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*10^99408+529)/9 | 99408 | Robert Price | 01/2017 |
9502 | (2^330216+3510)/2 | 99405 | Stefano Morozzi | 09/2020 |
9503 | 53^57646-2 | 99398 | Lelio R Paula | 09/2013 |
9504 | 2^330165+1870025 | 99390 | Enzo Doria | 05/2016 |
9505 | 7*10^99383+97 | 99384 | Robert Price | 09/2021 |
9506 | 2^330142-155 | 99383 | Lelio R Paula | 12/2012 |
9507 | 2^330088-115259 | 99367 | Enzo Doria | 06/2015 |
9508 | 24216^12671+12671^24216 | 99354 | Hans Havermann | 01/2018 |
9509 | Phi(200548,10)/3208769 | 99354 | Makoto Kamada | 04/2019 |
9510 | (746*10^99303+1)/9 | 99305 | Robert Price | 02/2017 |
9511 | 3^208107+40 | 99293 | Lelio R Paula | 08/2016 |
9512 | 2^329843+211343 | 99293 | Enzo Doria | 08/2023 |
9513 | (3939^27617-1)/3938 | 99291 | Serge Batalov | 06/2022 |
9514 | (5*2^329833-1)/369 | 99288 | Lelio R Paula | 02/2014 |
9515 | 8*3^208093+5 | 99287 | Lelio R Paula | 08/2014 |
9516 | 5^142043+56 | 99284 | Lelio R Paula | 08/2016 |
9517 | (13*10^99282-157)/9 | 99283 | Robert Price | 05/2021 |
9518 | (10^99284+7)/53/67 | 99281 | Alfred Reich | 06/2021 |
9519 | 22*10^99275+7 | 99277 | Robert Price | 04/2016 |
9520 | 2^329778+1876189 | 99274 | Enzo Doria | 07/2016 |
9521 | (149557^19321-1)/(149557^139-1) | 99264 | Ryan Propper | 11/2023 |
9522 | (6580^25997-1)/6579 | 99259 | Serge Batalov | 06/2022 |
9523 | (2249^29611-1)/2248 | 99253 | Serge Batalov | 06/2022 |
9524 | (10^99239-13)/3 | 99239 | Robert Price | 02/2017 |
9525 | (1199^32233-1)/1198 | 99237 | Serge Batalov | 06/2022 |
9527 | 3^207987+62 | 99236 | Lelio R Paula | 08/2016 |
9526 | (62*10^99235+91)/9 | 99236 | Robert Price | 07/2021 |
9528 | 2^329638+49 | 99231 | Lelio R Paula | 12/2012 |
9529 | 7^117413-18 | 99226 | Lelio R Paula | 08/2016 |
9530 | 2^329586-458903 | 99216 | Enzo Doria | 04/2015 |
9531 | (6290^26119-1)/6289 | 99214 | Serge Batalov | 06/2022 |
9533 | 2^329553+101 | 99206 | Lelio R Paula | 12/2012 |
9532 | (21297^22921-1)/21296 | 99206 | Serge Batalov | 06/2022 |
9534 | 2^329551-112897 | 99205 | Enzo Doria | 06/2014 |
9535 | (3^207925+7)/50 | 99204 | Lelio R Paula | 09/2013 |
9536 | 6^127458-17993 | 99182 | Enzo Doria | 03/2013 |
9537 | 27197^4428+4428^27197 | 99166 | Hans Havermann | 10/2017 |
9538 | (46*10^99143-1)/9 | 99144 | Robert Price | 12/2014 |
9539 | (64*10^99135+161)/9 | 99136 | Robert Price | 08/2022 |
9540 | 7*10^99133+69 | 99134 | Robert Price | 10/2017 |
9541 | (21701^22861-1)/21700 | 99132 | Serge Batalov | 06/2022 |
9542 | 18^78968-3065 | 99127 | Enzo Doria | 03/2013 |
