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 ? |
9251 | (2^297129+726)/2 | 89445 | Stefano Morozzi | 01/2020 |
9252 | 2^297122+99 | 89443 | Lelio R Paula | 10/2012 |
9253 | (3^187435-5)/1286 | 89427 | Lelio R Paula | 01/2013 |
9254 | (28*10^89425+101)/3 | 89426 | Robert Price | 09/2020 |
9255 | (2^297037+3522)/2 | 89417 | Stefano Morozzi | 01/2020 |
9256 | (25*10^89413+161)/3 | 89414 | Robert Price | 01/2017 |
9257 | (89*10^89411-737)/9 | 89412 | Robert Price | 01/2022 |
9258 | 29080^1187+1187^29080 | 89406 | Hans Havermann | 08/2020 |
9259 | 2^296990+1870699 | 89403 | Enzo Doria | 05/2016 |
9260 | 2^296972-1017 | 89398 | Lelio R Paula | 11/2013 |
9261 | (7*10^89383+167)/3 | 89384 | Robert Kinser | 10/2017 |
9262 | (85*10^89375-31)/9 | 89376 | Robert Price | 11/2015 |
9263 | 9*2^296870+7 | 89368 | Lelio R Paula | 08/2014 |
9264 | 100^44683-99^44683 | 89366 | Robert Price | 07/2013 |
9265 | 2^296810+76519 | 89349 | Enzo Doria | 05/2016 |
9266 | (2747^25981-1)/2746 | 89342 | Serge Batalov | 08/2020 |
9267 | 8^98928+7^(98928+1) | 89341 | Robert Price | 06/2016 |
9269 | (3^187247+4)/1027 | 89337 | Lelio R Paula | 01/2013 |
9268 | 21832^12359+12359^21832 | 89337 | Hans Havermann | 08/2020 |
9270 | (3^187235-2)/2045 | 89331 | Lelio R Paula | 01/2013 |
9271 | 2^296736-432149 | 89327 | Paul Bourdelais | 05/2013 |
9272 | 2^296737-112519 | 89327 | Enzo Doria | 06/2014 |
9273 | 2^296718-3645 | 89322 | Lelio R Paula | 11/2013 |
9274 | (2^296702+2338)/2 | 89316 | Stefano Morozzi | 01/2020 |
9275 | 2^296681-16221 | 89310 | Enzo Doria | 04/2013 |
9276 | 2^296669-16513 | 89307 | Enzo Doria | 04/2013 |
9277 | (2^296626+1282)/2 | 89294 | Stefano Morozzi | 01/2020 |
9278 | 2^296619-16357 | 89292 | Enzo Doria | 04/2013 |
9279 | 2^296614-110051 | 89290 | Enzo Doria | 04/2014 |
9280 | (2^296601+1730)/2 | 89286 | Stefano Morozzi | 01/2020 |
9281 | (3^187138+8)/83147 | 89283 | Lelio R Paula | 01/2013 |
9282 | (2*10^89279+457)/9 | 89279 | Robert Price | 01/2017 |
9283 | (2^296568+1222)/2 | 89276 | Stefano Morozzi | 01/2020 |
9284 | (133*10^89273+17)/3 | 89275 | Robert Price | 11/2021 |
9285 | 15^75907-28 | 89274 | Lelio R Paula | 11/2015 |
9286 | 2^296547-37 | 89270 | Yves Minet | 01/2012 |
9287 | 2^296485+191 | 89251 | Lelio R Paula | 10/2012 |
9288 | 2^296467-75 | 89246 | Lelio R Paula | 10/2012 |
9289 | (2^296462+1674)/2 | 89244 | Stefano Morozzi | 01/2020 |
9290 | 2^296416-109535 | 89231 | Enzo Doria | 04/2014 |
9291 | (23*2^296406+1)/3 | 89228 | Lelio R Paula | 01/2013 |
