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 | 2^1200021+149777 | 361243 | Stefano Morozzi | 10/2022 |
502 | (64*10^360173-1)/243 | 360173 | Serge Batalov | 09/2014 |
503 | (59^203309+1)/60 | 360029 | Paul Bourdelais | 09/2021 |
504 | (2*10^359860-11)/9 | 359860 | Rytis Slatkevicius | 01/2023 |
505 | 2^1195093+1882577 | 359759 | Enzo Doria | 12/2016 |
506 | 3^753451+5*2^753451 | 359488 | Jon Grantham | 08/2023 |
507 | (6^461108-11)/5 | 358812 | Paul Bourdelais | 04/2019 |
508 | 2^1191375+8543 | 358640 | Phil Moore | 06/2008 |
509 | 2^1190972+1880191 | 358519 | Enzo Doria | 11/2016 |
510 | (45^216551-1)/44 | 358004 | Paul Bourdelais | 08/2020 |
511 | (50^210319-1)/49 | 357324 | Paul Bourdelais | 08/2020 |
512 | (531^131072+25^131072)/2 | 357184 | Kellen Shenton | 03/2021 |
513 | 2^1184565+21227 | 356590 | Enzo Doria | 08/2023 |
514 | 2^1179836+1885021 | 355167 | Enzo Doria | 02/2017 |
515 | (19*10^353884-91)/9 | 353885 | Alfred Reich | 02/2023 |
516 | 20^271951+3 | 353817 | Paul Bourdelais | 01/2021 |
517 | 2^1173095+21 | 353137 | Stefano Morozzi | 10/2022 |
518 | (63^195893-1)/62 | 352477 | Paul Bourdelais | 08/2020 |
519 | 2^1170326+7 | 352304 | Stefano Morozzi | 10/2022 |
520 | numbpart(100000135540) | 352269 | Serge Batalov | 04/2017 |
521 | (10^352237-1)/9/10567111/950335427 | 352221 | Maksym Voznyy | 01/2015 |
522 | 5*3^737931-2^737931 | 352084 | Jon Grantham | 08/2023 |
523 | 2^1168554+783529 | 351770 | Enzo Doria | 01/2017 |
524 | (2^1168183-1)/54763676838381762583 | 351639 | Matthias Wurtinger | 01/2011 |
525 | 2^1166152+1882411 | 351047 | Enzo Doria | 12/2016 |
526 | 2^1163972+20971 | 350391 | Enzo Doria | 08/2023 |
527 | 10^350071+7 | 350072 | Alfred Reich | 05/2021 |
528 | 2^1162154+1890207 | 349844 | Enzo Doria | 08/2017 |
529 | (7*10^348724-61)/9 | 348724 | Erik Branger | 09/2016 |
530 | (4*10^348453+41)/9 | 348453 | Serge Batalov | 11/2022 |
531 | (127^165601-1)/126 | 348390 | Alexander Zhirkov | 09/2015 |
532 | 2^1155085+213377 | 347716 | Enzo Doria | 09/2023 |
533 | 2^1153083+1883693 | 347113 | Enzo Doria | 04/2017 |
534 | (2^1148729+2^574365+1)/5 | 345802 | Borys Jaworski | 03/2007 |
535 | 2^1145882+80779 | 344945 | Enzo Doria | 09/2016 |
536 | 2^1136462+1887289 | 342110 | Enzo Doria | 05/2017 |
537 | F(1636007) | 341905 | Henri Lifchitz | 03/2009 |
538 | (2^1135567+13)/3 | 341840 | Stefano Morozzi | 09/2022 |
539 | 2^1133678+782569 | 341272 | Enzo Doria | 11/2016 |
540 | (11^327001+1)/12 | 340536 | Paul Bourdelais | 09/2013 |
541 | (2^1127239+2^563620+1)/5 | 339333 | Borys Jaworski | 11/2006 |
