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 | 8*6^126830-7 | 98694 | Lelio R Paula | 11/2014 |
9502 | 11^94770-270 | 98693 | Enzo Doria | 08/2012 |
9503 | (2^327846+3250)/2 | 98692 | Stefano Morozzi | 01/2021 |
9504 | 2^327832+43 | 98688 | Lelio R Paula | 12/2012 |
9505 | 2^327795+761 | 98677 | Lelio R Paula | 02/2013 |
9506 | (2^327796+1522)/2 | 98677 | Stefano Morozzi | 09/2020 |
9507 | 10^98669+147411 | 98670 | Patrick De Geest | 11/2016 |
9509 | 10^98669-9^(98669-1) | 98669 | Robert Price | 05/2016 |
9508 | 10^98669-105249 | 98669 | Patrick De Geest | 11/2016 |
9510 | (7911^25309-1)/7910 | 98657 | Serge Batalov | 06/2022 |
9511 | 2^327671-109801 | 98639 | Enzo Doria | 04/2014 |
9512 | 24398^11037+11037^24398 | 98638 | Hans Havermann | 10/2019 |
9513 | 5^141090-14 | 98618 | Lelio R Paula | 08/2016 |
9514 | 20^75799+47 | 98617 | Lelio R Paula | 08/2016 |
9515 | (5452^26393-1)/5451 | 98616 | Serge Batalov | 06/2022 |
9516 | 28101^3226+3226^28101 | 98597 | Hans Havermann | 07/2020 |
9517 | 2^327515-1017 | 98592 | Lelio R Paula | 08/2016 |
9518 | 6^126692+97681 | 98586 | Mike Oakes | 04/2011 |
9519 | 2^327467-741 | 98578 | Lelio R Paula | 08/2016 |
9520 | (2947^28411-1)/2946 | 98566 | Serge Batalov | 06/2022 |
9521 | 22713^21800+21800^22713 | 98540 | Hans Havermann | 05/2020 |
9522 | (2^327329+9)/19 | 98535 | Lelio R Paula | 10/2013 |
9523 | 7^116577-66 | 98519 | Lelio R Paula | 08/2016 |
9524 | (3*2^327250+1)/7 | 98512 | Lelio R Paula | 11/2013 |
9525 | 23220^17479+17479^23220 | 98512 | Hans Havermann | 03/2020 |
9526 | (2^327240+3798)/2 | 98509 | Stefano Morozzi | 02/2021 |
9527 | (28514^22111-1)/28513 | 98502 | Serge Batalov | 06/2022 |
9528 | (19*2^327192+1)/275 | 98494 | Lelio R Paula | 11/2013 |
9529 | 2^327162-66945 | 98486 | Lelio R Paula | 08/2016 |
9530 | 30139^1852+1852^30139 | 98484 | Hans Havermann | 02/2020 |
9531 | 23182^17703+17703^23182 | 98479 | Hans Havermann | 01/2020 |
9532 | 4*10^98472-3 | 98473 | Erik Branger | 11/2013 |
9533 | 26167^5794+5794^26167 | 98466 | Hans Havermann | 01/2020 |
9534 | (6883*2^327047-1)/77 | 98453 | Lelio R Paula | 05/2015 |
9535 | (71*10^98446+1)/9 | 98447 | Serge Batalov | 11/2010 |
9536 | (2009^29803-1)/2008 | 98436 | Serge Batalov | 06/2022 |
9537 | 13^88363+94 | 98432 | Lelio R Paula | 08/2016 |
9538 | 48^58543-47^58543 | 98425 | Jean-Louis Charton | 11/2005 |
9539 | 6^126481+41 | 98422 | Lelio R Paula | 08/2016 |
9540 | 78*10^98413-7 | 98415 | Robert Price | 05/2017 |
9541 | 22634^22263+22263^22634 | 98404 | Hans Havermann | 07/2020 |
