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 ? |
6251 | (11*10^110496+49)/3 | 110497 | Robert Price | 07/2018 |
6252 | 25447^21958+21958^25447 | 110481 | Hans Havermann | 11/2020 |
6253 | (31*2^366976+1)/7 | 110472 | Lelio R Paula | 03/2014 |
6254 | 7*10^110468-23 | 110469 | Robert Price | 07/2019 |
6255 | 5*3^231464-2^231464 | 110438 | Jon Grantham | 08/2023 |
6256 | 2^366853+5711 | 110434 | Lelio R Paula | 05/2017 |
6257 | (21181*2^366833+1)/3 | 110432 | Lelio R Paula | 07/2014 |
6258 | 2^366816+315 | 110423 | Lelio R Paula | 05/2017 |
6259 | 2^366775-2787037 | 110411 | Enzo Doria | 03/2016 |
6260 | (112*10^110405-1)/3 | 110407 | Serge Batalov | 12/2014 |
6261 | 2^366751+2797553 | 110404 | Enzo Doria | 03/2016 |
6262 | 25276^23331+23331^25276 | 110404 | Hans Havermann | 10/2020 |
6263 | 2^366748+255 | 110403 | Lelio R Paula | 05/2017 |
6264 | 2^366738-2985 | 110400 | Yves Minet | 09/2014 |
6265 | (23*2^366739+1)/74575 | 110396 | Lelio R Paula | 03/2014 |
6266 | (2^366722+1314)/2 | 110395 | Stefano Morozzi | 02/2021 |
6267 | 2^366716+1871791 | 110393 | Enzo Doria | 05/2016 |
6268 | (239*23^81066-349)/22 | 110391 | Eric Chen | 08/2023 |
6269 | 2^366649+1871687 | 110373 | Enzo Doria | 05/2016 |
6270 | 2^366629-351205 | 110367 | Enzo Doria | 06/2015 |
6271 | (31*10^110363+257)/9 | 110364 | Robert Price | 12/2021 |
6272 | (17*19^86292-215)/18 | 110347 | Eric Chen | 01/2023 |
6274 | 2^366558+1874479 | 110345 | Enzo Doria | 06/2016 |
6273 | 25132^24581+24581^25132 | 110345 | Hans Havermann | 10/2020 |
6275 | (2^366539+770)/2 | 110339 | Stefano Morozzi | 09/2020 |
6276 | 26368^15285+15285^26368 | 110331 | Hans Havermann | 10/2020 |
6277 | (23*10^110290+19)/3 | 110291 | Robert Price | 07/2019 |
6278 | 18^87853-4409 | 110280 | Enzo Doria | 03/2013 |
6279 | 28490^7419+7419^28490 | 110267 | Hans Havermann | 11/2020 |
6280 | 2^366218-23 | 110243 | Lelio R Paula | 02/2013 |
6281 | (21*2^366193-1)/197 | 110235 | Lelio R Paula | 03/2014 |
6282 | 2^366178-343991 | 110231 | Enzo Doria | 07/2015 |
6283 | 12^102123-11 | 110210 | Riley Fisher | 09/2023 |
6284 | 25777^18798+18798^25777 | 110174 | Hans Havermann | 10/2020 |
6285 | 2^365944+1871341 | 110161 | Enzo Doria | 05/2016 |
6286 | 2^365838-461267 | 110129 | Enzo Doria | 07/2015 |
6287 | 30152^4491+4491^30152 | 110126 | Hans Havermann | 11/2020 |
6288 | 2^365802+23359 | 110118 | Enzo Doria | 10/2023 |
6289 | 10^110111+464019 | 110112 | Patrick De Geest | 05/2021 |
6290 | 10^110111-40001 | 110111 | Robert Price | 01/2021 |
6291 | 2^365763-6555 | 110106 | Lelio R Paula | 04/2017 |
