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 ? |
2501 | 2^633088-113879 | 190579 | Enzo Doria | 10/2014 |
2502 | 2^633064+1874911 | 190572 | Enzo Doria | 06/2016 |
2503 | 5*3^399402+2^399402 | 190564 | Jon Grantham | 08/2023 |
2504 | (43*10^190547+83)/9 | 190548 | Robert Price | 11/2018 |
2505 | 99*10^190543+7 | 190545 | Robert Price | 05/2021 |
2506 | (299*10^190533+1)/3 | 190535 | Robert Price | 04/2020 |
2507 | (1744*10^190519-1747)/3 | 190522 | Robert Price | 12/2020 |
2508 | 2^632815-15 | 190497 | Stefano Morozzi | 07/2022 |
2509 | 2^632795-351205 | 190491 | Enzo Doria | 06/2015 |
2510 | 41171^41272+41272^41171 | 190454 | Norbert Schneider | 05/2023 |
2511 | (16*10^190420-19)/3 | 190421 | Robert Price | 02/2019 |
2512 | (43*10^190351+101)/9 | 190352 | Robert Price | 11/2018 |
2513 | 2^632156+78391 | 190298 | Enzo Doria | 06/2016 |
2514 | 97*10^190295-3 | 190297 | Robert Price | 05/2020 |
2515 | 2^632082-1851971 | 190276 | Enzo Doria | 08/2015 |
2516 | 2^632020+1880311 | 190257 | Enzo Doria | 09/2016 |
2517 | (5*10^190242-77)/9 | 190242 | Robert Price | 04/2021 |
2518 | 7*10^190210+13 | 190211 | Robert Price | 03/2021 |
2519 | 2^631637-462373 | 190142 | Enzo Doria | 08/2015 |
2520 | 2*10^190126-21 | 190127 | Robert Price | 09/2020 |
2521 | (28*10^190105+191)/3 | 190106 | Robert Price | 02/2020 |
2522 | 2*10^190093+33 | 190094 | Robert Price | 11/2021 |
2523 | (68*10^190053-113)/9 | 190054 | Robert Price | 03/2021 |
2524 | 2^631183+213413 | 190006 | Enzo Doria | 09/2023 |
2525 | 2^631122-39251 | 189987 | Enzo Doria | 08/2015 |
2526 | (26*10^189928-89)/9 | 189929 | Robert Price | 06/2018 |
2527 | 2^630839-97 | 189902 | Yves Minet | 07/2017 |
2528 | 99*10^189876-7 | 189878 | Robert Price | 07/2020 |
2529 | 41147^41147+30^30 | 189867 | Henri Lifchitz | 09/2024 |
2530 | 2^630359+1788953 | 189757 | Enzo Doria | 05/2016 |
2531 | 2^630228+1871971 | 189718 | Enzo Doria | 05/2016 |
2532 | 2^630108+781 | 189682 | Lelio R Paula | 08/2013 |
2533 | 2^629870-1867811 | 189610 | Enzo Doria | 04/2016 |
2534 | (359^74189-1)/358 | 189558 | Alexander Zhirkov | 11/2017 |
2535 | 2^629326+2792929 | 189447 | Enzo Doria | 02/2016 |
2536 | 4*10^189401+13 | 189402 | Robert Price | 02/2021 |
2537 | (10^189385-61)/3 | 189385 | Serge Batalov | 12/2014 |
2538 | 2^629036+1874011 | 189359 | Enzo Doria | 06/2016 |
2539 | 2^628930+80779 | 189327 | Enzo Doria | 07/2016 |
2540 | (134*10^189309-11)/3 | 189311 | Robert Price | 01/2022 |
2541 | 9*10^189306-11 | 189307 | Mischa Rodermond | 04/2023 |
2542 | 3^396740-396740^3 | 189294 | Henri Lifchitz | 07/2024 |
2543 | 2^628582+1179439 | 189223 | Enzo Doria | 06/2016 |
2544 | (44*10^188999-503)/9 | 189000 | Robert Price | 12/2018 |
2545 | (16*10^188823+167)/3 | 188824 | Robert Price | 06/2019 |
2546 | (10^188796+7)/4967/71933 | 188788 | Alfred Reich | 08/2021 |
2547 | 2^627037-2785843 | 188757 | Enzo Doria | 10/2015 |
2548 | (10^188566+7)/311640753181661 | 188552 | Alfred Reich | 04/2022 |
2549 | (16*10^188457+161)/3 | 188458 | Robert Price | 02/2019 |
2550 | L(901657) | 188436 | Renaud Lifchitz | 11/2008 |
