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 ? |
8501 | 21713^17914+17914^21713 | 92350 | Hans Havermann | 09/2020 |
8502 | 2^306672-109059 | 92318 | Enzo Doria | 04/2014 |
8503 | (430463^16384+1)/2 | 92307 | Richard Fischer | 09/2020 |
8504 | 6^118604-17063 | 92292 | Enzo Doria | 02/2013 |
8505 | 21969^15878+15878^21969 | 92288 | Hans Havermann | 08/2020 |
8506 | 21690^17959+17959^21690 | 92276 | Hans Havermann | 08/2020 |
8507 | (7613^23773-1)/7612 | 92273 | Serge Batalov | 08/2020 |
8508 | (31*2^306487-1)/2291 | 92260 | Lelio R Paula | 02/2014 |
8509 | 2^306467-1835611 | 92256 | Enzo Doria | 04/2015 |
8510 | 2^306433+76427 | 92246 | Enzo Doria | 05/2016 |
8511 | 2^306430-113313 | 92245 | Enzo Doria | 07/2014 |
8512 | (23*10^92244-59)/9 | 92245 | Robert Price | 03/2017 |
8513 | (47*2^306398+1)/1384803 | 92231 | Lelio R Paula | 04/2015 |
8514 | 21436^20043+20043^21436 | 92217 | Hans Havermann | 09/2020 |
8515 | (25*10^92186+59)/3 | 92187 | Robert Price | 05/2016 |
8516 | 2^306213-286183 | 92180 | Enzo Doria | 06/2015 |
8517 | 10^92178+61 | 92179 | Robert Price | 04/2011 |
8518 | 10^92178-175509 | 92178 | Patrick De Geest | 07/2013 |
8519 | 2^306200+1185 | 92176 | Lelio R Paula | 02/2016 |
8520 | (2^306171+1514)/2 | 92167 | Stefano Morozzi | 09/2020 |
8521 | 2^306153+11 | 92162 | Lelio R Paula | 11/2012 |
8522 | (2^306141-7)/5 | 92157 | Lelio R Paula | 06/2013 |
8523 | 21725^17454+17454^21725 | 92156 | Hans Havermann | 09/2020 |
8524 | (26*10^92152-23)/3 | 92153 | Robert Price | 08/2016 |
8525 | (2*10^92152+67)/3 | 92152 | Robert Price | 04/2017 |
8526 | 21368^20457+20457^21368 | 92115 | Hans Havermann | 08/2020 |
8527 | (2^305996+3018)/2 | 92114 | Stefano Morozzi | 09/2020 |
8528 | 4*10^92099+3*7^92099 | 92100 | Mike Oakes | 12/2010 |
8529 | 30504^1045+1045^30504 | 92096 | Hans Havermann | 08/2020 |
8530 | 24334^6085+6085^24334 | 92087 | Hans Havermann | 08/2020 |
8531 | (2^305898+2262)/2 | 92085 | Stefano Morozzi | 01/2021 |
8532 | (4591^25147-1)/4590 | 92083 | Serge Batalov | 08/2020 |
8533 | 115^44684+114 | 92081 | Norbert Schneider | 07/2018 |
8534 | (416771^16384+1)/2 | 92077 | Richard Fischer | 08/2020 |
8535 | (2^305867-2^152934+1)/5 | 92075 | | 11/2006 |
8536 | 2^305851-109495 | 92071 | Enzo Doria | 04/2014 |
8537 | (416213^16384+1)/2 | 92067 | Richard Fischer | 08/2020 |
8539 | 12^85300+35 | 92055 | Lelio R Paula | 06/2016 |
8538 | ((10^46026-1)*1000/9+739)*10^46026+(10^46026-1)/9 | 92055 | Hans Havermann | 05/2019 |
8540 | 21732^17201+17201^21732 | 92048 | Hans Havermann | 08/2020 |