9544 | 25927^6654+6654^25927 | 99122 | Hans Havermann | 10/2017 |
9543 | 87915!4+16 | 99122 | Robert Price | 09/2019 |
9545 | 2^329259+189 | 99117 | Lelio R Paula | 12/2012 |
9546 | 8*10^99113+39 | 99114 | Robert Price | 01/2017 |
9547 | 6^127364+77 | 99109 | Lelio R Paula | 08/2016 |
9548 | (82*10^99103+143)/9 | 99104 | Robert Price | 06/2021 |
9549 | (11414^24421-1)/11413 | 99083 | Serge Batalov | 06/2022 |
9550 | (2^329135+1110)/2 | 99080 | Stefano Morozzi | 09/2020 |
9551 | (22089^22807-1)/22088 | 99074 | Serge Batalov | 06/2022 |
9552 | (59*10^99069+391)/9 | 99070 | Robert Price | 03/2021 |
9553 | (797*10^99061-41)/9 | 99063 | Robert Price | 04/2022 |
9554 | 2^329058-287681 | 99057 | Harrison Mesh | 06/2022 |
9555 | 6^127285+13 | 99047 | Lelio R Paula | 08/2016 |
9556 | (228581#+235862)/(2^3*11*71*151) | 99044 | Makoto Morimoto | 10/2017 |
9557 | 2^329007+73283 | 99041 | Enzo Doria | 04/2016 |
9558 | (145413^19321-1)/(145413^139-1) | 99030 | Ryan Propper | 11/2023 |
9559 | 5*3^207543-8 | 99024 | Lelio R Paula | 08/2014 |
9560 | (2^328930+570)/2 | 99018 | Stefano Morozzi | 09/2020 |
9561 | 2^328885+149 | 99005 | Lelio R Paula | 12/2012 |
9563 | 23392^17075+17075^23392 | 99004 | Hans Havermann | 09/2017 |
9562 | (7424^25579-1)/7423 | 99004 | Serge Batalov | 06/2022 |
9564 | (2^328879-9)/119 | 99001 | Lelio R Paula | 10/2013 |
9565 | 5^141618+54 | 98987 | Lelio R Paula | 08/2016 |
9566 | 2^328824+24391 | 98986 | Enzo Doria | 10/2023 |
9567 | 8*10^98982+51 | 98983 | Robert Price | 05/2017 |
9568 | (21*2^328809-1)/7709 | 98979 | Lelio R Paula | 03/2014 |
9569 | 2^328794-2797601 | 98977 | Enzo Doria | 04/2016 |
9570 | (88331^20011-1)/88330 | 98972 | Serge Batalov | 06/2022 |
9571 | 6^127176+5 | 98963 | Lelio R Paula | 11/2014 |
9572 | 2*10^98961-21 | 98962 | Robert Price | 09/2020 |
9573 | 24262^11989+11989^24262 | 98960 | Hans Havermann | 09/2017 |
9574 | (60138^20707-1)/60137 | 98958 | Serge Batalov | 06/2022 |
9575 | 2^328700-109709 | 98949 | Enzo Doria | 04/2014 |
9576 | (2^328698+1810)/2 | 98948 | Stefano Morozzi | 01/2021 |
9577 | (1014^32917-1)/1013 | 98947 | Serge Batalov | 06/2022 |
9578 | 87771!4-128 | 98944 | Robert Price | 11/2019 |
9579 | (4576^27031-1)/4575 | 98943 | Serge Batalov | 06/2022 |
9580 | (2^328658+3042)/2 | 98936 | Stefano Morozzi | 09/2020 |
9581 | (1933^30103-1)/1932 | 98923 | Serge Batalov | 06/2022 |
9582 | 46711!2-2^46711 | 98916 | Robert Price | 09/2015 |
9583 | 9*7^117037+4 | 98909 | Lelio R Paula | 11/2014 |
9584 | 127^47012+126 | 98905 | Norbert Schneider | 01/2019 |