9542 | (47*10^98386+241)/9 | 98387 | Robert Price | 06/2022 |
9543 | 7^116416+46 | 98383 | Lelio R Paula | 08/2016 |
9544 | 7^116407-24 | 98376 | Lelio R Paula | 07/2016 |
9545 | 439^37228+37228^439 | 98374 | Norbert Schneider | 08/2018 |
9546 | 2^326782-1860671 | 98372 | Enzo Doria | 03/2016 |
9547 | 17^79948+58 | 98372 | Lelio R Paula | 08/2016 |
9548 | 22499^22602-22602^22499 | 98368 | Norbert Schneider | 08/2020 |
9549 | 3^206143-98 | 98356 | Lelio R Paula | 08/2016 |
9550 | 8*5^140705+7 | 98350 | Lelio R Paula | 11/2014 |
9551 | (2^326632+1662)/2 | 98326 | Stefano Morozzi | 09/2020 |
9552 | (638^35053-1)/637 | 98315 | Serge Batalov | 06/2022 |
9553 | 2^326585+11 | 98312 | Lelio R Paula | 12/2012 |
9554 | 2^326582-459161 | 98311 | Enzo Doria | 05/2015 |
9555 | (1266^31687-1)/1265 | 98304 | Serge Batalov | 06/2022 |
9557 | (2^326548+9)/5 | 98301 | Lelio R Paula | 10/2013 |
9556 | (2^326549+18)/10 | 98301 | Stefano Morozzi | 03/2020 |
9558 | 2^326538+71659 | 98298 | Enzo Doria | 04/2016 |
9559 | (2^326520+2082)/2 | 98293 | Stefano Morozzi | 09/2020 |
9560 | (2^326521+1002)/2 | 98293 | Stefano Morozzi | 09/2020 |
9561 | 15^83575-58 | 98292 | Lelio R Paula | 08/2016 |
9562 | 10^98288+3 | 98289 | Robert Price | 03/2011 |
9563 | 10^98288-117017 | 98288 | Patrick De Geest | 11/2011 |
9564 | (44*10^98284-719)/9 | 98285 | Robert Price | 12/2017 |
9565 | (5985^26021-1)/5984 | 98280 | Serge Batalov | 06/2022 |
9566 | 23466^15419+15419^23466 | 98277 | Hans Havermann | 02/2020 |
9567 | 2^326421-2797573 | 98263 | Enzo Doria | 04/2016 |
9568 | 2^326401+16095 | 98257 | Lelio R Paula | 07/2016 |
9569 | (6494^25771-1)/6493 | 98249 | Serge Batalov | 06/2022 |
9570 | 7^116240-116240^7 | 98235 | Norbert Schneider | 05/2017 |
9571 | 17^79826-78 | 98222 | Lelio R Paula | 08/2016 |
9572 | 2^326268-809 | 98217 | Yves Minet | 04/2011 |
9573 | 22659^21590+21590^22659 | 98210 | Hans Havermann | 10/2019 |
9574 | 22268^22587-22587^22268 | 98202 | Norbert Schneider | 08/2020 |
9575 | (418*10^98199+23)/9 | 98201 | Robert Price | 08/2017 |
9576 | 12*10^98198-7 | 98200 | Robert Price | 11/2017 |
9577 | 15^83496-76 | 98199 | Lelio R Paula | 08/2016 |
9578 | (10223*2^326179+1)/6125 | 98190 | Lelio R Paula | 07/2014 |
9579 | 9*5^140471-4 | 98186 | Lelio R Paula | 11/2014 |
9580 | (35103^21601-1)/35102 | 98180 | Serge Batalov | 06/2022 |
9582 | 22774^20455+20455^22774 | 98175 | Hans Havermann | 07/2020 |
9581 | (4516^26863-1)/4515 | 98175 | Serge Batalov | 06/2022 |
9583 | 2^326069-2383 | 98157 | Yves Minet | 10/2013 |