9292 | 66^49037-65 | 89225 | Norbert Schneider | 11/2014 |
9293 | 2^296377-16245 | 89219 | Enzo Doria | 04/2013 |
9294 | 2^296372-64875 | 89217 | Lelio R Paula | 11/2013 |
9295 | 5*6^114648-7 | 89215 | Lelio R Paula | 10/2014 |
9296 | (2^296361+1262)/2 | 89214 | Stefano Morozzi | 01/2020 |
9297 | 3*10^89206+13 | 89207 | Robert Price | 08/2017 |
9298 | 7*3^186960-4 | 89204 | Lelio R Paula | 08/2014 |
9299 | (28*10^89203+47)/3 | 89204 | Robert Price | 02/2021 |
9300 | 2^296290-112133 | 89193 | Enzo Doria | 06/2014 |
9301 | 6^114620-17003 | 89192 | Enzo Doria | 02/2013 |
9302 | (22*10^89190+179)/3 | 89191 | Robert Price | 02/2022 |
9303 | (4*10^89189+41)/9 | 89189 | Robert Price | 11/2014 |
9304 | 2^296261-161181 | 89184 | Enzo Doria | 04/2013 |
9305 | (19*2^296269-1)/530623 | 89182 | Lelio R Paula | 02/2014 |
9306 | 3^186903-10 | 89176 | Robert Price | 09/2013 |
9307 | 7*10^89169-3 | 89170 | Dmitry Domanov | 11/2010 |
9308 | (2^296221-1017)/35 | 89170 | Lelio R Paula | 01/2013 |
9309 | (43*10^89169+47)/9 | 89170 | Robert Price | 12/2017 |
9310 | (10^89160+29)/3 | 89160 | Serge Batalov | 12/2014 |
9311 | (28*10^89141-97)/3 | 89142 | Robert Price | 07/2016 |
9312 | 2^296116-734639 | 89140 | Paul Bourdelais | 04/2013 |
9313 | (2^296101+1250)/2 | 89135 | Stefano Morozzi | 01/2020 |
9314 | 22574^8883+8883^22574 | 89135 | Hans Havermann | 08/2020 |
9315 | 6^114544-133 | 89133 | Lelio R Paula | 12/2015 |
9316 | 2^296076-113297 | 89128 | Enzo Doria | 07/2014 |
9317 | (11*2^296060-1)/215 | 89122 | Lelio R Paula | 02/2014 |
9318 | 5*10^89118+81 | 89119 | Robert Price | 08/2017 |
9319 | 2^296037-1411 | 89117 | Yves Minet | 01/2013 |
9320 | 2^296020-2604809 | 89111 | Enzo Doria | 05/2015 |
9321 | 2^296016+81 | 89110 | Henri Lifchitz | 02/2007 |
9322 | 2^296010-11 | 89108 | Lelio R Paula | 10/2012 |
9323 | 2^295998-111911 | 89105 | Enzo Doria | 05/2014 |
9324 | (2^296014+47)/93991149 | 89102 | Lelio R Paula | 01/2013 |
9325 | 29529^1040+1040^29529 | 89090 | Hans Havermann | 08/2020 |
9326 | 2^295939-16621 | 89087 | Enzo Doria | 04/2013 |
9327 | 2^295934-109691 | 89086 | Enzo Doria | 04/2014 |
9328 | (2^295921+2246)/2 | 89081 | Stefano Morozzi | 01/2020 |
9329 | (185*10^89070-11)/3 | 89072 | Robert Price | 10/2020 |
9330 | (185*10^89063+7)/3 | 89065 | Robert Price | 02/2017 |
9331 | (8*10^89055-59)/3 | 89056 | Robert Price | 08/2021 |
9332 | 2^295831+143 | 89055 | Lelio R Paula | 10/2012 |
9333 | 31^59713-30 | 89054 | Theodore Burton | 02/2007 |