9585 | (93826^19891-1)/93825 | 98900 | Serge Batalov | 06/2022 |
9586 | (2^328525+1250)/2 | 98896 | Stefano Morozzi | 09/2020 |
9587 | (10^98890+149)/3 | 98890 | Serge Batalov | 12/2014 |
9588 | 22775^21978+21978^22775 | 98889 | Hans Havermann | 08/2017 |
9589 | 2^328497-216933 | 98888 | Enzo Doria | 05/2014 |
9590 | 2^328487+32769 | 98885 | Lelio R Paula | 08/2016 |
9591 | 3^207250-112358 | 98884 | Enzo Doria | 06/2014 |
9592 | 13^88754-90 | 98867 | Lelio R Paula | 08/2016 |
9593 | 2^328406+709 | 98861 | Lelio R Paula | 02/2013 |
9594 | 2*10^98860-21 | 98861 | Robert Price | 09/2020 |
9595 | (17*2^328413-1)/16173 | 98860 | Lelio R Paula | 03/2014 |
9596 | 7^116976+76 | 98857 | Lelio R Paula | 08/2016 |
9597 | (2^328378+2710)/2 | 98852 | Stefano Morozzi | 09/2020 |
9598 | (25411^22441-1)/25410 | 98849 | Serge Batalov | 06/2022 |
9599 | (18668^23143-1)/18667 | 98842 | Serge Batalov | 06/2022 |
9601 | 2^328335+1155 | 98839 | Lelio R Paula | 08/2016 |
9600 | 2^328333-349003 | 98839 | Harrison Mesh | 06/2022 |
9603 | (3*2^328311+1)/5 | 98832 | Lelio R Paula | 11/2013 |
9602 | 287^40210+40210^287 | 98832 | Norbert Schneider | 11/2014 |
9604 | (2^328284+9)/1325 | 98821 | Lelio R Paula | 10/2013 |
9605 | (21*2^328258+1)/2125 | 98814 | Lelio R Paula | 11/2013 |
9606 | (28623^22171-1)/28622 | 98806 | Serge Batalov | 06/2022 |
9607 | 24746^9825+9825^24746 | 98795 | Hans Havermann | 07/2020 |
9608 | 26620^5143+5143^26620 | 98793 | Hans Havermann | 07/2020 |
9609 | 65^54491+48 | 98788 | Lelio R Paula | 04/2016 |
9610 | 3^207042+98 | 98785 | Lelio R Paula | 08/2016 |
9611 | (2^328121+3110)/2 | 98774 | Stefano Morozzi | 09/2020 |
9612 | 10^98754+73383 | 98755 | Patrick De Geest | 02/2021 |
9615 | 3^206977-112756 | 98754 | Enzo Doria | 07/2014 |
9614 | 10^98754-40000000001 | 98754 | Robert Price | 09/2020 |
9613 | 10^98754-169343 | 98754 | Patrick De Geest | 02/2021 |
9616 | 7^116846-50 | 98747 | Lelio R Paula | 08/2016 |
9617 | (19152^23059-1)/19151 | 98740 | Serge Batalov | 06/2022 |
9618 | 23267^17512+17512^23267 | 98730 | Hans Havermann | 04/2020 |
9619 | 2^327967-28635 | 98728 | Lelio R Paula | 08/2016 |
9620 | 707^34642+34642^707 | 98710 | Norbert Schneider | 12/2017 |
9621 | (10771^24481-1)/10770 | 98710 | Serge Batalov | 06/2022 |
9622 | 6^126850-47 | 98709 | Lelio R Paula | 08/2016 |
9623 | (2^327886+2170)/2 | 98704 | Stefano Morozzi | 09/2020 |
9624 | (22*10^98697+47)/3 | 98698 | Robert Price | 06/2021 |
9625 | (139706^19321-1)/(139706^139-1) | 98696 | Ryan Propper | 11/2023 |
