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) !!!
Submit your PRPs
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Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
501 | 2^1075425+1880117 | 323736 | Enzo Doria | 10/2016 |
502 | (28^223381+1)/29 | 323267 | Paul Bourdelais | 12/2020 |
503 | 10^322999+187071 | 323000 | Stefano Morozzi | 03/2019 |
504 | 10^322999-143151 | 322999 | Patrick De Geest | 05/2020 |
505 | 2^1072146+1886869 | 322749 | Enzo Doria | 04/2017 |
506 | 2^1072118-7321953 | 322740 | Enzo Doria | 02/2015 |
507 | 2^1071026+1886869 | 322411 | Enzo Doria | 04/2017 |
508 | 2^1070539+1880633 | 322265 | Enzo Doria | 10/2016 |
509 | 2^1069293-1147195 | 321890 | Enzo Doria | 05/2015 |
510 | 2^1065968+1880941 | 320889 | Enzo Doria | 10/2016 |
511 | 2^1065897-1147195 | 320867 | Enzo Doria | 05/2015 |
512 | 2^1063427+1882103 | 320124 | Enzo Doria | 12/2016 |
513 | 2^1061095-1061095 | 319422 | Henri Lifchitz | 10/2013 |
514 | 2^1060950+653119 | 319378 | Enzo Doria | 12/2016 |
515 | 2^1060363+782753 | 319202 | Enzo Doria | 11/2016 |
516 | (71^172259-1)/70 | 318895 | Paul Bourdelais | 08/2020 |
517 | 2^1054736+483061 | 317508 | Enzo Doria | 12/2016 |
518 | 2^1053965+1879667 | 317276 | Enzo Doria | 10/2016 |
519 | (43^194119-1)/42 | 317086 | Paul Bourdelais | 08/2020 |
520 | 2^1052186-458165 | 316740 | Enzo Doria | 03/2015 |
521 | 2^1046613-114685 | 315062 | Enzo Doria | 04/2015 |
522 | 2^1045625-459373 | 314765 | Enzo Doria | 06/2015 |
523 | (7*10^314564-61)/9 | 314564 | Erik Branger | 09/2016 |
524 | 2^1044206+7 | 314338 | Stefano Morozzi | 10/2022 |
525 | (7^371669-1)/6 | 314096 | Paul Bourdelais | 02/2010 |
526 | (53^181981-1)/52 | 313784 | Paul Bourdelais | 08/2020 |
527 | 2^1041944+283051 | 313657 | Enzo Doria | 12/2016 |
528 | 2^1041153-114391 | 313419 | Enzo Doria | 02/2015 |
529 | (2^1039700+9)/5 | 312981 | Stefano Morozzi | 09/2022 |
530 | (74^167311+1)/75 | 312742 | Paul Bourdelais | 12/2020 |
531 | 2^1038871+1866983 | 312732 | Enzo Doria | 04/2016 |
532 | (55^179573-1)/54 | 312521 | Paul Bourdelais | 08/2020 |
533 | 2^1037647+1881263 | 312363 | Enzo Doria | 11/2016 |
534 | 2^1037226+1879699 | 312237 | Enzo Doria | 10/2016 |
535 | 2^1036867+1865843 | 312129 | Enzo Doria | 02/2016 |
536 | (76^165947+1)/77 | 312114 | Paul Bourdelais | 01/2021 |
537 | 2^1036020-114539 | 311874 | Enzo Doria | 02/2015 |
538 | (90^159463-1)/89 | 311628 | Paul Bourdelais | 08/2020 |
539 | 2^1032240+1883431 | 310736 | Enzo Doria | 01/2017 |
540 | 2^1030352+1870021 | 310167 | Enzo Doria | 06/2016 |
541 | 2^1030046+1882843 | 310075 | Enzo Doria | 11/2016 |
542 | 3^648872+16 | 309591 | Robert Price | 04/2020 |
543 | 5^442281+2 | 309142 | Paul Bourdelais | 01/2021 |
544 | 2^1026190+15 | 308914 | Stefano Morozzi | 10/2022 |
545 | A352736(16) | 306985 | Kellen Shenton | 05/2022 |
546 | (11^293831-1)/10 | 305993 | Paul Bourdelais | 03/2010 |
547 | (67^167449-1)/66 | 305773 | Paul Bourdelais | 08/2020 |
548 | (62^170539+1)/63 | 305671 | Paul Bourdelais | 12/2020 |
549 | 2^1015321-1015321 | 305643 | Henri Lifchitz | 08/2013 |
550 | 2^1014450+7 | 305380 | Stefano Morozzi | 10/2022 |
