 | 700891*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7702 | 652320*2*((10^86453-1)/9)+1 | 86459 | Danilo Nitsche | 01/2016 |
7703 | 377920*2*((10^86453-1)/9)+1 | 86458 | Danilo Nitsche | 01/2016 |
7704 | 255060*2*((10^86453-1)/9)+1 | 86458 | Danilo Nitsche | 01/2016 |
7705 | 187111*2*((10^86453-1)/9)+1 | 86458 | Danilo Nitsche | 01/2016 |
7706 | (10^86453-1)/9 | 86453 | Lew Baxter | 10/2000 |
7707 | (2^287172+658)/2 | 86448 | Stefano Morozzi | 01/2020 |
7708 | (2^287146+2542)/2 | 86440 | Stefano Morozzi | 01/2020 |
7709 | (2^287136+1726)/2 | 86437 | Stefano Morozzi | 01/2020 |
7710 | (2^287128+958)/2 | 86434 | Stefano Morozzi | 01/2020 |
7711 | 87^44563-86^44563 | 86431 | Robert Price | 05/2015 |
7712 | 6^111057+61 | 86420 | Lelio R Paula | 12/2015 |
7713 | 2^287057-109909 | 86413 | Enzo Doria | 04/2014 |
7714 | 5^123608+22 | 86399 | Lelio R Paula | 10/2015 |
7715 | (2^286970+618)/2 | 86387 | Stefano Morozzi | 01/2020 |
7716 | (2^286916+2646)/2 | 86371 | Stefano Morozzi | 01/2020 |
7717 | (2^286907+3690)/2 | 86368 | Stefano Morozzi | 01/2020 |
7718 | (31*10^86364-121)/9 | 86365 | Robert Price | 06/2016 |
7719 | (2^286868+2686)/2 | 86356 | Stefano Morozzi | 01/2020 |
7720 | 2^286859+71663 | 86354 | Enzo Doria | 04/2016 |
7721 | (2^286845+1866)/2 | 86349 | Stefano Morozzi | 01/2020 |
7722 | 2^286740+29835 | 86318 | Lelio R Paula | 10/2013 |
7723 | 2*10^86310+9^86310 | 86311 | Mike Oakes | 05/2011 |
7724 | 11^82878-1860 | 86309 | Enzo Doria | 12/2012 |
7725 | 2223^25780+25780^2223 | 86285 | Norbert Schneider | 12/2019 |
7726 | 2^286547+16095 | 86260 | Lelio R Paula | 10/2013 |
7727 | (58*10^86242+329)/9 | 86243 | Robert Price | 11/2017 |
7728 | 2^286455-110427 | 86232 | Enzo Doria | 05/2014 |
7729 | (37*10^86230+53)/9 | 86231 | Robert Price | 05/2015 |
7730 | 8*3^180700+5 | 86217 | Lelio R Paula | 08/2014 |
7731 | 3^180697-40 | 86215 | Lelio R Paula | 12/2015 |
7732 | (2^286383+3386)/2 | 86210 | Stefano Morozzi | 01/2020 |
7733 | (13*2^286318-1)/13257 | 86188 | Lelio R Paula | 02/2014 |
7734 | (2^286295+1934)/2 | 86184 | Stefano Morozzi | 01/2020 |
7735 | (2^286251+3974)/2 | 86170 | Stefano Morozzi | 01/2020 |
7736 | 2^286165+77 | 86145 | Lelio R Paula | 08/2012 |
7737 | 9*2^286140+7 | 86138 | Lelio R Paula | 08/2014 |
7738 | (2^286128+1762)/2 | 86133 | Stefano Morozzi | 01/2020 |
7739 | (29*2^286108-1)/2002367 | 86123 | Lelio R Paula | 02/2014 |
7740 | (2^286077+1122)/2 | 86118 | Stefano Morozzi | 01/2020 |
7741 | 10^86110+161113 | 86111 | Patrick De Geest | 07/2013 |
7742 | 10^86110-69 | 86110 | Robert Price | 05/2011 |
7743 | 2^286036-869688105 | 86106 | Lelio R Paula | 07/2016 |
7744 | (2^285946+2358)/2 | 86079 | Stefano Morozzi | 01/2020 |
7746 | (77*10^86072+31)/9 | 86073 | Robert Price | 10/2015 |
7745 | (2^285926+3382)/2 | 86073 | Stefano Morozzi | 01/2020 |
7747 | 2^285923+345 | 86072 | Lelio R Paula | 09/2013 |
7748 | 13^77266-90 | 86070 | Lelio R Paula | 11/2015 |
7749 | 2^285910-458903 | 86068 | Enzo Doria | 04/2015 |
7750 | 2^285904-125 | 86066 | Lelio R Paula | 08/2012 |