2551 | (19*10^188375+137)/3 | 188376 | Robert Price | 03/2022 |
2552 | 2^625651+74213 | 188340 | Enzo Doria | 04/2016 |
2553 | 2^625584-460919 | 188320 | Enzo Doria | 06/2015 |
2554 | (8*10^188308-71)/9 | 188308 | Serge Batalov | 11/2022 |
2555 | 5^269399-3^269399-1 | 188302 | Jon Grantham | 08/2023 |
2556 | (14*10^188218+139)/9 | 188219 | Robert Price | 11/2021 |
2557 | 40760^40821-40821^40760 | 188195 | Norbert Schneider | 07/2024 |
2558 | (10^188164+359)/9 | 188164 | Serge Batalov | 12/2014 |
2559 | 2^624481-113611 | 187988 | Enzo Doria | 08/2014 |
2560 | (2^624389+125582)/2 | 187960 | Stefano Morozzi | 09/2019 |
2561 | (2^624389+166982)/2 | 187960 | Stefano Morozzi | 09/2019 |
2562 | (2^624389+179886)/2 | 187960 | Stefano Morozzi | 09/2019 |
2563 | (2^624387+64466)/2 | 187959 | Stefano Morozzi | 09/2019 |
2564 | (2^624378+112990)/2 | 187957 | Stefano Morozzi | 09/2019 |
2565 | (2^624380+161986)/2 | 187957 | Stefano Morozzi | 09/2019 |
2567 | (2^624376+55762)/2 | 187956 | Stefano Morozzi | 09/2019 |
2566 | (2^624377+169506)/2 | 187956 | Stefano Morozzi | 09/2019 |
2568 | (2^624355+80178)/2 | 187950 | Stefano Morozzi | 09/2019 |
2569 | (2^624351+61350)/2 | 187949 | Stefano Morozzi | 09/2019 |
2570 | (2^624341+63462)/2 | 187946 | Stefano Morozzi | 09/2019 |
2571 | (2^624344+135990)/2 | 187946 | Stefano Morozzi | 09/2019 |
2572 | (2^624344+183898)/2 | 187946 | Stefano Morozzi | 09/2019 |
2574 | (2^624340+126982)/2 | 187945 | Stefano Morozzi | 09/2019 |
2573 | (2^624338+196110)/2 | 187945 | Stefano Morozzi | 09/2019 |
2575 | (2^624337+105162)/2 | 187944 | Stefano Morozzi | 09/2019 |
2576 | (2^624333+44594)/2 | 187943 | Stefano Morozzi | 09/2019 |
2577 | (2^624321+88742)/2 | 187940 | Stefano Morozzi | 09/2019 |
2578 | (2^624324+135178)/2 | 187940 | Stefano Morozzi | 09/2019 |
2579 | (2^624317+14186)/2 | 187938 | Stefano Morozzi | 09/2019 |
2580 | (2^624313+24450)/2 | 187937 | Stefano Morozzi | 09/2019 |
2581 | (2^624309+184590)/2 | 187936 | Stefano Morozzi | 09/2019 |
2583 | (2^624307+152658)/2 | 187935 | Stefano Morozzi | 09/2019 |
2582 | (2^624307+185978)/2 | 187935 | Stefano Morozzi | 09/2019 |
2584 | (2^624303+40814)/2 | 187934 | Stefano Morozzi | 09/2019 |
2585 | (2^624304+99238)/2 | 187934 | Stefano Morozzi | 09/2019 |
2586 | 2^624077-112723 | 187866 | Enzo Doria | 07/2014 |
2587 | 2^624008+1874941 | 187846 | Enzo Doria | 06/2016 |
2588 | 2^623870+78679 | 187804 | Enzo Doria | 05/2016 |
2589 | 2^623714-114035 | 187757 | Enzo Doria | 11/2014 |
2590 | 2^623651-1834251 | 187738 | Enzo Doria | 07/2015 |
2591 | (7*10^187645+53)/3 | 187646 | Robert Price | 05/2018 |
2592 | 21650^43269+43269^21650 | 187591 | Norbert Schneider | 02/2023 |
2593 | 2^623146+1879609 | 187586 | Enzo Doria | 09/2016 |
2594 | (401*10^187512+13)/9 | 187514 | Robert Price | 09/2021 |
2595 | (5*10^187471+91)/3 | 187472 | Robert Price | 04/2018 |
2596 | 2^622500+81931 | 187392 | Enzo Doria | 08/2016 |
2597 | (13*10^187382-157)/9 | 187383 | Robert Price | 05/2021 |
2598 | 2^622300-1861739 | 187331 | Enzo Doria | 12/2015 |
2599 | (22*10^187317-91)/3 | 187318 | Robert Price | 02/2022 |