6292 | (2^365689+2^182845+1)/5 | 110083 | | 11/2006 |
6293 | 2^365666-459161 | 110077 | Enzo Doria | 05/2015 |
6294 | 3*10^110076-7 | 110077 | Robert Price | 08/2015 |
6295 | 13^98785-72 | 110041 | Lelio R Paula | 03/2017 |
6296 | (25^78707+1)/26 | 110027 | Robert Price | 06/2013 |
6297 | (7*10^110027-277)/9 | 110027 | Robert Price | 02/2022 |
6298 | (16*10^110014-73)/3 | 110015 | Robert Price | 07/2020 |
6299 | (7*10^110006-151)/9 | 110006 | Robert Price | 02/2018 |
6300 | 6^141361-7 | 110001 | Lelio R Paula | 11/2014 |
6301 | 20^84536+81 | 109984 | Lelio R Paula | 03/2017 |
6302 | 5*2^365347-9 | 109982 | Lelio R Paula | 09/2014 |
6303 | 9*6^141303+7 | 109957 | Lelio R Paula | 11/2014 |
6304 | 27635^9522+9522^27635 | 109953 | Hans Havermann | 11/2020 |
6305 | 78^58099+77 | 109929 | Norbert Schneider | 02/2015 |
6306 | 2*10^109928-7 | 109929 | Robert Price | 10/2015 |
6307 | 30580^3933+3933^30580 | 109927 | Hans Havermann | 11/2020 |
6308 | 26380^14689+14689^26380 | 109926 | Hans Havermann | 11/2020 |
6309 | (2^365161+1130)/2 | 109925 | Stefano Morozzi | 09/2020 |
6310 | (27*2^365137-1)/2021 | 109916 | Lelio R Paula | 03/2014 |
6311 | (4441*29^75152-24)/7 | 109905 | Curtis Bright | 05/2015 |
6312 | (2^365087+7)/6015 | 109899 | Lelio R Paula | 01/2014 |
6313 | (2^365069+3774)/2 | 109897 | Stefano Morozzi | 02/2021 |
6314 | 29652^5083+5083^29652 | 109894 | Mark Rodenkirch | 10/2020 |
6315 | (2^365049+722)/2 | 109891 | Stefano Morozzi | 09/2020 |
6316 | (35*10^109888-89)/9 | 109889 | Robert Price | 02/2022 |
6317 | 18^87533-11 | 109878 | Lelio R Paula | 04/2017 |
6318 | 2^364982-5 | 109871 | Henri Lifchitz | 04/2006 |
6319 | 3*10^109861+59 | 109862 | Robert Price | 08/2018 |
6320 | 5*10^109860-69 | 109861 | Robert Price | 02/2021 |
6321 | 2^364936-1867979 | 109857 | Enzo Doria | 04/2016 |
6322 | 27013^11652+11652^27013 | 109846 | Hans Havermann | 10/2020 |
6323 | 2^364867-112087 | 109836 | Enzo Doria | 06/2014 |
6324 | 4^182415-85061 | 109825 | Mike Oakes | 06/2011 |
6325 | (2^364809+1034)/2 | 109819 | Stefano Morozzi | 09/2020 |
6326 | (53*10^109809-287)/9 | 109810 | Robert Price | 02/2022 |
6327 | (52*10^109807-133)/9 | 109808 | Robert Price | 09/2021 |
6328 | 25708^18675+18675^25708 | 109806 | Hans Havermann | 10/2020 |
6329 | (17^89227+16^89227)/33 | 109788 | Robert Price | 08/2013 |
6330 | 2^364687-187 | 109782 | Lelio R Paula | 01/2012 |
6331 | 2^364644+67 | 109769 | Lelio R Paula | 01/2012 |
6332 | (9787*2^364625-1)/24947 | 109763 | Lelio R Paula | 05/2015 |
6333 | (2^364616+5)/9 | 109760 | Lelio R Paula | 01/2014 |