551 | 2^1014224-2791229 | 305312 | Enzo Doria | 01/2016 |
552 | (10^304986+7)/673 | 304984 | Alfred Reich | 11/2021 |
553 | 65473^45250+45250^65473 | 304818 | Hans Havermann | 02/2022 |
554 | 66012^41423+41423^66012 | 304794 | Hans Havermann | 03/2022 |
555 | 65073^48202+48202^65073 | 304742 | Hans Havermann | 04/2022 |
556 | 238176^19+19^238176 | 304569 | Hans Havermann | 03/2022 |
557 | 73122^14615+14615^73122 | 304539 | Hans Havermann | 05/2022 |
558 | 64492^52639+52639^64492 | 304487 | Hans Havermann | 03/2022 |
559 | 65955^41288+41288^65955 | 304437 | Hans Havermann | 04/2022 |
560 | 71729^17530+17530^71729 | 304403 | Hans Havermann | 03/2022 |
561 | 70170^21733+21733^70170 | 304336 | Hans Havermann | 03/2022 |
562 | 2^1010751-107709 | 304267 | Enzo Doria | 03/2014 |
563 | 84473^3994+3994^84473 | 304222 | Hans Havermann | 03/2022 |
564 | (2^1010623-1)/12602017578957977 | 304212 | Tony Prest / Never Odd Or Even | 02/2014 |
565 | 74442^12125+12125^74442 | 303998 | Hans Havermann | 02/2022 |
566 | 77842^8039+8039^77842 | 303989 | Hans Havermann | 01/2022 |
567 | 64209^54140+54140^64209 | 303935 | Hans Havermann | 02/2022 |
568 | 2^1009577+1869587 | 303913 | Enzo Doria | 06/2016 |
569 | 67389^32338+32338^67389 | 303906 | Hans Havermann | 02/2022 |
570 | 8^336419-7^336419 | 303817 | Jean-Louis Charton | 01/2008 |
571 | 2^1009168+15 | 303790 | Stefano Morozzi | 10/2022 |
572 | 80766^5771+5771^80766 | 303782 | Hans Havermann | 01/2022 |
573 | 66094^39385+39385^66094 | 303724 | Hans Havermann | 01/2022 |
574 | 69147^24574+24574^69147 | 303589 | Hans Havermann | 02/2022 |
575 | 78390^7459+7459^78390 | 303580 | Hans Havermann | 01/2022 |
576 | 2^1008301+1869377 | 303529 | Enzo Doria | 06/2016 |
577 | 65008^46615+46615^65008 | 303492 | Hans Havermann | 02/2022 |
578 | (2^1007838+5)/3 | 303389 | Serge Batalov | 08/2015 |
579 | 65018^46293+46293^65018 | 303343 | Hans Havermann | 01/2022 |
580 | 6^389756+5 | 303290 | Paul Bourdelais | 02/2021 |
581 | 71016^18649+18649^71016 | 303285 | Hans Havermann | 01/2022 |
582 | 80160^6047+6047^80160 | 303129 | Hans Havermann | 01/2022 |
583 | 67434^31237+31237^67434 | 303094 | Hans Havermann | 03/2022 |
584 | 65537^42102+42102^65537 | 303063 | Hans Havermann | 02/2022 |
585 | 71790^16553+16553^71790 | 302874 | Hans Havermann | 11/2021 |
586 | 65208^44129+44129^65208 | 302874 | Hans Havermann | 11/2021 |
587 | 84181^3960+3960^84181 | 302858 | Hans Havermann | 11/2021 |
588 | 71956^16087+16087^71956 | 302682 | Hans Havermann | 11/2021 |
589 | 2^1005326+383059 | 302634 | Enzo Doria | 12/2016 |
590 | 2^1004929+1877567 | 302514 | Enzo Doria | 08/2016 |
591 | 69341^22978+22978^69341 | 302418 | Hans Havermann | 12/2021 |
592 | 111443^516+516^111443 | 302306 | Hans Havermann | 12/2021 |
593 | 67638^29387+29387^67638 | 302218 | Hans Havermann | 11/2021 |
594 | 2^1003494+1874899 | 302082 | Enzo Doria | 06/2016 |
595 | 75189^10400+10400^75189 | 302037 | Hans Havermann | 12/2021 |
596 | 67237^31042+31042^67237 | 302026 | Hans Havermann | 12/2021 |
597 | 64540^47791+47791^64540 | 302005 | Hans Havermann | 11/2021 |
598 | 2^1003046+60529 | 301947 | Enzo Doria | 04/2016 |