2600 | (38*10^187281+349)/9 | 187282 | Robert Price | 10/2018 |
2601 | 2^622052+76771 | 187257 | Enzo Doria | 05/2016 |
2602 | 2^622005-24091 | 187243 | Enzo Doria | 08/2013 |
2603 | 7*10^187205-57 | 187206 | Robert Price | 06/2020 |
2604 | 2^621879-351787 | 187205 | Enzo Doria | 07/2015 |
2605 | (5*10^187168-53)/3 | 187169 | Robert Price | 09/2020 |
2606 | (23*10^187164-143)/3 | 187165 | Robert Price | 12/2019 |
2607 | 2^621700+79411 | 187151 | Enzo Doria | 06/2016 |
2608 | 2^621597+39 | 187120 | Jon Grantham | 08/2023 |
2609 | (7*10^187101+131)/3 | 187102 | Robert Price | 06/2020 |
2610 | (2*10^187062-71)/3 | 187062 | Robert Price | 12/2017 |
2611 | V(1118570) | 187013 | Serge Batalov | 07/2024 |
2612 | 2^620929-113203 | 186919 | Enzo Doria | 07/2014 |
2613 | 10^186842+168223 | 186843 | Patrick De Geest | 07/2018 |
2614 | 10^186842-8001 | 186842 | Robert Price | 03/2018 |
2615 | V(1117640)/13411681/26823361/575709775681/989889277441 | 186814 | Serge Batalov | 07/2024 |
2616 | 2^620347-1777 | 186744 | Lelio R Paula | 07/2013 |
2617 | (11*10^186659+133)/9 | 186660 | Robert Price | 04/2022 |
2618 | 11^179143-4170 | 186559 | Enzo Doria | 12/2012 |
2619 | (5^266863+1)/6 | 186529 | Paul Bourdelais | 07/2010 |
2620 | 2*10^186517+33 | 186518 | Robert Price | 11/2021 |
2621 | 2^619297-2792353 | 186427 | Enzo Doria | 02/2016 |
2622 | (49*10^186409+383)/9 | 186410 | Robert Price | 04/2019 |
2623 | 2^618940-460139 | 186320 | Enzo Doria | 06/2015 |
2624 | 2^618046+74179 | 186051 | Enzo Doria | 04/2016 |
2625 | 3*10^186019+11 | 186020 | Robert Price | 07/2018 |
2626 | (16*10^185978-19)/3 | 185979 | Robert Price | 02/2019 |
2627 | (22*10^185950-7)/3 | 185951 | Robert Price | 12/2022 |
2628 | 2*10^185930-3 | 185931 | Robert Price | 07/2015 |
2629 | (82^97151+1)/83 | 185928 | Paul Bourdelais | 10/2017 |
2630 | 4*10^185905+91 | 185906 | Robert Price | 08/2018 |
2631 | (34*10^185902-601)/9 | 185903 | Robert Price | 07/2022 |
2632 | (19*10^185873+131)/3 | 185874 | Robert Price | 06/2020 |
2633 | (37*10^185811+377)/9 | 185812 | Robert Price | 11/2018 |
2634 | 2^617250+9385 | 185811 | Lelio R Paula | 07/2013 |
2635 | 2^617130+9 | 185775 | Stefano Morozzi | 05/2021 |
2636 | 39^116749+2 | 185756 | Robert Price | 04/2015 |
2637 | 2^616924-39719 | 185713 | Enzo Doria | 09/2015 |
2638 | 2^616712+1867621 | 185649 | Enzo Doria | 04/2016 |
2639 | (3^388897+2^388897)/5 | 185551 | Jean-Louis Charton | 09/2011 |
2640 | 8*10^185529-67 | 185530 | Robert Price | 08/2019 |
2641 | (5*10^185520-11)/3 | 185521 | Tyler Busby | 02/2023 |
2642 | 5*2^616124+3 | 185473 | Jon Grantham | 07/2023 |
2643 | 2^615916+26791 | 185410 | Enzo Doria | 12/2023 |
2644 | 4*10^185355+9 | 185356 | Robert Price | 05/2015 |
2645 | (16*10^185347+89)/3 | 185348 | Robert Price | 03/2021 |
2646 | 40247^40247-56^56 | 185327 | Henri Lifchitz | 08/2024 |
2647 | 2^615622+76639 | 185321 | Enzo Doria | 05/2016 |
2648 | 2^615591+1879223 | 185312 | Enzo Doria | 08/2016 |
2649 | 2^615587+217013 | 185311 | Enzo Doria | 12/2023 |
2650 | 2^615503+1876643 | 185285 | Enzo Doria | 07/2016 |