6334 | (55*10^109759+89)/9 | 109760 | Robert Price | 03/2019 |
6335 | (14*32^72921+4171)/31 | 109757 | Eric Chen | 04/2023 |
6336 | 12^101694+73 | 109747 | Lelio R Paula | 03/2017 |
6337 | 13^98518-66 | 109744 | Lelio R Paula | 03/2017 |
6338 | (8*10^109721-269)/9 | 109721 | Robert Price | 09/2021 |
6339 | 2^364478-113261 | 109719 | Enzo Doria | 07/2014 |
6340 | (2^364513+1)/(3*44290516579) | 109719 | Gord Palameta | 08/2018 |
6341 | (24737*2^364454+1)/3 | 109716 | Lelio R Paula | 07/2014 |
6342 | 27814^8777+8777^27814 | 109681 | Hans Havermann | 10/2020 |
6343 | (62*10^109673+1)/9 | 109674 | Serge Batalov | 11/2010 |
6344 | 3^229824+340 | 109654 | Lelio R Paula | 03/2017 |
6345 | (16623*33^72206-8095)/16 | 109649 | Eric Chen | 05/2023 |
6346 | (2*10^109645-71)/3 | 109645 | Robert Price | 12/2017 |
6347 | 2^364169-112813 | 109626 | Enzo Doria | 06/2014 |
6348 | 28822^6359+6359^28822 | 109622 | Hans Havermann | 11/2020 |
6349 | 10^109615+21 | 109616 | Lelio R Paula | 03/2017 |
6350 | 10^109615-456267 | 109615 | Patrick De Geest | 03/2017 |
6351 | 2^364094-113 | 109604 | Lelio R Paula | 01/2012 |
6352 | 25251^21898+21898^25251 | 109600 | Hans Havermann | 11/2020 |
6353 | (4^182026+11)/3 | 109591 | Robert Price | 02/2017 |
6354 | 2^364041-6883 | 109588 | Lelio R Paula | 09/2011 |
6355 | 12^101545+11 | 109586 | Lelio R Paula | 03/2017 |
6356 | 26536^13475+13475^26536 | 109582 | Hans Havermann | 11/2020 |
6357 | 2^363842-2787101 | 109528 | Enzo Doria | 03/2016 |
6358 | (2^363812+3570)/2 | 109519 | Stefano Morozzi | 09/2020 |
6359 | (22699*2^363782+1)/763 | 109511 | Lelio R Paula | 07/2014 |
6360 | 2^363769+5711 | 109506 | Lelio R Paula | 04/2017 |
6361 | (31*2^363748+1)/49 | 109499 | Lelio R Paula | 03/2014 |
6362 | (2^363670+100280245065) | 109476 | Paul Bourdelais | 05/2022 |
6363 | p(22245)#/2-8 | 109462 | Robert Price | 01/2017 |
6364 | 2^363556+55 | 109442 | Lelio R Paula | 01/2012 |
6365 | (509*10^109414+31)/9 | 109416 | Robert Price | 08/2022 |
6366 | (2^363437+1682)/2 | 109406 | Stefano Morozzi | 09/2020 |
6367 | (55459*2^363411+1)/81 | 109401 | Lelio R Paula | 07/2014 |
6368 | 2^363406+1871569 | 109397 | Enzo Doria | 05/2016 |
6369 | (129^51829-1)/128 | 109388 | Anonymous | 06/2024 |
6370 | (2^363370+1338)/2 | 109385 | Stefano Morozzi | 02/2021 |
6371 | (2^363349+7)/9 | 109378 | Lelio R Paula | 01/2014 |
6372 | (13869^26569-1)/(13869^163-1) | 109375 | Ryan Propper | 11/2023 |
6373 | 3^229236+100 | 109374 | Lelio R Paula | 03/2017 |
6374 | (2^363315+3258)/2 | 109369 | Stefano Morozzi | 02/2021 |
6375 | 2^363268-58695 | 109355 | Lelio R Paula | 04/2017 |