8541 | 7*10^92043-39 | 92044 | Robert Price | 07/2020 |
8542 | 3^192911+92 | 92042 | Lelio R Paula | 04/2016 |
8543 | 2^305723-276271 | 92032 | Enzo Doria | 07/2015 |
8544 | (2^305714+3738)/2 | 92029 | Stefano Morozzi | 09/2020 |
8545 | (2^305707+798)/2 | 92027 | Stefano Morozzi | 01/2021 |
8546 | (2^305682+3202)/2 | 92020 | Stefano Morozzi | 09/2020 |
8547 | 2^305676+93 | 92018 | Donovan Johnson | 06/2008 |
8548 | 3^192819+4 | 91999 | Lelio R Paula | 04/2013 |
8549 | (38*10^91997-83)/99 | 91997 | Ray Chandler | 07/2011 |
8550 | 2^305539+72911 | 91977 | Enzo Doria | 04/2016 |
8551 | Phi(212014,10) | 91960 | Makoto Kamada | 04/2019 |
8552 | (2^305423+2666)/2 | 91942 | Stefano Morozzi | 09/2020 |
8553 | (2^305427-5)/1297998213 | 91934 | Lelio R Paula | 06/2013 |
8554 | 82077!4-128 | 91928 | Robert Price | 11/2019 |
8555 | 85^47642+84 | 91922 | Norbert Schneider | 10/2015 |
8556 | 10^91920+187651 | 91921 | Patrick De Geest | 07/2013 |
8557 | 10^91920-39 | 91920 | Robert Price | 05/2011 |
8558 | (3975^25537-1)/3974 | 91913 | Serge Batalov | 08/2020 |
8559 | (22*10^91903+41)/9 | 91904 | Robert Price | 02/2015 |
8561 | 2^305275-27 | 91897 | Lelio R Paula | 11/2012 |
8560 | 3^192607-7726 | 91897 | Enzo Doria | 05/2014 |
8562 | (2^305243+2274)/2 | 91887 | Stefano Morozzi | 02/2021 |
8563 | (2^305220+1278)/2 | 91881 | Stefano Morozzi | 01/2021 |
8565 | (6958^23911-1)/6957 | 91874 | Serge Batalov | 08/2020 |
8564 | 9*10^91873-91 | 91874 | Robert Price | 06/2021 |
8566 | (7*10^91856-61)/9 | 91856 | Erik Branger | 12/2009 |
8567 | (2^305107+1274)/2 | 91847 | Stefano Morozzi | 01/2021 |
8568 | (89*10^91839-53)/9 | 91840 | Robert Price | 11/2015 |
8569 | 1678^28479+28479^1678 | 91839 | Norbert Schneider | 08/2020 |
8570 | (2^304996+406)/2 | 91813 | Stefano Morozzi | 02/2021 |
8571 | 8*5^131351+7 | 91812 | Lelio R Paula | 10/2014 |
8572 | 3^192426-68 | 91811 | Lelio R Paula | 04/2016 |
8573 | 2^304979-375607 | 91808 | Harrison Mesh | 06/2022 |
8575 | 2^304973+117 | 91807 | Lelio R Paula | 11/2012 |
8574 | (2^304974+234)/2 | 91807 | Stefano Morozzi | 09/2020 |
8576 | 2^304939-1863727 | 91796 | Enzo Doria | 03/2016 |
8577 | 2^304902+8415 | 91785 | Lelio R Paula | 02/2016 |
8578 | 18^73119+61 | 91785 | Lelio R Paula | 06/2016 |
8579 | (78557*2^304740+1)/159 | 91739 | Lelio R Paula | 06/2014 |
8580 | 2^304747+71355 | 91738 | Lelio R Paula | 02/2016 |
8581 | (2^304738+2262)/2 | 91735 | Stefano Morozzi | 01/2021 |
8582 | 8*3^192251+5 | 91728 | Lelio R Paula | 08/2014 |