2651 | 5^265007-4^265007 | 185232 | Jean-Louis Charton | 02/2013 |
2652 | (4*10^185221-91)/3 | 185222 | Robert Price | 03/2018 |
2653 | (7*10^185221+131)/3 | 185222 | Robert Price | 06/2020 |
2654 | (62*10^185216-413)/9 | 185217 | Robert Price | 09/2021 |
2655 | (232*10^185180+17)/3 | 185182 | Robert Price | 04/2021 |
2656 | 56*10^185139-3 | 185141 | Robert Price | 03/2020 |
2657 | (28*10^185105+47)/3 | 185106 | Robert Price | 02/2021 |
2658 | 2^614783-2790097 | 185069 | Enzo Doria | 12/2015 |
2659 | 2^614745-29323 | 185057 | Enzo Doria | 07/2015 |
2660 | (((((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840 | 185052 | Serge Batalov | 07/2013 |
2661 | (49*10^184993-67)/9 | 184994 | Robert Price | 03/2019 |
2662 | (5*10^184954-191)/3 | 184955 | Robert Price | 05/2021 |
2663 | (14*10^184950+211)/9 | 184951 | Robert Price | 04/2018 |
2664 | 2^614360+215221 | 184941 | Enzo Doria | 11/2023 |
2665 | 2^614315+81353 | 184928 | Enzo Doria | 08/2016 |
2666 | 2^614294-1849961 | 184921 | Enzo Doria | 08/2015 |
2667 | 2^614266+1877809 | 184913 | Enzo Doria | 08/2016 |
2668 | (185*10^184878-11)/3 | 184880 | Robert Price | 10/2020 |
2669 | 91*10^184867-9 | 184869 | Robert Price | 06/2021 |
2670 | 39803^40170-40170^39803 | 184779 | Norbert Schneider | 05/2024 |
2671 | (14789^44521-1)/(14789^211-1) | 184770 | Ryan Propper | 11/2023 |
2672 | (85*10^184768+77)/9 | 184769 | Robert Price | 07/2022 |
2673 | 2^613715-1860727 | 184747 | Enzo Doria | 11/2015 |
2674 | (4*10^184730+89)/3 | 184731 | Robert Price | 08/2021 |
2675 | 2*10^184723+41 | 184724 | Robert Price | 05/2018 |
2676 | (49*10^184703+293)/9 | 184704 | Robert Price | 03/2021 |
2677 | 2^613487-349987 | 184678 | Enzo Doria | 07/2015 |
2678 | (95^93377+1)/96 | 184672 | Paul Bourdelais | 11/2017 |
2679 | 14^161089-161089^14 | 184629 | Serge Batalov | 05/2016 |
2680 | 40008^40109-40109^40008 | 184588 | Norbert Schneider | 05/2024 |
2681 | 2^613033-114621 | 184542 | Enzo Doria | 05/2015 |
2682 | 3*2^612759+5 | 184460 | Jon Grantham | 07/2023 |
2683 | 2^612587+213413 | 184408 | Enzo Doria | 09/2023 |
2684 | (9^193247+1)/10 | 184404 | Paul Bourdelais | 02/2010 |
2685 | 2^612424+212341 | 184358 | Enzo Doria | 08/2023 |
2686 | 2^612341-1604809 | 184334 | Enzo Doria | 05/2015 |
2687 | (11*10^184308-161)/3 | 184309 | Robert Price | 11/2021 |
2688 | (2^611999-1)/18464214225958267477777390354183 | 184199 | Dario Alpern | 06/2017 |
2689 | (14*10^184150+37)/3 | 184151 | Robert Price | 12/2018 |
2690 | 2^611731-4837 | 184150 | Yves Minet | 08/2018 |
2691 | 3*10^184124-7 | 184125 | Robert Price | 08/2015 |
2692 | 2^611644+29371 | 184124 | Enzo Doria | 02/2024 |
2693 | 9*10^184122-11 | 184123 | Mischa Rodermond | 04/2023 |
2694 | 2^611628-24537 | 184119 | Enzo Doria | 08/2013 |
2695 | 3*10^184110+13 | 184111 | Robert Price | 07/2018 |
2696 | (14280^44521-1)/(14280^211-1) | 184097 | Ryan Propper | 11/2023 |
2697 | (194*10^184093+1)/3 | 184095 | Robert Price | 01/2022 |
2698 | 2^611460+71881 | 184068 | Enzo Doria | 04/2016 |
2699 | 4*10^184004-63 | 184005 | Robert Price | 10/2018 |