542 | Tau(907^20848) | 339132 | Nik Lygeros / Olivier Rozier | 01/2016 |
543 | 2^1126380-3 | 339075 | Paul Bourdelais | 06/2019 |
544 | 2^1125446+1885099 | 338794 | Enzo Doria | 08/2017 |
545 | (63^188147+1)/64 | 338539 | Paul Bourdelais | 12/2020 |
546 | 2^1124549+143807 | 338523 | Enzo Doria | 10/2016 |
547 | (4*10^338476+23)/9 | 338476 | Serge Batalov | 11/2022 |
548 | 2^1121187+1886783 | 337511 | Enzo Doria | 04/2017 |
549 | (48^200443-1)/47 | 336992 | Paul Bourdelais | 08/2020 |
550 | (2*10^336405+43)/9 | 336405 | Rytis Slatkevicius | 01/2023 |
551 | (94^170371-1)/93 | 336162 | Paul Bourdelais | 09/2020 |
552 | 2^1116184+1883521 | 336005 | Enzo Doria | 02/2017 |
553 | 2^1114517+1880087 | 335504 | Enzo Doria | 10/2016 |
554 | (5*10^335480+31)/9 | 335480 | Serge Batalov | 11/2022 |
555 | 5*3^701688+2^701688 | 334791 | Jon Grantham | 08/2023 |
556 | (101^166849+1)/102 | 334418 | Paul Bourdelais | 12/2020 |
557 | (3^700897+1)/4 | 334413 | Paul Bourdelais | 04/2010 |
558 | 2^1104408-114539 | 332460 | Enzo Doria | 03/2015 |
559 | 2^1101921+82085 | 331712 | Enzo Doria | 10/2016 |
560 | 2^1100655+683513 | 331331 | Enzo Doria | 01/2017 |
561 | (10^330906+71)/9 | 330906 | Serge Batalov | 07/2015 |
562 | 2^1098346+482899 | 330636 | Enzo Doria | 11/2016 |
563 | (2^1096212+19)/5 | 329992 | Stefano Morozzi | 09/2022 |
564 | 2^1095679+1878713 | 329833 | Enzo Doria | 09/2016 |
565 | 2^1094740-1851539 | 329550 | Enzo Doria | 08/2015 |
566 | 2^1092242+19 | 328798 | Stefano Morozzi | 10/2022 |
567 | 2^1091469-379369 | 328565 | Enzo Doria | 03/2014 |
568 | (69^178093-1)/68 | 327485 | Paul Bourdelais | 08/2020 |
569 | (10^326884+7)/421 | 326882 | Alfred Reich | 12/2021 |
570 | 2^1085750+582349 | 326844 | Enzo Doria | 11/2016 |
571 | 2^1083596+483061 | 326195 | Enzo Doria | 12/2016 |
572 | (2^541456-1)*10^162995+2^541455-1 | 325990 | paolo galliani | 01/2019 |
573 | 2^1082908+583561 | 325988 | Enzo Doria | 05/2017 |
574 | 30^220411-29^220411 | 325574 | Jean-Louis Charton | 10/2018 |
575 | 2^1081308+7 | 325507 | Stefano Morozzi | 10/2022 |
576 | 2^1078242+1881649 | 324584 | Enzo Doria | 11/2016 |
577 | 2^1077737+1880117 | 324432 | Enzo Doria | 10/2016 |
578 | 3^679017-5*2^679017 | 323974 | Jon Grantham | 08/2023 |
579 | 2^1075425+1880117 | 323736 | Enzo Doria | 10/2016 |
580 | (28^223381+1)/29 | 323267 | Paul Bourdelais | 12/2020 |
581 | 10^322999+187071 | 323000 | Stefano Morozzi | 03/2019 |
582 | 10^322999-143151 | 322999 | Patrick De Geest | 05/2020 |
583 | 2^1072146+1886869 | 322749 | Enzo Doria | 04/2017 |
584 | 2^1072118-7321953 | 322740 | Enzo Doria | 02/2015 |