8583 | (2^304693+3746)/2 | 91722 | Stefano Morozzi | 01/2021 |
8584 | 2^304674-87 | 91717 | Lelio R Paula | 11/2012 |
8585 | (28*10^91715+701)/9 | 91716 | Robert Price | 11/2017 |
8586 | (2^304698+9)/1362809953 | 91715 | Lelio R Paula | 06/2013 |
8587 | (2^304671+1046)/2 | 91715 | Stefano Morozzi | 09/2020 |
8588 | (2^304667+3614)/2 | 91714 | Stefano Morozzi | 09/2020 |
8589 | (5*10^91707-119)/3 | 91708 | Robert Price | 04/2016 |
8590 | 2^304633-112771 | 91704 | Enzo Doria | 06/2014 |
8591 | (395347^16384+1)/2 | 91701 | Richard Fischer | 06/2020 |
8592 | 9*10^91693+23 | 91694 | Robert Price | 11/2017 |
8593 | 2^304578-16713 | 91688 | Enzo Doria | 05/2013 |
8594 | 6^117824-18713 | 91685 | Enzo Doria | 03/2013 |
8595 | (17^74509-1)/16 | 91679 | Paul Bourdelais | 03/2010 |
8596 | (2^304551-7)/8912231 | 91673 | Lelio R Paula | 06/2013 |
8597 | (2^304532-3)/3013 | 91670 | Lelio R Paula | 06/2013 |
8598 | 2^304507-16645 | 91666 | Enzo Doria | 04/2013 |
8599 | (29*2^304489+1)/31 | 91661 | Lelio R Paula | 09/2013 |
8600 | 21221^20860+20860^21221 | 91661 | Hans Havermann | 08/2020 |
8601 | 2^304471-1836127 | 91655 | Enzo Doria | 04/2015 |
8602 | 2^304433+32769 | 91644 | Lelio R Paula | 02/2016 |
8603 | 2^304410-16625 | 91637 | Enzo Doria | 04/2013 |
8604 | 183^40500-2 | 91630 | Lelio R Paula | 10/2017 |
8605 | (2^304373+1850)/2 | 91626 | Stefano Morozzi | 02/2021 |
8606 | (203*10^91613+1)/3 | 91615 | Robert Price | 09/2017 |
8607 | 4*5^131052+7 | 91603 | Lelio R Paula | 10/2014 |
8609 | 21725^16458+16458^21725 | 91601 | Hans Havermann | 08/2020 |
8608 | (2^304290+3270)/2 | 91601 | Stefano Morozzi | 02/2021 |
8610 | 2^304278-4001 | 91597 | Yves Minet | 04/2017 |
8611 | 2^304272-1866377 | 91595 | Enzo Doria | 03/2016 |
8612 | 2^304268+19683 | 91594 | Lelio R Paula | 02/2016 |
8613 | 21718^16497+16497^21718 | 91594 | Hans Havermann | 08/2020 |
8614 | 22759^10576+10576^22759 | 91590 | Hans Havermann | 08/2020 |
8615 | (29*10^91584-119)/9 | 91585 | Robert Price | 12/2021 |
8616 | 2^304162-161 | 91562 | Lelio R Paula | 11/2012 |
8617 | 5*7^108331+6 | 91552 | Lelio R Paula | 10/2014 |
8618 | 2^304082-2787821 | 91538 | Enzo Doria | 03/2016 |
8619 | 2^304015+64133 | 91518 | David Broadhurst | 06/2002 |
8620 | (7*2^303963+1)/3 | 91503 | Lelio R Paula | 09/2013 |
8621 | (2^303945-9)/3617 | 91494 | Lelio R Paula | 06/2013 |
8622 | (384007^16384+1)/2 | 91494 | Richard Fischer | 05/2020 |
8623 | 2^303930+1880527 | 91493 | Enzo Doria | 10/2016 |
8624 | 24810^4859+4859^24810 | 91464 | Hans Havermann | 08/2020 |