6376 | 2^363191-112321 | 109332 | Enzo Doria | 06/2014 |
6377 | 26231^14718+14718^26231 | 109327 | Hans Havermann | 10/2020 |
6378 | 25461^19672+19672^25461 | 109326 | Hans Havermann | 10/2020 |
6379 | 4*10^109316-63 | 109317 | Robert Price | 10/2018 |
6380 | 2^363120+3 | 109311 | Donovan Johnson | 04/2006 |
6381 | (10^109297-1)/9 | 109297 | Paul Bourdelais, Harvey Dubner | 04/2007 |
6382 | 2^363042-114641 | 109287 | Enzo Doria | 04/2015 |
6383 | 2^363020-112793 | 109280 | Enzo Doria | 06/2014 |
6384 | (8732^27889-1)/(8732^167-1) | 109256 | Ryan Propper | 11/2023 |
6385 | (85*10^109253+203)/9 | 109254 | Robert Price | 09/2020 |
6386 | 24990^23431+23431^24990 | 109202 | Hans Havermann | 10/2020 |
6387 | (2^362746+3738)/2 | 109198 | Stefano Morozzi | 09/2020 |
6388 | (2^362716+1282)/2 | 109189 | Stefano Morozzi | 01/2021 |
6389 | (67607*2^362695+1)/70499 | 109183 | Lelio R Paula | 07/2014 |
6390 | 25060^22737+22737^25060 | 109180 | Hans Havermann | 10/2020 |
6391 | (9*2^362644+1)/5 | 109167 | Lelio R Paula | 03/2014 |
6392 | 5^156181+38 | 109166 | Lelio R Paula | 02/2017 |
6393 | (2^362623+3618)/2 | 109161 | Stefano Morozzi | 09/2020 |
6394 | 25138^21993+21993^25138 | 109157 | Hans Havermann | 10/2020 |
6395 | 4*10^109142+91 | 109143 | Robert Price | 08/2018 |
6396 | 3^228750+68 | 109142 | Lelio R Paula | 03/2017 |
6397 | (2^362561+786)/2 | 109142 | Stefano Morozzi | 09/2020 |
6398 | (23*10^109113+61)/3 | 109114 | Robert Price | 01/2022 |
6399 | 2*10^109100-49 | 109101 | Robert Price | 09/2020 |
6400 | (10223*2^362376+1)/9 | 109090 | Lelio R Paula | 07/2014 |
6401 | 65^60167-64^60167 | 109078 | Robert Price | 08/2013 |
6402 | 7*3^228589-2 | 109066 | Lelio R Paula | 09/2014 |
6403 | 24874^24243+24243^24874 | 109063 | Hans Havermann | 11/2020 |
6404 | (17*10^109059-161)/9 | 109060 | Robert Price | 02/2021 |
6405 | 29158^5495+5495^29158 | 109050 | Hans Havermann | 11/2020 |
6406 | (78557*2^362188+1)/189 | 109033 | Lelio R Paula | 07/2014 |
6407 | 2*7^129008+9 | 109025 | Lelio R Paula | 11/2014 |
6408 | 88*10^109022+7 | 109024 | Robert Price | 03/2020 |
6409 | 28194^7337+7337^28194 | 108985 | Hans Havermann | 10/2020 |
6410 | (5*10^108960+91)/3 | 108961 | Robert Price | 04/2018 |
6411 | 2^361957+76367 | 108960 | Enzo Doria | 05/2016 |
6412 | 2^361873-1411 | 108935 | Yves Minet | 01/2013 |
6413 | (24737*2^361857+1)/35 | 108933 | Lelio R Paula | 07/2014 |
6414 | 115^52842-114 | 108892 | Norbert Schneider | 07/2018 |
6415 | 13^97745-24 | 108883 | Lelio R Paula | 03/2017 |
6416 | (2^361696+2562)/2 | 108882 | Stefano Morozzi | 09/2020 |