2700 | 2^611111+84083 | 183963 | Boyan Hu | 05/2024 |
2701 | 2^610982-1867301 | 183924 | Enzo Doria | 03/2016 |
2702 | (10^183917+1)/11/9952852373 | 183906 | Alfred Reich | 02/2024 |
2703 | 3*10^183872-49 | 183873 | Robert Price | 07/2018 |
2704 | 2^610597+1178897 | 183809 | Enzo Doria | 05/2016 |
2705 | (8*10^183773-269)/9 | 183773 | Robert Price | 09/2021 |
2706 | (65*10^183768-29)/9 | 183769 | Robert Price | 01/2021 |
2707 | (17*10^183764+31)/3 | 183765 | Robert Price | 01/2019 |
2708 | (28*10^183760-1)/9 | 183761 | Serge Batalov | 10/2014 |
2709 | 2^610401+1277 | 183750 | Lelio R Paula | 07/2013 |
2710 | 2^610356-1849019 | 183736 | Enzo Doria | 08/2015 |
2711 | (63^102103+1)/64 | 183717 | Paul Bourdelais | 10/2017 |
2712 | 23*10^183710-7 | 183712 | Robert Price | 02/2020 |
2713 | (2^610289+1)/(3*528882550291) | 183704 | Gord Palameta | 08/2018 |
2714 | (4*10^183659+149)/9 | 183659 | Robert Price | 04/2018 |
2715 | (4*10^183657+41)/9 | 183657 | Serge Batalov | 11/2022 |
2716 | 7*10^183596+39 | 183597 | Robert Price | 01/2023 |
2717 | 2^609846-1834251 | 183582 | Enzo Doria | 07/2015 |
2718 | 10^183546+1547323 | 183547 | Patrick De Geest | 02/2021 |
2720 | 10^183546-800000001 | 183546 | Robert Price | 09/2020 |
2719 | 10^183546-584297 | 183546 | Patrick De Geest | 02/2021 |
2721 | 7*10^183511+39 | 183512 | Robert Price | 01/2023 |
2722 | (23*10^183477-71)/3 | 183478 | Robert Price | 07/2020 |
2723 | 11^176176-240 | 183469 | Enzo Doria | 11/2012 |
2724 | (41^113749+1)/42 | 183451 | Paul Bourdelais | 11/2017 |
2725 | 6*10^183429+53 | 183430 | Robert Price | 05/2020 |
2726 | (28*10^183403+191)/3 | 183404 | Robert Price | 02/2020 |
2727 | U(1096855) | 183383 | Serge Batalov | 07/2024 |
2728 | (101^91463+1)/102 | 183320 | Paul Bourdelais | 11/2017 |
2729 | 2^608965-113989 | 183317 | Enzo Doria | 10/2014 |
2730 | A005165(43592) | 183312 | Serge Batalov | 07/2017 |
2731 | (61*10^183294+119)/9 | 183295 | Robert Price | 05/2019 |
2732 | 2^608814-461261 | 183272 | Enzo Doria | 06/2015 |
2733 | (163*10^183227-1)/9 | 183229 | Serge Batalov | 12/2014 |
2734 | 2^608541-608541^2 | 183190 | Henri Lifchitz | 02/2018 |
2735 | 2^608293+143807 | 183115 | Enzo Doria | 10/2016 |
2736 | 2^608164-462419 | 183076 | Enzo Doria | 08/2015 |
2737 | 2^607950-1862303 | 183012 | Enzo Doria | 12/2015 |
2738 | 5*3^383376+2^383376 | 182918 | Jon Grantham | 08/2023 |
2739 | (46*10^182896+683)/9 | 182897 | Robert Price | 03/2019 |
2740 | (2*10^182787+73)/3 | 182787 | Robert Price | 09/2020 |
2741 | 2^607101+1871987 | 182756 | Enzo Doria | 05/2016 |
2742 | 2^606869-69175 | 182686 | Enzo Doria | 11/2015 |
2743 | 8*10^182671+71 | 182672 | Robert Price | 05/2022 |
2744 | (23*10^182666-71)/3 | 182667 | Robert Price | 07/2020 |
2745 | 3*10^182614-11 | 182615 | Robert Price | 07/2018 |
2746 | 2^606560-357 | 182593 | Enzo Doria | 05/2014 |
2747 | 66*10^182420+7 | 182422 | Robert Price | 07/2020 |
2748 | 2^605687+1866983 | 182330 | Enzo Doria | 03/2016 |
2749 | 2^605655-114445 | 182321 | Enzo Doria | 02/2015 |
2750 | 94*10^182291-3 | 182293 | Robert Price | 05/2020 |