599 | 83278^4209+4209^83278 | 301815 | Hans Havermann | 09/2021 |
600 | 64431^48250+48250^64431 | 301763 | Hans Havermann | 09/2021 |
601 | 67237^30714+30714^67237 | 301716 | Hans Havermann | 09/2021 |
602 | 64641^46288+46288^64641 | 301581 | Hans Havermann | 09/2021 |
603 | 71513^16482+16482^71513 | 301572 | Hans Havermann | 10/2021 |
604 | 72017^15396+15396^72017 | 301565 | Hans Havermann | 09/2021 |
605 | 65534^39927+39927^65534 | 301540 | Hans Havermann | 10/2021 |
606 | (23^221411-1)/22 | 301501 | Paul Bourdelais | 08/2020 |
607 | 63169^59152+59152^63169 | 301441 | Hans Havermann | 09/2021 |
608 | 75055^10348+10348^75055 | 301336 | Hans Havermann | 09/2021 |
609 | 2^1000647+1881353 | 301225 | Enzo Doria | 11/2016 |
610 | 83686^3961+3961^83686 | 301086 | Hans Havermann | 10/2021 |
611 | 68529^24742+24742^68529 | 301078 | Hans Havermann | 09/2021 |
612 | 65207^41416+41416^65207 | 301072 | Hans Havermann | 10/2021 |
613 | 2^1000000+821751 | 301030 | Tony Prest / Never Odd Or Even | 10/2015 |
614 | 90762^2071+2071^90762 | 300984 | Hans Havermann | 06/2021 |
615 | 63909^51142+51142^63909 | 300934 | Hans Havermann | 08/2021 |
616 | 2^999646+1874029 | 300924 | Enzo Doria | 06/2016 |
617 | 66280^34607+34607^66280 | 300856 | Hans Havermann | 06/2021 |
618 | 65534^38925+38925^65534 | 300817 | Hans Havermann | 07/2021 |
619 | 63296^56517+56517^63296 | 300794 | Hans Havermann | 06/2021 |
620 | 110610^523+523^110610 | 300694 | Hans Havermann | 06/2021 |
621 | 66428^33609+33609^66428 | 300684 | Hans Havermann | 06/2021 |
622 | 74247^11210+11210^74247 | 300672 | Hans Havermann | 06/2021 |
623 | (59^169777-1)/58 | 300649 | Paul Bourdelais | 08/2020 |
624 | 10^300476+3 | 300477 | Ed Trice | 10/2012 |
625 | 10^300476-530913 | 300476 | Patrick De Geest | 03/2016 |
626 | 66703^31960+31960^66703 | 300471 | Hans Havermann | 06/2021 |
627 | 63722^51915+51915^63722 | 300468 | Hans Havermann | 06/2021 |
628 | 2^998053+1879667 | 300444 | Enzo Doria | 09/2016 |
629 | 77018^7959+7959^77018 | 300437 | Hans Havermann | 07/2021 |
630 | 314738^9+9^314738 | 300337 | Anatoly F. Selevich | 02/2011 |
631 | 69478^20987+20987^69478 | 300281 | Hans Havermann | 07/2021 |
632 | 64088^48393+48393^64088 | 300239 | Hans Havermann | 07/2021 |
633 | 64605^44146+44146^64605 | 300084 | Hans Havermann | 08/2021 |
634 | 7^355039-6^355039 | 300043 | Jean-Louis Charton | 05/2011 |
635 | 67594^27465+27465^67594 | 300035 | Hans Havermann | 06/2021 |
636 | 10^300000+287389 | 300001 | Patrick De Geest | 10/2019 |
637 | 10^300000-83657 | 300000 | Patrick De Geest | 10/2019 |
638 | 70599^17756+17756^70599 | 300000 | Hans Havermann | 06/2021 |
639 | numbpart(72526161452) | 299999 | Serge Batalov | 04/2017 |
640 | 2^996182-114431 | 299881 | Enzo Doria | 02/2015 |
641 | 2^996034-17 | 299837 | Stefano Morozzi | 07/2022 |
642 | (19*10^299640+77)/3 | 299641 | Robert Price | 03/2023 |
643 | 2^995214-2788361 | 299590 | Enzo Doria | 12/2015 |
644 | 2^994791-114319 | 299462 | Enzo Doria | 01/2015 |
645 | 2^994490-1863791 | 299372 | Enzo Doria | 01/2016 |
646 | 2^993322+1885729 | 299020 | Enzo Doria | 03/2017 |