6417 | 17^88475+8 | 108864 | Lelio R Paula | 04/2017 |
6418 | 2^361567+51 | 108843 | Lelio R Paula | 01/2012 |
6419 | 2^361563-109501 | 108842 | Enzo Doria | 04/2014 |
6420 | 12^100852+83 | 108838 | Lelio R Paula | 03/2017 |
6421 | 2^361497+767 | 108822 | Lelio R Paula | 01/2012 |
6422 | L(520691)/78103651/1057002731/42794551909 | 108791 | Serge Batalov | 07/2024 |
6423 | 28019^7630+7630^28019 | 108785 | Hans Havermann | 10/2020 |
6424 | 24723^24758-24758^24723 | 108765 | Norbert Schneider | 11/2020 |
6425 | 2^361280-112349 | 108757 | Enzo Doria | 06/2014 |
6426 | 2^361271-5955 | 108754 | Lelio R Paula | 04/2017 |
6427 | 12^100757+19 | 108736 | Lelio R Paula | 03/2017 |
6428 | 25368^19333+19333^25368 | 108735 | Hans Havermann | 11/2020 |
6429 | (2^361195+3618)/2 | 108731 | Stefano Morozzi | 09/2020 |
6430 | 2^361153-111546435 | 108718 | Paul Bourdelais | 10/2021 |
6431 | 14^94849-61 | 108710 | Lelio R Paula | 05/2017 |
6432 | 27267^9700+9700^27267 | 108708 | Hans Havermann | 11/2020 |
6433 | (19^85009-5^85009)/14 | 108705 | Robert Price | 07/2013 |
6434 | 24832^23853+23853^24832 | 108704 | Hans Havermann | 11/2020 |
6435 | 27502^8963+8963^27502 | 108701 | Hans Havermann | 11/2020 |
6436 | (130^51419-1)/129 | 108695 | Anonymous | 06/2024 |
6437 | 25807^16278+16278^25807 | 108689 | Hans Havermann | 11/2020 |
6438 | (11*2^361027+1)/40463 | 108677 | Lelio R Paula | 03/2014 |
6439 | 2^360973+1299 | 108664 | Lelio R Paula | 04/2017 |
6440 | 2^360959-109101 | 108660 | Enzo Doria | 04/2014 |
6441 | 6*5^155454+7 | 108659 | Lelio R Paula | 11/2014 |
6442 | 10^108656+79 | 108657 | Lelio R Paula | 02/2017 |
6443 | 10^108656-50001 | 108656 | Patrick De Geest | 03/2017 |
6444 | 2^360918-35 | 108648 | Lelio R Paula | 01/2012 |
6445 | 4*10^108646-47 | 108647 | Robert Price | 10/2021 |
6446 | (11*2^360886-1)/7 | 108638 | Lelio R Paula | 03/2014 |
6447 | 3*10^108624-49 | 108625 | Robert Price | 07/2018 |
6448 | (2^360831+666)/2 | 108621 | Stefano Morozzi | 09/2020 |
6449 | (134*10^108610+7)/3 | 108612 | Robert Price | 02/2020 |
6450 | 2*10^108609+29 | 108610 | Robert Price | 02/2021 |
6451 | (10^108604+7)/71/191 | 108600 | Alfred Reich | 06/2021 |
6452 | (2^360742+3222)/2 | 108594 | Stefano Morozzi | 09/2020 |
6453 | 2^360706-458903 | 108584 | Enzo Doria | 04/2015 |
6454 | 31112^3087+3087^31112 | 108567 | Hans Havermann | 11/2020 |
6455 | 2^360641-113185 | 108564 | Enzo Doria | 07/2014 |
6456 | (29*10^108559-71)/3 | 108560 | Robert Price | 07/2021 |
6457 | 2^360618+75019 | 108557 | Enzo Doria | 04/2016 |