9584 | (9661^24631-1)/9660 | 98152 | Serge Batalov | 06/2022 |
9585 | (4175^27109-1)/4174 | 98149 | Serge Batalov | 06/2022 |
9586 | (57801^20611-1)/57800 | 98144 | Serge Batalov | 06/2022 |
9587 | 2^325973+1185 | 98128 | Lelio R Paula | 07/2016 |
9588 | (25*2^325949-1)/17 | 98121 | Lelio R Paula | 02/2014 |
9589 | 2^325945-1847269 | 98120 | Enzo Doria | 07/2015 |
9590 | 8*10^98111-67 | 98112 | Robert Price | 10/2017 |
9591 | (2^325902+1138)/2 | 98106 | Stefano Morozzi | 09/2020 |
9592 | (215^42061-1)/214 | 98103 | Serge Batalov | 06/2022 |
9593 | (6775^25609-1)/6774 | 98102 | Serge Batalov | 06/2022 |
9594 | (6718^25633-1)/6717 | 98100 | Serge Batalov | 06/2022 |
9595 | 2^325866+15 | 98096 | Lelio R Paula | 12/2012 |
9596 | 22591^21952+21952^22591 | 98079 | Hans Havermann | 02/2020 |
9597 | 2^325765+1974305 | 98066 | Enzo Doria | 06/2016 |
9598 | 2^325733-58695 | 98056 | Lelio R Paula | 07/2016 |
9599 | (7*10^98055-529)/9 | 98055 | Robert Price | 08/2021 |
9600 | 2^325717-3645 | 98051 | Lelio R Paula | 07/2016 |
9601 | (2^325704+718)/2 | 98047 | Stefano Morozzi | 09/2020 |
9602 | (15679^23371-1)/15678 | 98045 | Serge Batalov | 06/2022 |
9603 | (23342^22441-1)/23341 | 98022 | Serge Batalov | 06/2022 |
9604 | 23968^12293+12293^23968 | 98021 | Hans Havermann | 11/2019 |
9606 | (23*2^325583-1)/21 | 98011 | Lelio R Paula | 02/2014 |
9605 | (128678^19321-1)/(128678^139-1) | 98011 | Ryan Propper | 11/2023 |
9607 | (7*10^98007-13)/3 | 98008 | Robert Price | 06/2016 |
9608 | (2^325544+1110)/2 | 97999 | Stefano Morozzi | 09/2020 |
9609 | 2^325512-459959 | 97989 | Enzo Doria | 05/2015 |
9610 | (2^325492+682)/2 | 97983 | Stefano Morozzi | 09/2020 |
9611 | (3^205354-8)/17 | 97978 | Lelio R Paula | 09/2013 |
9612 | 2^325433+76817 | 97966 | Enzo Doria | 05/2016 |
9613 | 5*10^97963+81 | 97964 | Robert Price | 08/2017 |
9614 | (9996^24649-1)/(9996^157-1) | 97964 | Ryan Propper | 11/2023 |
9615 | (13208^23773-1)/13207 | 97961 | Serge Batalov | 06/2022 |
9616 | 2^325390-109061 | 97953 | Enzo Doria | 04/2014 |
9617 | (55*10^97940+323)/9 | 97941 | Robert Price | 10/2021 |
9618 | 13^87921+24 | 97940 | Lelio R Paula | 08/2016 |
9619 | 10^97930+188277 | 97931 | Patrick De Geest | 09/2022 |
9621 | 10^97930-7000001 | 97930 | Robert Price | 06/2021 |
9620 | 10^97930-66951 | 97930 | Patrick De Geest | 09/2022 |
9622 | 961^32831-960^32831 | 97926 | Jean-Louis Charton | 02/2011 |
9623 | (17968^23017-1)/17967 | 97922 | Serge Batalov | 06/2022 |
9624 | (3337^27793-1)/3336 | 97922 | Serge Batalov | 06/2022 |
9626 | 22600^21499+21499^22600 | 97913 | Hans Havermann | 05/2020 |