585 | 2^1071026+1886869 | 322411 | Enzo Doria | 04/2017 |
586 | 2^1070539+1880633 | 322265 | Enzo Doria | 10/2016 |
587 | 2^1069293-1147195 | 321890 | Enzo Doria | 05/2015 |
588 | 2^1065968+1880941 | 320889 | Enzo Doria | 10/2016 |
589 | 2^1065897-1147195 | 320867 | Enzo Doria | 05/2015 |
590 | 2^1063427+1882103 | 320124 | Enzo Doria | 12/2016 |
591 | 2^1061095-1061095 | 319422 | Henri Lifchitz | 10/2013 |
592 | 2^1060950+653119 | 319378 | Enzo Doria | 12/2016 |
593 | 2^1060363+782753 | 319202 | Enzo Doria | 11/2016 |
594 | (71^172259-1)/70 | 318895 | Paul Bourdelais | 08/2020 |
595 | 2^1054736+483061 | 317508 | Enzo Doria | 12/2016 |
596 | 2^1053965+1879667 | 317276 | Enzo Doria | 10/2016 |
597 | (43^194119-1)/42 | 317086 | Paul Bourdelais | 08/2020 |
598 | 2^1052186-458165 | 316740 | Enzo Doria | 03/2015 |
599 | 11^303909+2 | 316489 | Paul Bourdelais | 08/2023 |
600 | 2^1046613-114685 | 315062 | Enzo Doria | 04/2015 |
601 | 2^1045625-459373 | 314765 | Enzo Doria | 06/2015 |
602 | 10^(2*157363+1)-8*10^157363-1 | 314727 | Robert Price | 07/2023 |
603 | (7*10^314564-61)/9 | 314564 | Erik Branger | 09/2016 |
604 | 2^1044206+7 | 314338 | Stefano Morozzi | 10/2022 |
605 | (7^371669-1)/6 | 314096 | Paul Bourdelais | 02/2010 |
606 | (53^181981-1)/52 | 313784 | Paul Bourdelais | 08/2020 |
607 | (8^347437-5^347437)/3 | 313767 | Jon Grantham | 08/2023 |
608 | 2^1041944+283051 | 313657 | Enzo Doria | 12/2016 |
609 | 2^1041153-114391 | 313419 | Enzo Doria | 02/2015 |
610 | (2^1039700+9)/5 | 312981 | Stefano Morozzi | 09/2022 |
611 | (74^167311+1)/75 | 312742 | Paul Bourdelais | 12/2020 |
612 | 2^1038871+1866983 | 312732 | Enzo Doria | 04/2016 |
613 | 2^1038545+71657 | 312634 | Enzo Doria | 08/2023 |
614 | (55^179573-1)/54 | 312521 | Paul Bourdelais | 08/2020 |
615 | 2^1037647+1881263 | 312363 | Enzo Doria | 11/2016 |
616 | 2^1037226+1879699 | 312237 | Enzo Doria | 10/2016 |
617 | 2^1036867+1865843 | 312129 | Enzo Doria | 02/2016 |
618 | (76^165947+1)/77 | 312114 | Paul Bourdelais | 01/2021 |
619 | 2^1036020-114539 | 311874 | Enzo Doria | 02/2015 |
620 | (90^159463-1)/89 | 311628 | Paul Bourdelais | 08/2020 |
621 | 2^1032240+1883431 | 310736 | Enzo Doria | 01/2017 |
622 | 2^1030352+1870021 | 310167 | Enzo Doria | 06/2016 |
623 | 2^1030046+1882843 | 310075 | Enzo Doria | 11/2016 |
624 | 3^648872+16 | 309591 | Robert Price | 04/2020 |
625 | 64221^64366+64366^64221 | 309451 | Norbert Schneider | 06/2023 |
626 | 2^1026999+20513 | 309158 | Enzo Doria | 07/2023 |
627 | 5^442281+2 | 309142 | Paul Bourdelais | 01/2021 |
628 | 2^1026204+22741 | 308919 | Enzo Doria | 09/2023 |