9334 | (7*2^295834-1)/27663 | 89052 | Lelio R Paula | 02/2014 |
9335 | 79731!4-32 | 89049 | Robert Price | 11/2019 |
9337 | 123^42608+2 | 89047 | Lelio R Paula | 05/2013 |
9336 | 6^114433-114433^6 | 89047 | Norbert Schneider | 04/2017 |
9338 | 2^295800-459659 | 89045 | Enzo Doria | 05/2015 |
9339 | (52*10^89037-1)/3 | 89039 | Robert Price | 02/2017 |
9340 | (2^295782+1870)/2 | 89039 | Stefano Morozzi | 01/2020 |
9341 | (6^114398-11)/5 | 89019 | Paul Bourdelais | 11/2011 |
9342 | 11^85470-5850 | 89008 | Enzo Doria | 02/2013 |
9343 | (2^295650+522)/2 | 89000 | Stefano Morozzi | 01/2020 |
9344 | (35*10^88998+37)/9 | 88999 | Robert Price | 05/2022 |
9345 | (27*2^295634+1)/4687 | 88993 | Lelio R Paula | 01/2013 |
9346 | (p(18424)#-4)/2 | 88985 | Robert Price | 09/2017 |
9347 | 2^295564-189 | 88974 | Lelio R Paula | 10/2012 |
9348 | 2^295557+73097 | 88972 | Enzo Doria | 04/2016 |
9349 | 11^85433-5760 | 88970 | Enzo Doria | 01/2013 |
9350 | 23074^7165+7165^23074 | 88956 | Hans Havermann | 08/2020 |
9351 | (2^295479+3050)/2 | 88948 | Stefano Morozzi | 01/2020 |
9352 | 6*10^88938-91 | 88939 | Robert Price | 11/2017 |
9353 | 15^75619-88 | 88935 | Lelio R Paula | 11/2015 |
9354 | (2^295433+2234)/2 | 88934 | Stefano Morozzi | 01/2020 |
9355 | (15589^21211-1)/15588 | 88930 | Serge Batalov | 08/2020 |
9356 | 23776^5499+5499^23776 | 88929 | Hans Havermann | 08/2020 |
9357 | 21112^16291+16291^21112 | 88923 | Hans Havermann | 08/2020 |
9358 | 2^295385-85 | 88920 | Lelio R Paula | 10/2012 |
9359 | 2^295385-16333 | 88920 | Enzo Doria | 04/2013 |
9360 | (89*10^88916-377)/9 | 88917 | Robert Price | 07/2021 |
9361 | 79617!4+32 | 88909 | Robert Price | 09/2019 |
9362 | 22077^10640+10640^22077 | 88903 | Hans Havermann | 08/2020 |
9363 | 2^295309+59 | 88897 | Lelio R Paula | 10/2012 |
9364 | 2^295288-109799 | 88891 | Enzo Doria | 04/2014 |
9365 | 10^88888+170373 | 88889 | Stefano Morozzi | 07/2018 |
9366 | 10^88888+271399 | 88889 | Stefano Morozzi | 07/2018 |
9367 | 10^88888+375397 | 88889 | Stefano Morozzi | 07/2018 |
9368 | 10^88888+548823 | 88889 | Stefano Morozzi | 07/2018 |
9369 | 10^88888+820861 | 88889 | Stefano Morozzi | 07/2018 |
9370 | 10^88888+1206349 | 88889 | Stefano Morozzi | 07/2018 |
9371 | 10^88888+1292047 | 88889 | Stefano Morozzi | 07/2018 |
9372 | 10^88888+1456029 | 88889 | Stefano Morozzi | 07/2018 |
9373 | 10^88888-490287 | 88888 | Patrick De Geest | 09/2018 |
9374 | 2^295245+65 | 88878 | Lelio R Paula | 10/2012 |
9375 | 5*10^88874-37 | 88875 | Robert Price | 11/2021 |