9625 | (18703^22921-1)/18702 | 97913 | Serge Batalov | 06/2022 |
9627 | 2^325215+12909 | 97900 | Lelio R Paula | 07/2016 |
9628 | 22619^21286+21286^22619 | 97898 | Hans Havermann | 04/2020 |
9629 | (126852^19321-1)/(126852^139-1) | 97892 | Ryan Propper | 11/2023 |
9630 | (85*10^97876-49)/9 | 97877 | Robert Price | 12/2020 |
9631 | 7*10^97874-69 | 97875 | Robert Price | 01/2017 |
9632 | 5*10^97869-63 | 97870 | Robert Price | 04/2021 |
9633 | (643*32^65022-23)/31 | 97870 | Eric Chen | 04/2023 |
9635 | (17^79537+5^79537)/22 | 97865 | Robert Price | 06/2013 |
9634 | 15^83212+22 | 97865 | Lelio R Paula | 08/2016 |
9636 | 2^325087-109057 | 97861 | Enzo Doria | 04/2014 |
9637 | 2^325074+1870657 | 97858 | Enzo Doria | 05/2016 |
9639 | 2^325066-2111 | 97855 | Yves Minet | 08/2013 |
9638 | CopelandErdos(97855) | 97855 | Eric W. Weisstein | 10/2015 |
9640 | (8694^24841-1)/8693 | 97851 | Serge Batalov | 06/2022 |
9641 | (21159^22621-1)/21158 | 97843 | Serge Batalov | 06/2022 |
9642 | 26660^4677+4677^26660 | 97842 | Hans Havermann | 12/2019 |
9643 | 18^77944-89 | 97841 | Lelio R Paula | 07/2016 |
9644 | 119^47136+118 | 97833 | Norbert Schneider | 09/2018 |
9645 | 6^125720-17663 | 97830 | Enzo Doria | 02/2013 |
9646 | (73640^20101-1)/73639 | 97829 | Serge Batalov | 06/2022 |
9647 | 2^324972-2069 | 97827 | Yves Minet | 05/2013 |
9648 | 2^324958-4691 | 97823 | Yves Minet | 05/2019 |
9649 | (8087^25033-1)/8086 | 97820 | Serge Batalov | 06/2022 |
9650 | 2^324944-2788319 | 97818 | Enzo Doria | 03/2016 |
9651 | 22845^19112+19112^22845 | 97807 | Hans Havermann | 10/2019 |
9652 | (2^324898+1270)/2 | 97804 | Stefano Morozzi | 09/2020 |
9654 | 4*10^97799-9 | 97800 | Erik Branger | 11/2013 |
9653 | 2^324882+577294575 | 97800 | Lelio R Paula | 09/2016 |
9655 | 2^324863-1867921 | 97794 | Enzo Doria | 04/2016 |
9656 | 6^125662-18473 | 97785 | Enzo Doria | 03/2013 |
9657 | 12^90604+11 | 97779 | Lelio R Paula | 07/2016 |
9658 | (29*10^97751+49)/3 | 97752 | Robert Price | 09/2021 |
9659 | (37108^21391-1)/37107 | 97741 | Serge Batalov | 06/2022 |
9660 | 5^139830+24 | 97737 | Lelio R Paula | 04/2016 |
9661 | (2^324672+1198)/2 | 97736 | Stefano Morozzi | 09/2020 |
9662 | (101*10^97724+7)/9 | 97726 | Robert Price | 02/2022 |
9663 | 2^324612+1155 | 97718 | Lelio R Paula | 07/2016 |
9664 | (2^324597+3542)/2 | 97714 | Stefano Morozzi | 09/2020 |
9665 | 19^76404-44 | 97702 | Lelio R Paula | 07/2016 |
9666 | (5939^25889-1)/5938 | 97694 | Serge Batalov | 06/2022 |
9668 | 2^324505-2983 | 97686 | Yves Minet | 04/2011 |