647 | (36395^65536+36393^65536)/2 | 298913 | Kellen Shenton | 04/2022 |
648 | 2^992711-2788807 | 298836 | Enzo Doria | 12/2015 |
649 | 2^992525-114199 | 298780 | Enzo Doria | 12/2014 |
650 | 2^992329+1884707 | 298721 | Enzo Doria | 02/2017 |
651 | 2^991968-23 | 298613 | Stefano Morozzi | 07/2022 |
652 | 2^991715-19 | 298536 | Stefano Morozzi | 07/2022 |
653 | (10^298414+7)/191/823/56921/1334467 | 298398 | Alfred Reich | 10/2021 |
654 | 2^989562+25 | 297888 | Stefano Morozzi | 05/2021 |
655 | (19*10^297497+77)/3 | 297498 | Robert Price | 03/2023 |
656 | (2*10^297488-23)/3 | 297488 | Robert Price | 10/2022 |
657 | (2^986191+1)/3 | 296873 | Vincent Diepeveen | 06/2008 |
658 | 2^985217-114553 | 296580 | Enzo Doria | 03/2015 |
659 | (76^157649-1)/75 | 296507 | Paul Bourdelais | 07/2019 |
660 | 2^984131-1828009 | 296253 | Enzo Doria | 01/2015 |
661 | 2^983620+60451 | 296100 | Phil Moore | 03/2008 |
662 | 2^982721-2683 | 295829 | Yves Minet | 02/2016 |
663 | 2^981298-2788421 | 295401 | Enzo Doria | 12/2015 |
664 | 2^981163-2790967 | 295360 | Enzo Doria | 12/2015 |
665 | 2^981016-15 | 295316 | Stefano Morozzi | 07/2022 |
666 | 2^979594+1865839 | 294888 | Enzo Doria | 02/2016 |
667 | 2*10^294527-91 | 294528 | Robert Price | 02/2023 |
668 | 10^294475+49 | 294476 | Robert Price | 02/2023 |
669 | 2^976760+1885021 | 294035 | Enzo Doria | 02/2017 |
670 | 2^974441+1878407 | 293336 | Enzo Doria | 08/2016 |
671 | 2^974180+1876831 | 293258 | Enzo Doria | 07/2016 |
672 | 2^974170+453319 | 293255 | Enzo Doria | 01/2017 |
673 | 2^972744-114419 | 292826 | Enzo Doria | 03/2015 |
674 | 11^281143+6 | 292781 | Paul Bourdelais | 03/2021 |
675 | (4*10^292619-91)/3 | 292620 | Robert Price | 12/2022 |
676 | CopelandErdos(292447) | 292447 | Mark Rodenkirch | 12/2015 |
677 | 3*10^292344-97 | 292345 | Robert Price | 01/2023 |
678 | 2^970911+1873373 | 292274 | Enzo Doria | 07/2016 |
679 | 2^970670+383059 | 292201 | Enzo Doria | 12/2016 |
680 | 2^970141-100363 | 292042 | Enzo Doria | 02/2014 |
681 | 2^969945+1874717 | 291983 | Enzo Doria | 06/2016 |
682 | 2^969384+1880701 | 291814 | Enzo Doria | 10/2016 |
683 | (3^611333-2^611333)/(99035947*87109884549431) | 291659 | Mike Oakes | 02/2017 |
684 | 2^962194+60139 | 289650 | Enzo Doria | 02/2016 |
685 | 2^959307-1861507 | 288781 | Enzo Doria | 11/2015 |
686 | 14^251797-3 | 288592 | Paul Bourdelais | 12/2020 |
687 | (2*10^288371-23)/3 | 288371 | Robert Price | 10/2022 |
688 | (31*10^288238-1)/3 | 288240 | Robert Price | 02/2023 |
689 | 2^957372+283531 | 288198 | Enzo Doria | 02/2017 |
690 | 2^957192-1864049 | 288144 | Enzo Doria | 01/2016 |
691 | 2^956446+1875199 | 287919 | Enzo Doria | 06/2016 |
692 | 7*10^287854+39 | 287855 | Robert Price | 01/2023 |
693 | (17^233689+1)/18 | 287542 | Paul Bourdelais | 09/2013 |
694 | 2^955178+1883587 | 287538 | Enzo Doria | 12/2016 |
695 | 2^954127-954127^2 | 287221 | Henri Lifchitz | 01/2021 |
696 | 2^954001+1875257 | 287183 | Enzo Doria | 07/2016 |
697 | (23891^65536+1)/2 | 286932 | Serge Batalov | 03/2018 |
698 | 2^952268+1878601 | 286662 | Enzo Doria | 09/2016 |
699 | 2^950801+81557 | 286220 | Enzo Doria | 08/2016 |