9377 | Padovan(727734) | 88874 | Eric W. Weisstein | 04/2011 |
9376 | (2^295231+1554)/2 | 88874 | Stefano Morozzi | 01/2020 |
9378 | 79585!4-32 | 88870 | Robert Price | 11/2019 |
9379 | 18^70793-17 | 88865 | Lelio R Paula | 11/2015 |
9380 | 6^114196-17573 | 88862 | Enzo Doria | 02/2013 |
9381 | 302^35829+35829^302 | 88857 | Norbert Schneider | 12/2014 |
9382 | 3^186223+16 | 88851 | Robert Price | 10/2013 |
9383 | 3*2^295139+7 | 88847 | Lelio R Paula | 08/2014 |
9384 | 3^186212-113008 | 88846 | Enzo Doria | 07/2014 |
9385 | 2*10^88844+23 | 88845 | Robert Price | 12/2020 |
9386 | 23415^6224+6224^23415 | 88839 | Hans Havermann | 08/2020 |
9387 | 87^45803-86 | 88836 | Norbert Schneider | 11/2015 |
9388 | 2^295094-110273 | 88833 | Enzo Doria | 05/2014 |
9389 | 2^295080+1869151 | 88828 | Enzo Doria | 05/2016 |
9390 | (2^295061+2174)/2 | 88822 | Stefano Morozzi | 01/2020 |
9391 | 2^295027-113407 | 88812 | Enzo Doria | 07/2014 |
9392 | (2*10^88808-71)/3 | 88808 | Robert Price | 01/2017 |
9393 | (2^295009+2102)/2 | 88807 | Stefano Morozzi | 01/2020 |
9394 | 13^79722-80 | 88806 | Lelio R Paula | 11/2015 |
9395 | 21112^16077+16077^21112 | 88802 | Hans Havermann | 08/2020 |
9396 | 3^186103+104 | 88794 | Lelio R Paula | 12/2015 |
9397 | (3*2^294971+1)/25565 | 88792 | Lelio R Paula | 01/2013 |
9398 | (4025^24631-1)/4024 | 88786 | Serge Batalov | 08/2020 |
9399 | 2^294937-115003 | 88785 | Enzo Doria | 06/2015 |
9400 | (2^294923+3134)/2 | 88781 | Stefano Morozzi | 01/2020 |
9401 | (15326^21211-1)/15325 | 88773 | Serge Batalov | 08/2020 |
9402 | 2551^26056+26056^2551 | 88766 | Norbert Schneider | 01/2020 |
9403 | 2^294868-109709 | 88765 | Enzo Doria | 04/2014 |
9404 | 2^294864-2985 | 88763 | Yves Minet | 09/2014 |
9405 | (17*10^88761-41)/3 | 88762 | Robert Price | 09/2017 |
9406 | (2^294834+2854)/2 | 88754 | Stefano Morozzi | 01/2020 |
9407 | (2^294823+2466)/2 | 88751 | Stefano Morozzi | 01/2020 |
9408 | (2^294764+642)/2 | 88733 | Stefano Morozzi | 01/2020 |
9409 | 2^294754+15015 | 88730 | Lelio R Paula | 10/2013 |
9410 | (3^185951-7)/20 | 88720 | Lelio R Paula | 01/2013 |
9411 | (2^294688+3186)/2 | 88710 | Stefano Morozzi | 01/2020 |
9412 | (2^294673+3906)/2 | 88706 | Stefano Morozzi | 01/2020 |
9413 | (2^294661+5)/443693383 | 88694 | Lelio R Paula | 01/2013 |
9414 | (2^294631+2514)/2 | 88693 | Stefano Morozzi | 01/2020 |
9415 | (10^88680+11)/3 | 88680 | Robert Price | 11/2014 |
9416 | 2*7^104928-9 | 88675 | Lelio R Paula | 10/2014 |
9417 | 11^85150+12 | 88675 | Lelio R Paula | 11/2015 |