9667 | (7740^25121-1)/7739 | 97686 | Serge Batalov | 06/2022 |
9669 | (144^45259+1)/145 | 97684 | Paul Bourdelais | 08/2019 |
9670 | (31*2^324496-1)/45 | 97683 | Lelio R Paula | 02/2014 |
9671 | (11234^24109-1)/11233 | 97651 | Serge Batalov | 06/2022 |
9672 | 6^125488-17843 | 97649 | Enzo Doria | 02/2013 |
9673 | (2^324350+3198)/2 | 97639 | Stefano Morozzi | 09/2020 |
9674 | (75638^20011-1)/75637 | 97624 | Serge Batalov | 06/2022 |
9675 | 7^115515-2 | 97622 | Lelio R Paula | 09/2013 |
9676 | (15672^23269-1)/15671 | 97613 | Serge Batalov | 06/2022 |
9677 | (89^50069-1)/88 | 97603 | Paul Bourdelais | 08/2017 |
9678 | 6^125418-5 | 97595 | Lelio R Paula | 11/2014 |
9679 | (2^324193+3522)/2 | 97592 | Stefano Morozzi | 09/2020 |
9680 | (2422^28837-1)/2421 | 97587 | Serge Batalov | 06/2022 |
9681 | (2^324172+666)/2 | 97586 | Stefano Morozzi | 09/2020 |
9682 | 2^324091+111546435 | 97562 | Paul Bourdelais | 10/2021 |
9683 | 2^324076+1155 | 97557 | Lelio R Paula | 07/2016 |
9684 | (2^324074+9)/13 | 97555 | Lelio R Paula | 10/2013 |
9685 | (11*10^97539-71)/3 | 97540 | Robert Price | 04/2022 |
9686 | (55459*2^323993+1)/21 | 97536 | Lelio R Paula | 07/2014 |
9687 | 13^87557-12^87557 | 97534 | Jean-Louis Charton | 12/2008 |
9688 | 2^323946+73939 | 97518 | Enzo Doria | 04/2016 |
9689 | 2^323937+1866947 | 97515 | Enzo Doria | 03/2016 |
9690 | (98339^19531-1)/98338 | 97508 | Serge Batalov | 06/2022 |
9692 | 6^125290-18533 | 97495 | Enzo Doria | 03/2013 |
9691 | (2^323870+1650)/2 | 97495 | Stefano Morozzi | 09/2020 |
9693 | 2^323845+101 | 97488 | Lelio R Paula | 12/2012 |
9694 | 18^77662+65 | 97487 | Lelio R Paula | 07/2016 |
9695 | 22634^20265+20265^22634 | 97479 | Hans Havermann | 04/2020 |
9696 | (5533^26041-1)/5532 | 97467 | Serge Batalov | 06/2022 |
9697 | 2^323772+76771 | 97466 | Enzo Doria | 05/2016 |
9698 | 2^323765-112213 | 97463 | Enzo Doria | 06/2014 |
9699 | 22730^19389+19389^22730 | 97457 | Hans Havermann | 02/2020 |
9700 | 2^323720-6555 | 97450 | Lelio R Paula | 07/2016 |
9701 | 2^323694-1860401 | 97442 | Enzo Doria | 03/2016 |
9702 | 20^74886+3 | 97429 | Lelio R Paula | 06/2016 |
9703 | 24476^9561+9561^24476 | 97427 | Hans Havermann | 01/2020 |
9704 | (2^323623+1766)/2 | 97420 | Stefano Morozzi | 09/2020 |
9705 | (3250^27739-1)/3249 | 97413 | Serge Batalov | 06/2022 |
9707 | 3*10^97399+73 | 97400 | Robert Price | 04/2016 |
9706 | 7^115252-50 | 97400 | Lelio R Paula | 07/2016 |
9708 | (4715^26513-1)/4714 | 97392 | Serge Batalov | 06/2022 |
9709 | 2^323521+213317 | 97390 | Enzo Doria | 09/2023 |