700 | 22845^65536+22844^65536 | 285658 | Kellen Shenton | 03/2021 |
701 | (11*2^948635-13)/15 | 285568 | Renaud Lifchitz | 11/2011 |
702 | (54^164839+1)/55 | 285565 | Paul Bourdelais | 07/2019 |
703 | 2^948510+782569 | 285530 | Enzo Doria | 10/2016 |
704 | 2^948374-114137 | 285490 | Enzo Doria | 01/2015 |
705 | 2^947739+1872923 | 285298 | Enzo Doria | 06/2016 |
706 | 2^944561-21 | 284342 | Stefano Morozzi | 07/2022 |
707 | (4*10^284330-31)/9 | 284330 | Serge Batalov | 11/2022 |
708 | (32*10^283826-23)/9 | 283827 | Serge Batalov | 01/2023 |
709 | 2^942533+82067 | 283731 | Enzo Doria | 09/2016 |
710 | (10^283342-31)/3 | 283342 | Robert Price | 03/2023 |
711 | 2^940950+1874989 | 283255 | Enzo Doria | 06/2016 |
712 | 2^940426-2786141 | 283097 | Enzo Doria | 11/2015 |
713 | (2*10^282948-11)/9 | 282948 | Serge Batalov | 11/2022 |
714 | 2^939881-114115 | 282933 | Enzo Doria | 12/2014 |
715 | 10^282493+632551 | 282494 | Patrick De Geest | 02/2017 |
717 | 10^282493-9^282493 | 282493 | Jean-Louis Charton | 03/2007 |
716 | 10^282493-132327 | 282493 | Patrick De Geest | 02/2017 |
718 | 3^591827-2^591827 | 282374 | Mike Oakes | 08/2009 |
719 | Tau(509^18946) | 282048 | Nik Lygeros / Olivier Rozier | 05/2015 |
720 | 2^936716-1835519 | 281980 | Enzo Doria | 04/2015 |
721 | 2^936588+79411 | 281942 | Enzo Doria | 06/2016 |
722 | 2^936047+1877423 | 281779 | Enzo Doria | 08/2016 |
723 | 2^935713+1869377 | 281678 | Enzo Doria | 05/2016 |
724 | 2^933082+1877059 | 280886 | Enzo Doria | 08/2016 |
725 | (5*10^280710+1)/3 | 280711 | Serge Batalov | 10/2014 |
726 | 2^931976+25 | 280553 | Stefano Morozzi | 05/2021 |
727 | (42^172663+1)/43 | 280274 | Paul Bourdelais | 07/2019 |
728 | (10^279978+11)/3 | 279978 | Serge Batalov | 07/2015 |
729 | 2^929524-1864829 | 279815 | Enzo Doria | 01/2016 |
730 | 2^929152+60181 | 279703 | Enzo Doria | 02/2016 |
731 | 2^928621-2787883 | 279543 | Enzo Doria | 11/2015 |
732 | (2^927763+lucasV(-1,2,927763)+1)/2 | 279285 | Mike Oakes | 04/2017 |
733 | 2^927429+1882277 | 279184 | Enzo Doria | 12/2016 |
734 | 2^926766-11 | 278985 | Stefano Morozzi | 07/2022 |
735 | (71^150697-1)/70 | 278978 | Paul Bourdelais | 07/2019 |
736 | (33^183661-1)/32 | 278891 | Paul Bourdelais | 12/2018 |
737 | (166*10^278633-7)/3 | 278635 | Robert Price | 02/2023 |
738 | (2^925391+lucasV(-1,2,925391)+1)/2 | 278571 | Mike Oakes | 03/2017 |
739 | 2^925330-1301 | 278553 | Yves Minet | 06/2015 |
740 | (299*10^278546+1)/3 | 278548 | Robert Price | 01/2023 |
741 | (21^210599+1)/22 | 278457 | Paul Bourdelais | 08/2013 |
742 | (266*10^278325+1)/3 | 278327 | Robert Price | 01/2023 |
743 | 2^923261-69175 | 277930 | Enzo Doria | 11/2015 |
744 | 757^96487-756^96487 | 277796 | Jean-Louis Charton | 12/2012 |
745 | 2^922301+57 | 277641 | Stefano Morozzi | 05/2021 |
746 | ((((((((((((((291487327^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2 | 277369 | Serge Batalov | 07/2016 |
747 | (5^396413-1)/4 | 277081 | Paul Bourdelais | 06/2010 |
748 | 2^920318+2792299 | 277044 | Enzo Doria | 02/2016 |
749 | 2^918596-2786939 | 276525 | Enzo Doria | 11/2015 |
750 | 2^916639+60563 | 275936 | Enzo Doria | 04/2016 |