9418 | 8*6^113951+5 | 88673 | Lelio R Paula | 10/2014 |
9419 | (2^294549+3110)/2 | 88668 | Stefano Morozzi | 01/2020 |
9420 | 6*10^88665-79 | 88666 | Robert Price | 03/2022 |
9421 | 6^113940-18563 | 88663 | Enzo Doria | 03/2013 |
9422 | 38^56123-56123^38 | 88663 | Norbert Schneider | 02/2017 |
9423 | 2^294492-65 | 88651 | Lelio R Paula | 10/2012 |
9424 | (2^294481+2822)/2 | 88648 | Stefano Morozzi | 01/2020 |
9425 | 23874^5165+5165^23874 | 88646 | Hans Havermann | 08/2020 |
9426 | (2^294469+1466)/2 | 88644 | Stefano Morozzi | 01/2020 |
9427 | 7*10^88634-27 | 88635 | Dylan Delgado | 08/2019 |
9428 | 20^68120-53 | 88627 | Lelio R Paula | 01/2016 |
9429 | 122^42477-121 | 88623 | Norbert Schneider | 10/2018 |
9430 | 2^294391-113257 | 88621 | Enzo Doria | 07/2014 |
9431 | 5^126784-15476 | 88619 | Enzo Doria | 02/2013 |
9432 | (2^294335+554)/2 | 88604 | Stefano Morozzi | 01/2020 |
9433 | 2^294327+9 | 88602 | Lelio R Paula | 10/2012 |
9434 | 2^294320+18915 | 88600 | Lelio R Paula | 10/2013 |
9435 | (98*10^88595-89)/99 | 88595 | Ray Chandler | 07/2011 |
9436 | (5*10^88595+103)/9 | 88595 | Robert Price | 04/2022 |
9437 | 20723^18834+18834^20723 | 88590 | Hans Havermann | 08/2020 |
9438 | (68*10^88586-329)/9 | 88587 | Robert Price | 03/2021 |
9439 | (2^294268+1402)/2 | 88584 | Stefano Morozzi | 01/2020 |
9440 | (17*2^294262-1)/13 | 88582 | Lelio R Paula | 02/2014 |
9441 | 6*10^88571+77 | 88572 | Robert Price | 06/2020 |
9442 | 2^294224-933075 | 88571 | Paul Bourdelais | 04/2021 |
9443 | (2^294145+1346)/2 | 88547 | Stefano Morozzi | 01/2020 |
9444 | (2*10^88545+7)/9 | 88545 | Erik Branger | 03/2011 |
9445 | 2^294137-133 | 88545 | Lelio R Paula | 10/2012 |
9446 | (2^294109+3494)/2 | 88536 | Stefano Morozzi | 01/2020 |
9447 | 23515^5818+5818^23515 | 88529 | Hans Havermann | 08/2020 |
9448 | (2^294092-1017)/1171 | 88528 | Lelio R Paula | 01/2013 |
9449 | 4^147036+147036^4+1 | 88525 | Robert Price | 04/2014 |
9450 | 22541^8458+8458^22541 | 88525 | Hans Havermann | 08/2020 |
9451 | (14^77213-5^77213)/9 | 88496 | Robert Price | 04/2013 |
9452 | 2^293965+2793377 | 88493 | Enzo Doria | 03/2016 |
9453 | (2^293956+3306)/2 | 88490 | Stefano Morozzi | 01/2020 |
9454 | 11^84971-150 | 88489 | Enzo Doria | 08/2012 |
9455 | (2^293944+1066)/2 | 88486 | Stefano Morozzi | 01/2020 |
9456 | 9*10^88485+91 | 88486 | Robert Price | 10/2020 |
9457 | 48^52626-47 | 88478 | Norbert Schneider | 08/2018 |
9458 | (2^293956-1017)/39851376696677 | 88476 | Lelio R Paula | 01/2013 |
9459 | 11^84958-2190 | 88475 | Enzo Doria | 12/2012 |