9710 | 224737#+393581 | 97389 | Nicolas PALAU | 12/2016 |
9711 | (119374^19321-1)/(119374^139-1) | 97386 | Ryan Propper | 11/2023 |
9712 | (2^323468+3186)/2 | 97374 | Stefano Morozzi | 09/2020 |
9713 | (2^323456+538)/2 | 97370 | Stefano Morozzi | 09/2020 |
9714 | 23277^15236+15236^23277 | 97365 | Hans Havermann | 07/2020 |
9716 | (26*10^97353+31)/3 | 97354 | Robert Price | 07/2016 |
9715 | (2^323403+3510)/2 | 97354 | Stefano Morozzi | 09/2020 |
9717 | 23756^12519+12519^23756 | 97342 | Hans Havermann | 07/2020 |
9718 | 10^97335+271641 | 97336 | Patrick De Geest | 09/2022 |
9720 | 10^97335-4000001 | 97335 | Robert Price | 07/2021 |
9719 | 10^97335-98193 | 97335 | Patrick De Geest | 09/2022 |
9721 | (637*10^97331+11)/9 | 97333 | Robert Price | 10/2021 |
9722 | 23991^11402+11402^23991 | 97332 | Hans Havermann | 07/2020 |
9723 | 61^54517-60^54517 | 97331 | Jean-Louis Charton | 09/2005 |
9724 | (38*10^97329+187)/9 | 97330 | Robert Price | 11/2017 |
9725 | (31147^21661-1)/31146 | 97328 | Serge Batalov | 06/2022 |
9726 | 86455!4+8 | 97319 | Robert Price | 09/2019 |
9727 | 22591^20310+20310^22591 | 97316 | Hans Havermann | 06/2020 |
9728 | 2^323261-431173 | 97312 | Harrison Mesh | 06/2022 |
9729 | (3996^27017-1)/3995 | 97302 | Serge Batalov | 06/2022 |
9730 | 25101^7520+7520^25101 | 97297 | Hans Havermann | 05/2020 |
9731 | (233*10^97294+7)/3 | 97296 | Robert Price | 03/2022 |
9732 | p(19982)#/2-1024 | 97293 | Robert Price | 09/2017 |
9733 | (65*10^97288+277)/9 | 97289 | Robert Price | 08/2021 |
9734 | 6^125021-85 | 97286 | Lelio R Paula | 08/2016 |
9735 | 2^323164-47355 | 97283 | Lelio R Paula | 07/2016 |
9736 | (3^203902+4)/36051015157 | 97276 | Lelio R Paula | 09/2013 |
9737 | 2^323054-6555 | 97249 | Lelio R Paula | 07/2016 |
9738 | 22822^18247+18247^22822 | 97249 | Hans Havermann | 02/2020 |
9739 | 9*10^97247-29 | 97248 | Robert Price | 05/2017 |
9740 | 24382^9735+9735^24382 | 97244 | Hans Havermann | 02/2020 |
9741 | (31*10^97242+41)/9 | 97243 | Robert Price | 04/2015 |
9742 | GoldenRatioPrime[97241] | 97241 | Eric W. Weisstein | 06/2009 |
9743 | 12^90098-41 | 97233 | Lelio R Paula | 07/2016 |
9744 | 2^322928-216929 | 97212 | Enzo Doria | 05/2014 |
9745 | (16731^23017-1)/16730 | 97209 | Serge Batalov | 06/2022 |
9746 | 3*10^97205-61 | 97206 | Robert Price | 04/2021 |
9747 | (7*2^322896-1)/57 | 97201 | Lelio R Paula | 02/2014 |
9748 | (35*10^97194-341)/9 | 97195 | Robert Price | 08/2021 |
9749 | (4872^26357-1)/4871 | 97194 | Serge Batalov | 06/2022 |
9750 | (2^322897+5)/3146107489 | 97193 | Lelio R Paula | 10/2013 |