8625 | 2*10^91457+29 | 91458 | Robert Price | 02/2021 |
8626 | 3^191684-68 | 91457 | Lelio R Paula | 04/2016 |
8627 | (21*2^303813-1)/169 | 91456 | Lelio R Paula | 02/2014 |
8628 | (298*10^91446+11)/3 | 91448 | Robert Price | 11/2020 |
8629 | 11^87797-120 | 91432 | Enzo Doria | 07/2012 |
8630 | 2^303698-1847231 | 91423 | Enzo Doria | 07/2015 |
8631 | (38^28933-1)^2-2 | 91416 | Norbert Schneider | 06/2016 |
8632 | 25660^3643+3643^25660 | 91388 | Hans Havermann | 08/2020 |
8633 | 2^303554+77299 | 91379 | Enzo Doria | 05/2016 |
8634 | 2^303542+12705 | 91376 | Lelio R Paula | 02/2016 |
8635 | 31^61269-2 | 91375 | Lelio R Paula | 09/2013 |
8636 | (2^303525+2774)/2 | 91370 | Stefano Morozzi | 01/2021 |
8637 | 2^303516-64875 | 91368 | Lelio R Paula | 02/2016 |
8638 | (2^303497+1266)/2 | 91362 | Stefano Morozzi | 09/2020 |
8639 | (2^303497+3710)/2 | 91362 | Stefano Morozzi | 09/2020 |
8640 | (31*10^91352+257)/9 | 91353 | Robert Price | 12/2021 |
8641 | 21856^15121+15121^21856 | 91349 | Hans Havermann | 09/2020 |
8642 | (24737*2^303428+1)/3 | 91345 | Lelio R Paula | 06/2014 |
8643 | (21181*2^303357+1)/32361 | 91320 | Lelio R Paula | 06/2014 |
8644 | 23081^9046+9046^23081 | 91319 | Hans Havermann | 09/2020 |
8645 | 2^303344-2786969 | 91316 | Enzo Doria | 03/2016 |
8646 | 6^117346+65 | 91313 | Lelio R Paula | 05/2016 |
8647 | 2^303321+855 | 91309 | Lelio R Paula | 02/2016 |
8648 | (2^303315+3746)/2 | 91307 | Stefano Morozzi | 01/2021 |
8649 | 12^84606+47 | 91306 | Lelio R Paula | 06/2016 |
8650 | 2^303288+1874311 | 91299 | Enzo Doria | 06/2016 |
8651 | 197^39788-2 | 91293 | Lelio R Paula | 10/2017 |
8652 | 11^87663-1710 | 91292 | Enzo Doria | 12/2012 |
8653 | (4*10^91284-61)/3 | 91285 | Robert Price | 12/2021 |
8655 | 2^303224+2533 | 91280 | Henri Lifchitz | 05/2005 |
8654 | (2^303226+1830)/2 | 91280 | Stefano Morozzi | 02/2021 |
8656 | 6^117302+143 | 91279 | Lelio R Paula | 05/2016 |
8657 | 3^191306-112880 | 91277 | Enzo Doria | 07/2014 |
8658 | (10052^22807-1)/10051 | 91276 | Serge Batalov | 08/2020 |
8659 | 2^303207+12915 | 91275 | Lelio R Paula | 02/2016 |
8660 | 2^303191-111259 | 91270 | Enzo Doria | 05/2014 |
8661 | 81539!4+32 | 91267 | Robert Price | 09/2019 |
8662 | (2^303116+3570)/2 | 91247 | Stefano Morozzi | 09/2020 |
8663 | 10^91230+9^(91230+1) | 91231 | Robert Price | 08/2016 |
8664 | 10^91230+7921 | 91231 | Patrick De Geest | 11/2016 |
8665 | 10^91230-911141 | 91230 | Patrick De Geest | 11/2016 |
8666 | 10^91229+348793 | 91230 | Patrick De Geest | 07/2019 |
8667 | 10^91229-2471 | 91229 | Patrick De Geest | 01/2019 |