629 | 2^1026190+15 | 308914 | Stefano Morozzi | 10/2022 |
630 | 2^1025108+20851 | 308589 | Enzo Doria | 08/2023 |
631 | A352736(16) | 306985 | Kellen Shenton | 05/2022 |
632 | (11^293831-1)/10 | 305993 | Paul Bourdelais | 03/2010 |
633 | (67^167449-1)/66 | 305773 | Paul Bourdelais | 08/2020 |
634 | (62^170539+1)/63 | 305671 | Paul Bourdelais | 12/2020 |
635 | 2^1015321-1015321 | 305643 | Henri Lifchitz | 08/2013 |
636 | 2^1014450+7 | 305380 | Stefano Morozzi | 10/2022 |
637 | 2^1014224-2791229 | 305312 | Enzo Doria | 01/2016 |
638 | (10^304986+7)/673 | 304984 | Alfred Reich | 11/2021 |
639 | 65473^45250+45250^65473 | 304818 | Hans Havermann | 02/2022 |
640 | 66012^41423+41423^66012 | 304794 | Hans Havermann | 03/2022 |
641 | 65073^48202+48202^65073 | 304742 | Hans Havermann | 04/2022 |
642 | 238176^19+19^238176 | 304569 | Hans Havermann | 03/2022 |
643 | 73122^14615+14615^73122 | 304539 | Hans Havermann | 05/2022 |
644 | 64492^52639+52639^64492 | 304487 | Hans Havermann | 03/2022 |
645 | 65955^41288+41288^65955 | 304437 | Hans Havermann | 04/2022 |
646 | 71729^17530+17530^71729 | 304403 | Hans Havermann | 03/2022 |
647 | 70170^21733+21733^70170 | 304336 | Hans Havermann | 03/2022 |
648 | 2^1010751-107709 | 304267 | Enzo Doria | 03/2014 |
649 | 84473^3994+3994^84473 | 304222 | Hans Havermann | 03/2022 |
650 | (2^1010623-1)/12602017578957977 | 304212 | Tony Prest / Never Odd Or Even | 02/2014 |
651 | 74442^12125+12125^74442 | 303998 | Hans Havermann | 02/2022 |
652 | 77842^8039+8039^77842 | 303989 | Hans Havermann | 01/2022 |
653 | 64209^54140+54140^64209 | 303935 | Hans Havermann | 02/2022 |
654 | 2^1009577+1869587 | 303913 | Enzo Doria | 06/2016 |
655 | 67389^32338+32338^67389 | 303906 | Hans Havermann | 02/2022 |
656 | 8^336419-7^336419 | 303817 | Jean-Louis Charton | 01/2008 |
657 | 2^1009168+15 | 303790 | Stefano Morozzi | 10/2022 |
658 | 80766^5771+5771^80766 | 303782 | Hans Havermann | 01/2022 |
659 | 66094^39385+39385^66094 | 303724 | Hans Havermann | 01/2022 |
660 | 69147^24574+24574^69147 | 303589 | Hans Havermann | 02/2022 |
661 | 78390^7459+7459^78390 | 303580 | Hans Havermann | 01/2022 |
662 | 2^1008301+1869377 | 303529 | Enzo Doria | 06/2016 |
663 | 65008^46615+46615^65008 | 303492 | Hans Havermann | 02/2022 |
664 | (2^1007838+5)/3 | 303389 | Serge Batalov | 08/2015 |
665 | 65018^46293+46293^65018 | 303343 | Hans Havermann | 01/2022 |
666 | 6^389756+5 | 303290 | Paul Bourdelais | 02/2021 |
667 | 71016^18649+18649^71016 | 303285 | Hans Havermann | 01/2022 |
668 | 80160^6047+6047^80160 | 303129 | Hans Havermann | 01/2022 |
669 | 67434^31237+31237^67434 | 303094 | Hans Havermann | 03/2022 |