6458 | 2^360607-216367 | 108554 | Harrison Mesh | 06/2022 |
6459 | (14*10^108545-221)/9 | 108546 | Robert Price | 04/2018 |
6460 | (44*10^108544-233)/9 | 108545 | Robert Price | 01/2019 |
6461 | 3*5^155286+2 | 108541 | Lelio R Paula | 11/2014 |
6462 | 7^128422+4 | 108530 | Lelio R Paula | 11/2014 |
6463 | 7^128421-80 | 108529 | Lelio R Paula | 03/2017 |
6464 | 2^360497+53807 | 108521 | Yves Minet | 07/2010 |
6465 | 2^360468-4559 | 108512 | Yves Minet | 03/2019 |
6466 | (2^360451+578)/2 | 108507 | Stefano Morozzi | 09/2020 |
6467 | 2^360366+1459 | 108481 | Lelio R Paula | 09/2011 |
6468 | 2^360364-1499 | 108481 | Yves Minet | 05/2013 |
6469 | 25662^16871+16871^25662 | 108477 | Hans Havermann | 10/2020 |
6470 | 25009^21748+21748^25009 | 108475 | Hans Havermann | 10/2020 |
6471 | 14^94642-23 | 108472 | Lelio R Paula | 05/2017 |
6472 | (197*10^108446-11)/3 | 108448 | Robert Price | 03/2020 |
6473 | (2^360229+1290)/2 | 108440 | Stefano Morozzi | 01/2021 |
6474 | 4*10^108412+9 | 108413 | Robert Price | 05/2015 |
6475 | (259*10^108409-1)/3 | 108411 | Robert Price | 12/2020 |
6476 | (2^360122+1194)/2 | 108408 | Stefano Morozzi | 09/2020 |
6477 | 133^51042-132 | 108406 | Norbert Schneider | 02/2019 |
6478 | 2^360093-351103 | 108399 | Enzo Doria | 06/2015 |
6479 | 2^360042+723 | 108384 | Lelio R Paula | 01/2012 |
6480 | 85^56164-2 | 108364 | Lelio R Paula | 10/2013 |
6481 | 2^359937-29323 | 108352 | Enzo Doria | 07/2015 |
6482 | (2^359922+1762)/2 | 108348 | Stefano Morozzi | 09/2020 |
6483 | (2^359903+758)/2 | 108342 | Stefano Morozzi | 09/2020 |
6484 | 2^359898-19997 | 108341 | Enzo Doria | 06/2013 |
6485 | (2^359886+2122)/2 | 108337 | Stefano Morozzi | 09/2020 |
6486 | 2*5^154948-7 | 108305 | Lelio R Paula | 11/2014 |
6487 | 10^108303+13 | 108304 | Lelio R Paula | 02/2017 |
6488 | 10^108303-207959 | 108303 | Patrick De Geest | 03/2017 |
6489 | 7*10^108300+69 | 108301 | Robert Price | 07/2019 |
6490 | 2^359730+753 | 108290 | Lelio R Paula | 12/2011 |
6491 | (8*10^108273+13)/3 | 108274 | Robert Price | 06/2018 |
6492 | 2^359650-19797 | 108266 | Enzo Doria | 06/2013 |
6493 | (47*10^108261+259)/9 | 108262 | Robert Price | 03/2022 |
6494 | 91^55259-90^55259 | 108255 | Robert Price | 07/2015 |
6495 | 10^108251+85749 | 108252 | Patrick De Geest | 02/2021 |
6496 | 10^108251-50000000001 | 108251 | Robert Price | 12/2020 |
6497 | 10^108251-252081 | 108251 | Patrick De Geest | 02/2021 |
6498 | 4*10^108238-71 | 108239 | Robert Price | 12/2018 |
6499 | 3^226845+68 | 108233 | Lelio R Paula | 03/2017 |
6500 | 31836^2509+2509^31836 | 108227 | Hans Havermann | 10/2020 |