8668 | 2996^26241+26241^2996 | 91228 | Norbert Schneider | 01/2020 |
8669 | 2^303249-494953 | 91228 | Harrison Mesh | 06/2022 |
8671 | 10^91225+5271 | 91226 | Patrick De Geest | 01/2019 |
8670 | 2816^26445+26445^2816 | 91226 | Norbert Schneider | 02/2020 |
8673 | 6^117232-17423 | 91225 | Enzo Doria | 02/2013 |
8672 | 10^91225-147219 | 91225 | Patrick De Geest | 07/2019 |
8674 | 7*10^91215+3 | 91216 | Erik Branger | 11/2009 |
8675 | 10^91215+230611 | 91216 | Patrick De Geest | 07/2019 |
8676 | 10^91215-4539 | 91215 | Patrick De Geest | 01/2019 |
8677 | (2^303007+606)/2 | 91214 | Stefano Morozzi | 09/2020 |
8678 | 9*10^91212-47 | 91213 | Robert Price | 11/2021 |
8679 | (2^302998+3102)/2 | 91212 | Stefano Morozzi | 09/2020 |
8680 | 10^91210+2421 | 91211 | Patrick De Geest | 01/2019 |
8681 | 10^91210-1737 | 91210 | Patrick De Geest | 01/2019 |
8682 | 97*10^91205-3 | 91207 | Robert Price | 01/2017 |
8683 | 2^302979-161181 | 91206 | Enzo Doria | 04/2013 |
8684 | (166*10^91199-7)/3 | 91201 | Robert Price | 11/2020 |
8685 | 2^302949+81 | 91197 | Henri Lifchitz | 02/2007 |
8686 | 10^91194+200361 | 91195 | Patrick De Geest | 07/2019 |
8688 | 4*6^117192-5 | 91194 | Lelio R Paula | 10/2014 |
8687 | 10^91194-8663 | 91194 | Patrick De Geest | 01/2019 |
8689 | 2^302918+99 | 91188 | Lelio R Paula | 11/2012 |
8690 | 10^91186+268111 | 91187 | Patrick De Geest | 07/2019 |
8691 | 10^91186-7763 | 91186 | Patrick De Geest | 01/2019 |
8692 | 10^91181+550471 | 91182 | Patrick De Geest | 07/2019 |
8693 | (26*10^91181-83)/3 | 91182 | Robert Price | 12/2021 |
8694 | 10^91181-1367 | 91181 | Patrick De Geest | 01/2019 |
8695 | 10^91178+71883 | 91179 | Patrick De Geest | 07/2019 |
8696 | 10^91178-4719 | 91178 | Patrick De Geest | 01/2019 |
8697 | 24686^4935+4935^24686 | 91173 | Hans Havermann | 08/2020 |
8698 | 10^91165+3043 | 91166 | Patrick De Geest | 01/2019 |
8699 | 10^91165-399413 | 91165 | Patrick De Geest | 07/2019 |
8700 | 10^91156+3861 | 91157 | Patrick De Geest | 01/2019 |
8701 | 10^91156-231497 | 91156 | Patrick De Geest | 07/2019 |
8702 | 2^302797+1869587 | 91151 | Enzo Doria | 05/2016 |
8703 | 265^37614+37614^265 | 91148 | Norbert Schneider | 01/2015 |
8704 | 7^107852+82 | 91146 | Lelio R Paula | 04/2016 |
8705 | (38*10^91144-11)/9 | 91145 | Robert Price | 05/2015 |
8706 | (55*10^91142+377)/9 | 91143 | Robert Price | 10/2017 |
8707 | 10^91128+9961 | 91129 | Patrick De Geest | 01/2019 |
8709 | 249^38030+38030^249 | 91128 | Norbert Schneider | 01/2015 |