670 | 65537^42102+42102^65537 | 303063 | Hans Havermann | 02/2022 |
671 | 71790^16553+16553^71790 | 302874 | Hans Havermann | 11/2021 |
672 | 65208^44129+44129^65208 | 302874 | Hans Havermann | 11/2021 |
673 | 84181^3960+3960^84181 | 302858 | Hans Havermann | 11/2021 |
674 | 71956^16087+16087^71956 | 302682 | Hans Havermann | 11/2021 |
675 | 2^1005326+383059 | 302634 | Enzo Doria | 12/2016 |
676 | 2^1004929+1877567 | 302514 | Enzo Doria | 08/2016 |
677 | 69341^22978+22978^69341 | 302418 | Hans Havermann | 12/2021 |
678 | 111443^516+516^111443 | 302306 | Hans Havermann | 12/2021 |
679 | 67638^29387+29387^67638 | 302218 | Hans Havermann | 11/2021 |
680 | 2^1003494+1874899 | 302082 | Enzo Doria | 06/2016 |
681 | 75189^10400+10400^75189 | 302037 | Hans Havermann | 12/2021 |
682 | 67237^31042+31042^67237 | 302026 | Hans Havermann | 12/2021 |
683 | 64540^47791+47791^64540 | 302005 | Hans Havermann | 11/2021 |
684 | 2^1003046+60529 | 301947 | Enzo Doria | 04/2016 |
685 | 83278^4209+4209^83278 | 301815 | Hans Havermann | 09/2021 |
686 | 64431^48250+48250^64431 | 301763 | Hans Havermann | 09/2021 |
687 | 67237^30714+30714^67237 | 301716 | Hans Havermann | 09/2021 |
688 | 64641^46288+46288^64641 | 301581 | Hans Havermann | 09/2021 |
689 | 71513^16482+16482^71513 | 301572 | Hans Havermann | 10/2021 |
690 | 72017^15396+15396^72017 | 301565 | Hans Havermann | 09/2021 |
691 | 65534^39927+39927^65534 | 301540 | Hans Havermann | 10/2021 |
692 | (23^221411-1)/22 | 301501 | Paul Bourdelais | 08/2020 |
693 | 63169^59152+59152^63169 | 301441 | Hans Havermann | 09/2021 |
694 | 75055^10348+10348^75055 | 301336 | Hans Havermann | 09/2021 |
695 | 2^1000647+1881353 | 301225 | Enzo Doria | 11/2016 |
696 | 83686^3961+3961^83686 | 301086 | Hans Havermann | 10/2021 |
697 | 68529^24742+24742^68529 | 301078 | Hans Havermann | 09/2021 |
698 | 65207^41416+41416^65207 | 301072 | Hans Havermann | 10/2021 |
700 | 2^1000000+821751 | 301030 | Tony Prest / Never Odd Or Even | 10/2015 |
699 | 2^999999+77151 | 301030 | Norman Luhn | 09/2023 |
701 | 90762^2071+2071^90762 | 300984 | Hans Havermann | 06/2021 |
702 | 63909^51142+51142^63909 | 300934 | Hans Havermann | 08/2021 |
703 | 2^999646+1874029 | 300924 | Enzo Doria | 06/2016 |
704 | 66280^34607+34607^66280 | 300856 | Hans Havermann | 06/2021 |
705 | 65534^38925+38925^65534 | 300817 | Hans Havermann | 07/2021 |
706 | 63296^56517+56517^63296 | 300794 | Hans Havermann | 06/2021 |
707 | 110610^523+523^110610 | 300694 | Hans Havermann | 06/2021 |
708 | 66428^33609+33609^66428 | 300684 | Hans Havermann | 06/2021 |