9626 | 8*6^126830-7 | 98694 | Lelio R Paula | 11/2014 |
9627 | 11^94770-270 | 98693 | Enzo Doria | 08/2012 |
9628 | (2^327846+3250)/2 | 98692 | Stefano Morozzi | 01/2021 |
9629 | 2^327832+43 | 98688 | Lelio R Paula | 12/2012 |
9630 | 2^327795+761 | 98677 | Lelio R Paula | 02/2013 |
9631 | (2^327796+1522)/2 | 98677 | Stefano Morozzi | 09/2020 |
9632 | 10^98669+147411 | 98670 | Patrick De Geest | 11/2016 |
9634 | 10^98669-9^(98669-1) | 98669 | Robert Price | 05/2016 |
9633 | 10^98669-105249 | 98669 | Patrick De Geest | 11/2016 |
9635 | (7911^25309-1)/7910 | 98657 | Serge Batalov | 06/2022 |
9636 | 2^327671-109801 | 98639 | Enzo Doria | 04/2014 |
9637 | 24398^11037+11037^24398 | 98638 | Hans Havermann | 10/2019 |
9638 | 5^141090-14 | 98618 | Lelio R Paula | 08/2016 |
9639 | 20^75799+47 | 98617 | Lelio R Paula | 08/2016 |
9640 | (5452^26393-1)/5451 | 98616 | Serge Batalov | 06/2022 |
9641 | 28101^3226+3226^28101 | 98597 | Hans Havermann | 07/2020 |
9642 | 2^327515-1017 | 98592 | Lelio R Paula | 08/2016 |
9643 | 6^126692+97681 | 98586 | Mike Oakes | 04/2011 |
9644 | 2^327467-741 | 98578 | Lelio R Paula | 08/2016 |
9645 | (2947^28411-1)/2946 | 98566 | Serge Batalov | 06/2022 |
9646 | 22713^21800+21800^22713 | 98540 | Hans Havermann | 05/2020 |
9647 | (2^327329+9)/19 | 98535 | Lelio R Paula | 10/2013 |
9648 | 7^116577-66 | 98519 | Lelio R Paula | 08/2016 |
9649 | (3*2^327250+1)/7 | 98512 | Lelio R Paula | 11/2013 |
9650 | 23220^17479+17479^23220 | 98512 | Hans Havermann | 03/2020 |
9651 | (2^327240+3798)/2 | 98509 | Stefano Morozzi | 02/2021 |
9652 | (28514^22111-1)/28513 | 98502 | Serge Batalov | 06/2022 |
9653 | (19*2^327192+1)/275 | 98494 | Lelio R Paula | 11/2013 |
9654 | 2^327162-66945 | 98486 | Lelio R Paula | 08/2016 |
9655 | 30139^1852+1852^30139 | 98484 | Hans Havermann | 02/2020 |
9656 | 23182^17703+17703^23182 | 98479 | Hans Havermann | 01/2020 |
9657 | 4*10^98472-3 | 98473 | Erik Branger | 11/2013 |
9658 | 26167^5794+5794^26167 | 98466 | Hans Havermann | 01/2020 |
9659 | (6883*2^327047-1)/77 | 98453 | Lelio R Paula | 05/2015 |
9660 | (71*10^98446+1)/9 | 98447 | Serge Batalov | 11/2010 |
9661 | (2009^29803-1)/2008 | 98436 | Serge Batalov | 06/2022 |
9662 | 13^88363+94 | 98432 | Lelio R Paula | 08/2016 |
9663 | 48^58543-47^58543 | 98425 | Jean-Louis Charton | 11/2005 |
9664 | 6^126481+41 | 98422 | Lelio R Paula | 08/2016 |
9665 | 78*10^98413-7 | 98415 | Robert Price | 05/2017 |
9666 | 22634^22263+22263^22634 | 98404 | Hans Havermann | 07/2020 |
9667 | (47*10^98386+241)/9 | 98387 | Robert Price | 06/2022 |