9460 | (2^293902+3070)/2 | 88474 | Stefano Morozzi | 01/2020 |
9461 | (35*10^88471-89)/9 | 88472 | Robert Price | 02/2022 |
9462 | 2^293880-111347 | 88467 | Enzo Doria | 05/2014 |
9463 | (106*10^88459-1)/3 | 88461 | Serge Batalov | 12/2014 |
9464 | 10^88457+278733 | 88458 | Patrick De Geest | 07/2017 |
9465 | 10^88457-401 | 88457 | Robert Price | 06/2017 |
9466 | (2^293834+3018)/2 | 88453 | Stefano Morozzi | 01/2020 |
9467 | (2^293747+1626)/2 | 88427 | Stefano Morozzi | 01/2020 |
9468 | 22080^10109+10109^22080 | 88424 | Hans Havermann | 08/2020 |
9469 | 11^84908-1680 | 88423 | Enzo Doria | 12/2012 |
9470 | (2^293703+3486)/2 | 88414 | Stefano Morozzi | 01/2020 |
9471 | 2^293679-112075 | 88407 | Enzo Doria | 06/2014 |
9472 | 6^113608-65 | 88405 | Lelio R Paula | 12/2015 |
9473 | 2^293624+82875 | 88390 | Lelio R Paula | 10/2013 |
9474 | (2^293618+918)/2 | 88388 | Stefano Morozzi | 01/2020 |
9475 | 2^293612-112337 | 88387 | Enzo Doria | 06/2014 |
9476 | 758^30693+30693^758 | 88386 | Norbert Schneider | 05/2017 |
9477 | gcd(V(1062695),5*F(212539)*(F(212539)-1)+1) | 88383 | Serge Batalov | 04/2015 |
9478 | 5*10^88377+77 | 88378 | Robert Price | 06/2016 |
9479 | (2^293584+1558)/2 | 88378 | Stefano Morozzi | 01/2020 |
9480 | (5*2^293526+1)/303 | 88359 | Lelio R Paula | 01/2013 |
9481 | 20918^16743+16743^20918 | 88355 | Hans Havermann | 08/2020 |
9482 | (2^293499+2390)/2 | 88352 | Stefano Morozzi | 01/2020 |
9483 | 20727^18292+18292^20727 | 88344 | Hans Havermann | 08/2020 |
9485 | 2^293461-2785843 | 88341 | Enzo Doria | 03/2016 |
9484 | (31*10^88340+257)/9 | 88341 | Robert Price | 12/2021 |
9487 | 2^293416-1037 | 88328 | Yves Minet | 11/2012 |
9486 | 6*10^88327+43 | 88328 | Robert Price | 07/2020 |
9488 | (58*10^88321-49)/9 | 88322 | Robert Price | 05/2022 |
9489 | 25109^3292+3292^25109 | 88320 | Hans Havermann | 08/2020 |
9490 | (2^293384-7)/3 | 88317 | Lelio R Paula | 01/2013 |
9491 | (3^185092-5)/4 | 88311 | Lelio R Paula | 01/2013 |
9492 | (5*10^88303-173)/3 | 88304 | Robert Price | 11/2017 |
9493 | 14^77039-57 | 88297 | Lelio R Paula | 11/2015 |
9494 | 3*10^88293-59 | 88294 | Robert Price | 10/2017 |
9495 | 5^126303+42 | 88283 | Lelio R Paula | 10/2015 |
9496 | (88*10^88269+83)/9 | 88270 | Robert Price | 03/2022 |
9498 | 19^69025-18 | 88266 | Theodore Burton | 01/2006 |
9497 | 140^41128+139 | 88266 | Norbert Schneider | 04/2019 |
9499 | 5*10^88261-3 | 88262 | Dmitry Domanov | 11/2010 |
9500 | (2^293195+2238)/2 | 88261 | Stefano Morozzi | 01/2020 |