8708 | 10^91128-189153 | 91128 | Patrick De Geest | 07/2019 |
8710 | (2^302718+2814)/2 | 91127 | Stefano Morozzi | 02/2021 |
8711 | 6^117094-211049 | 91117 | Mike Oakes | 05/2011 |
8712 | 21353^18472+18472^21353 | 91103 | Hans Havermann | 05/2020 |
8714 | 2^302602-1867181 | 91093 | Enzo Doria | 03/2016 |
8713 | 10^91092+56127 | 91093 | Patrick De Geest | 07/2019 |
8715 | 10^91092-9107 | 91092 | Patrick De Geest | 01/2019 |
8716 | (1017*2^302592-1)/53 | 91091 | Lelio R Paula | 04/2015 |
8717 | 15^77445-16 | 91083 | Lelio R Paula | 07/2016 |
8718 | 21656^16053+16053^21656 | 91076 | Hans Havermann | 05/2020 |
8719 | 2^302505+81 | 91064 | Henri Lifchitz | 02/2007 |
8720 | 2^302496-112349 | 91061 | Enzo Doria | 06/2014 |
8721 | (2^302475+1790)/2 | 91054 | Stefano Morozzi | 01/2021 |
8722 | 17^73999-72 | 91052 | Lelio R Paula | 07/2016 |
8723 | 10^91047+9331 | 91048 | Patrick De Geest | 01/2019 |
8724 | 10^91047-671169 | 91047 | Patrick De Geest | 07/2019 |
8725 | 10^91044+2077 | 91045 | Patrick De Geest | 01/2019 |
8726 | 10^91044-136359 | 91044 | Patrick De Geest | 07/2019 |
8727 | 10^91041+9349 | 91042 | Patrick De Geest | 01/2019 |
8728 | 10^91041-7823 | 91041 | Patrick De Geest | 01/2019 |
8729 | 10^91040+4077 | 91041 | Patrick De Geest | 01/2019 |
8730 | 10^91040-119421 | 91040 | Patrick De Geest | 07/2019 |
8731 | 10^91037+2769 | 91038 | Patrick De Geest | 01/2019 |
8732 | 10^91037-22649 | 91037 | Patrick De Geest | 07/2019 |
8733 | (16*10^91032+41)/3 | 91033 | Robert Price | 10/2017 |
8734 | 10^91023+58039 | 91024 | Patrick De Geest | 07/2019 |
8735 | 10^91023-6471 | 91023 | Patrick De Geest | 01/2019 |
8736 | 10^91018+7873 | 91019 | Patrick De Geest | 01/2019 |
8737 | 10^91018-333177 | 91018 | Patrick De Geest | 07/2019 |
8738 | 10^91017+68767 | 91018 | Patrick De Geest | 07/2019 |
8739 | 10^91017-428327 | 91017 | Patrick De Geest | 07/2019 |
8740 | 10^91016+73111 | 91017 | Patrick De Geest | 07/2019 |
8741 | 10^91016-7973 | 91016 | Patrick De Geest | 01/2019 |
8742 | 1949^27666+27666^1949 | 91016 | Norbert Schneider | 04/2020 |
8743 | 2^302332+577294575 | 91012 | Lelio R Paula | 08/2016 |
8744 | 10^91004+9073 | 91005 | Patrick De Geest | 01/2019 |
8745 | 10^91004-491519 | 91004 | Patrick De Geest | 07/2019 |
8746 | 10^91003+74269 | 91004 | Patrick De Geest | 07/2019 |
8747 | 10^91003-3903 | 91003 | Patrick De Geest | 01/2019 |
8748 | 2^302274+74209 | 90994 | Enzo Doria | 04/2016 |
8749 | 10^90991+81147 | 90992 | Patrick De Geest | 07/2019 |
8750 | 10^90991-6279 | 90991 | Patrick De Geest | 01/2019 |