9668 | 7^116416+46 | 98383 | Lelio R Paula | 08/2016 |
9669 | 7^116407-24 | 98376 | Lelio R Paula | 07/2016 |
9670 | 439^37228+37228^439 | 98374 | Norbert Schneider | 08/2018 |
9671 | 2^326782-1860671 | 98372 | Enzo Doria | 03/2016 |
9672 | 17^79948+58 | 98372 | Lelio R Paula | 08/2016 |
9673 | 22499^22602-22602^22499 | 98368 | Norbert Schneider | 08/2020 |
9674 | 3^206143-98 | 98356 | Lelio R Paula | 08/2016 |
9675 | 8*5^140705+7 | 98350 | Lelio R Paula | 11/2014 |
9676 | (2^326632+1662)/2 | 98326 | Stefano Morozzi | 09/2020 |
9677 | (638^35053-1)/637 | 98315 | Serge Batalov | 06/2022 |
9678 | 2^326585+11 | 98312 | Lelio R Paula | 12/2012 |
9679 | 2^326582-459161 | 98311 | Enzo Doria | 05/2015 |
9680 | (1266^31687-1)/1265 | 98304 | Serge Batalov | 06/2022 |
9682 | (2^326548+9)/5 | 98301 | Lelio R Paula | 10/2013 |
9681 | (2^326549+18)/10 | 98301 | Stefano Morozzi | 03/2020 |
9683 | 2^326538+71659 | 98298 | Enzo Doria | 04/2016 |
9684 | (2^326520+2082)/2 | 98293 | Stefano Morozzi | 09/2020 |
9685 | (2^326521+1002)/2 | 98293 | Stefano Morozzi | 09/2020 |
9686 | 15^83575-58 | 98292 | Lelio R Paula | 08/2016 |
9687 | 10^98288+3 | 98289 | Robert Price | 03/2011 |
9688 | 10^98288-117017 | 98288 | Patrick De Geest | 11/2011 |
9689 | (44*10^98284-719)/9 | 98285 | Robert Price | 12/2017 |
9690 | (5985^26021-1)/5984 | 98280 | Serge Batalov | 06/2022 |
9691 | 23466^15419+15419^23466 | 98277 | Hans Havermann | 02/2020 |
9692 | 2^326421-2797573 | 98263 | Enzo Doria | 04/2016 |
9693 | 2^326401+16095 | 98257 | Lelio R Paula | 07/2016 |
9694 | (6494^25771-1)/6493 | 98249 | Serge Batalov | 06/2022 |
9695 | 7^116240-116240^7 | 98235 | Norbert Schneider | 05/2017 |
9696 | 17^79826-78 | 98222 | Lelio R Paula | 08/2016 |
9697 | 2^326268-809 | 98217 | Yves Minet | 04/2011 |
9698 | 22659^21590+21590^22659 | 98210 | Hans Havermann | 10/2019 |
9699 | 22268^22587-22587^22268 | 98202 | Norbert Schneider | 08/2020 |
9700 | (418*10^98199+23)/9 | 98201 | Robert Price | 08/2017 |
9701 | 12*10^98198-7 | 98200 | Robert Price | 11/2017 |
9702 | 15^83496-76 | 98199 | Lelio R Paula | 08/2016 |
9703 | (10223*2^326179+1)/6125 | 98190 | Lelio R Paula | 07/2014 |
9704 | 9*5^140471-4 | 98186 | Lelio R Paula | 11/2014 |
9705 | (35103^21601-1)/35102 | 98180 | Serge Batalov | 06/2022 |
9707 | 22774^20455+20455^22774 | 98175 | Hans Havermann | 07/2020 |
9706 | (4516^26863-1)/4515 | 98175 | Serge Batalov | 06/2022 |
9708 | 2^326069-2383 | 98157 | Yves Minet | 10/2013 |
9709 | (9661^24631-1)/9660 | 98152 | Serge Batalov | 06/2022 |