709 | 74247^11210+11210^74247 | 300672 | Hans Havermann | 06/2021 |
710 | (59^169777-1)/58 | 300649 | Paul Bourdelais | 08/2020 |
711 | 10^300476+3 | 300477 | Ed Trice | 10/2012 |
712 | 10^300476-530913 | 300476 | Patrick De Geest | 03/2016 |
713 | 66703^31960+31960^66703 | 300471 | Hans Havermann | 06/2021 |
714 | 63722^51915+51915^63722 | 300468 | Hans Havermann | 06/2021 |
715 | 2^998053+1879667 | 300444 | Enzo Doria | 09/2016 |
716 | 77018^7959+7959^77018 | 300437 | Hans Havermann | 07/2021 |
717 | 314738^9+9^314738 | 300337 | Anatoly F. Selevich | 02/2011 |
718 | 69478^20987+20987^69478 | 300281 | Hans Havermann | 07/2021 |
719 | 64088^48393+48393^64088 | 300239 | Hans Havermann | 07/2021 |
720 | 64605^44146+44146^64605 | 300084 | Hans Havermann | 08/2021 |
721 | 7^355039-6^355039 | 300043 | Jean-Louis Charton | 05/2011 |
722 | 67594^27465+27465^67594 | 300035 | Hans Havermann | 06/2021 |
723 | 10^300000+287389 | 300001 | Patrick De Geest | 10/2019 |
724 | 10^300000-83657 | 300000 | Patrick De Geest | 10/2019 |
725 | 70599^17756+17756^70599 | 300000 | Hans Havermann | 06/2021 |
726 | numbpart(72526161452) | 299999 | Serge Batalov | 04/2017 |
727 | 2^996182-114431 | 299881 | Enzo Doria | 02/2015 |
728 | 3*2^996126+5 | 299865 | Jon Grantham | 07/2023 |
729 | 2^996034-17 | 299837 | Stefano Morozzi | 07/2022 |
730 | (19*10^299640+77)/3 | 299641 | Robert Price | 03/2023 |
731 | 2^995214-2788361 | 299590 | Enzo Doria | 12/2015 |
732 | 2^994791-114319 | 299462 | Enzo Doria | 01/2015 |
733 | 2^994490-1863791 | 299372 | Enzo Doria | 01/2016 |
734 | 2^993322+1885729 | 299020 | Enzo Doria | 03/2017 |
735 | (36395^65536+36393^65536)/2 | 298913 | Kellen Shenton | 04/2022 |
736 | 2^992711-2788807 | 298836 | Enzo Doria | 12/2015 |
737 | 2^992525-114199 | 298780 | Enzo Doria | 12/2014 |
738 | (7^353501-3^353501)/4 | 298743 | Jon Grantham | 08/2023 |
739 | 2^992329+1884707 | 298721 | Enzo Doria | 02/2017 |
740 | 2^991968-23 | 298613 | Stefano Morozzi | 07/2022 |
741 | 2^991715-19 | 298536 | Stefano Morozzi | 07/2022 |
742 | (10^298414+7)/191/823/56921/1334467 | 298398 | Alfred Reich | 10/2021 |
743 | 2^989562+25 | 297888 | Stefano Morozzi | 05/2021 |
744 | (19*10^297497+77)/3 | 297498 | Robert Price | 03/2023 |
745 | (2*10^297488-23)/3 | 297488 | Robert Price | 10/2022 |
746 | (2^986191+1)/3 | 296873 | Vincent Diepeveen | 06/2008 |
747 | 2^985217-114553 | 296580 | Enzo Doria | 03/2015 |
748 | (76^157649-1)/75 | 296507 | Paul Bourdelais | 07/2019 |
749 | 2^984650+21529 | 296410 | Enzo Doria | 08/2023 |
750 | 2^984131-1828009 | 296253 | Enzo Doria | 01/2015 |