9710 | (4175^27109-1)/4174 | 98149 | Serge Batalov | 06/2022 |
9711 | (57801^20611-1)/57800 | 98144 | Serge Batalov | 06/2022 |
9712 | 2^325973+1185 | 98128 | Lelio R Paula | 07/2016 |
9713 | (25*2^325949-1)/17 | 98121 | Lelio R Paula | 02/2014 |
9714 | 2^325945-1847269 | 98120 | Enzo Doria | 07/2015 |
9715 | 8*10^98111-67 | 98112 | Robert Price | 10/2017 |
9716 | (2^325902+1138)/2 | 98106 | Stefano Morozzi | 09/2020 |
9717 | (215^42061-1)/214 | 98103 | Serge Batalov | 06/2022 |
9718 | (6775^25609-1)/6774 | 98102 | Serge Batalov | 06/2022 |
9719 | (6718^25633-1)/6717 | 98100 | Serge Batalov | 06/2022 |
9720 | 2^325866+15 | 98096 | Lelio R Paula | 12/2012 |
9721 | 22591^21952+21952^22591 | 98079 | Hans Havermann | 02/2020 |
9722 | 2^325765+1974305 | 98066 | Enzo Doria | 06/2016 |
9723 | 2^325733-58695 | 98056 | Lelio R Paula | 07/2016 |
9724 | (7*10^98055-529)/9 | 98055 | Robert Price | 08/2021 |
9725 | 2^325717-3645 | 98051 | Lelio R Paula | 07/2016 |
9726 | (2^325704+718)/2 | 98047 | Stefano Morozzi | 09/2020 |
9727 | (15679^23371-1)/15678 | 98045 | Serge Batalov | 06/2022 |
9728 | (23342^22441-1)/23341 | 98022 | Serge Batalov | 06/2022 |
9729 | 23968^12293+12293^23968 | 98021 | Hans Havermann | 11/2019 |
9731 | (23*2^325583-1)/21 | 98011 | Lelio R Paula | 02/2014 |
9730 | (128678^19321-1)/(128678^139-1) | 98011 | Ryan Propper | 11/2023 |
9732 | (7*10^98007-13)/3 | 98008 | Robert Price | 06/2016 |
9733 | (2^325544+1110)/2 | 97999 | Stefano Morozzi | 09/2020 |
9734 | 2^325512-459959 | 97989 | Enzo Doria | 05/2015 |
9735 | (2^325492+682)/2 | 97983 | Stefano Morozzi | 09/2020 |
9736 | (3^205354-8)/17 | 97978 | Lelio R Paula | 09/2013 |
9737 | 2^325433+76817 | 97966 | Enzo Doria | 05/2016 |
9738 | 5*10^97963+81 | 97964 | Robert Price | 08/2017 |
9739 | (9996^24649-1)/(9996^157-1) | 97964 | Ryan Propper | 11/2023 |
9740 | (13208^23773-1)/13207 | 97961 | Serge Batalov | 06/2022 |
9741 | 2^325390-109061 | 97953 | Enzo Doria | 04/2014 |
9742 | (55*10^97940+323)/9 | 97941 | Robert Price | 10/2021 |
9743 | 13^87921+24 | 97940 | Lelio R Paula | 08/2016 |
9744 | 10^97930+188277 | 97931 | Patrick De Geest | 09/2022 |
9746 | 10^97930-7000001 | 97930 | Robert Price | 06/2021 |
9745 | 10^97930-66951 | 97930 | Patrick De Geest | 09/2022 |
9747 | 961^32831-960^32831 | 97926 | Jean-Louis Charton | 02/2011 |
9748 | (17968^23017-1)/17967 | 97922 | Serge Batalov | 06/2022 |
9749 | (3337^27793-1)/3336 | 97922 | Serge Batalov | 06/2022 |
9750 | (18703^22921-1)/18702 | 97913 | Serge Batalov | 06/2022 |