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 ? |
5251 | 10^118009+664447 | 118010 | Patrick De Geest | 07/2018 |
5252 | 10^118009-20001 | 118009 | Robert Price | 01/2018 |
5253 | (14*10^117991+61)/3 | 117992 | Robert Price | 12/2018 |
5254 | 2^391933+5897 | 117984 | Yves Minet | 02/2011 |
5255 | 2^391912-1849259 | 117978 | Enzo Doria | 08/2015 |
5256 | (2*34^77029+1043)/11 | 117968 | Eric Chen | 08/2023 |
5257 | 5*2^391836-3 | 117956 | Jon Grantham | 07/2023 |
5258 | 28357^14440+14440^28357 | 117953 | Hans Havermann | 02/2021 |
5259 | (2^391812+5)/14181 | 117944 | Lelio R Paula | 04/2014 |
5260 | (7*10^117930-79)/9 | 117930 | Robert Price | 01/2021 |
5261 | 27531^19168+19168^27531 | 117904 | Hans Havermann | 02/2021 |
5262 | (46*10^117902+683)/9 | 117903 | Robert Price | 03/2019 |
5263 | 27670^18189+18189^27670 | 117869 | Hans Havermann | 02/2021 |
5264 | (7*2^391521+1)/285 | 117858 | Lelio R Paula | 04/2014 |
5265 | (56*10^117856+583)/9 | 117857 | Robert Price | 08/2021 |
5266 | 32169^4606+4606^32169 | 117846 | Hans Havermann | 02/2021 |
5267 | (71*10^117803-143)/9 | 117804 | Robert Price | 10/2021 |
5268 | 2^391324+90901 | 117801 | Yves Minet | 02/2011 |
5269 | 2^391301-1859923 | 117794 | Enzo Doria | 03/2016 |
5270 | 2^391261-4131 | 117782 | Yves Minet | 02/2018 |
5271 | (2^391211+554)/2 | 117766 | Stefano Morozzi | 09/2020 |
5272 | 38434^1157+1157^38434 | 117737 | Hans Havermann | 01/2021 |
5273 | (2^391115-7)/11791 | 117734 | Lelio R Paula | 04/2014 |
5274 | (85*10^117719+473)/9 | 117720 | Robert Price | 01/2020 |
5275 | (2^391049+582)/2 | 117718 | Stefano Morozzi | 09/2020 |
5276 | (21*23^86445-197)/22 | 117715 | Eric Chen | 08/2023 |
5277 | 31083^6122+6122^31083 | 117709 | Hans Havermann | 02/2021 |
5278 | 2^390960+1880881 | 117691 | Enzo Doria | 11/2016 |
5279 | (71*10^117685+91)/9 | 117686 | Robert Price | 04/2022 |
5281 | 2^390897-1860553 | 117672 | Enzo Doria | 11/2015 |
5280 | (61*10^117671+47)/9 | 117672 | Robert Price | 07/2022 |
5282 | 2^390889-115003 | 117670 | Enzo Doria | 06/2015 |
5283 | 12^109010+35 | 117642 | Riley Fisher | 08/2023 |
5284 | 2^390779-30337 | 117637 | Paul Bourdelais | 02/2012 |
5285 | (2^390735+378)/2 | 117623 | Stefano Morozzi | 02/2021 |
5286 | 11^112916-780 | 117590 | Enzo Doria | 12/2012 |
5287 | 27641^17916+17916^27641 | 117564 | Hans Havermann | 02/2021 |
5288 | 27906^16321+16321^27906 | 117561 | Hans Havermann | 02/2021 |
5289 | 27294^20275+20275^27294 | 117555 | Hans Havermann | 02/2021 |
5290 | (2^390473+1002)/2 | 117544 | Stefano Morozzi | 09/2020 |
5291 | 35912^1875+1875^35912 | 117541 | Hans Havermann | 02/2021 |
5292 | (16^97613-5^97613)/11 | 117537 | Robert Price | 07/2013 |
5293 | (2^390438+1774)/2 | 117534 | Stefano Morozzi | 01/2021 |
5294 | (2^390418+1722)/2 | 117528 | Stefano Morozzi | 09/2020 |
5295 | (2^390403+654)/2 | 117523 | Stefano Morozzi | 09/2020 |
5296 | (2^390343+2246)/2 | 117505 | Stefano Morozzi | 01/2021 |
5297 | 28043^15478+15478^28043 | 117493 | Hans Havermann | 02/2021 |
5298 | (17*2^390272+1)/1983 | 117482 | Lelio R Paula | 04/2014 |
5299 | 37077^1474+1474^37077 | 117479 | Hans Havermann | 01/2021 |
5300 | 31^78759-30 | 117459 | Theodore Burton | 02/2007 |
5301 | (2^390176+1222)/2 | 117455 | Stefano Morozzi | 09/2020 |
5302 | (2^390133+1242)/2 | 117442 | Stefano Morozzi | 09/2020 |
5303 | (2^390013+2634)/2 | 117406 | Stefano Morozzi | 09/2020 |
5304 | 3*10^117402+61 | 117403 | Robert Price | 09/2020 |
5305 | 2^389998+37 | 117402 | Lelio R Paula | 03/2013 |
5306 | (2^389977+314)/2 | 117395 | Stefano Morozzi | 09/2020 |
5307 | (2^389921+342)/2 | 117378 | Stefano Morozzi | 01/2021 |
5308 | 71830^43+43^71830 | 117333 | Hans Havermann | 01/2021 |
5309 | (2^389768+1582)/2 | 117332 | Stefano Morozzi | 09/2020 |
5310 | 33802^2951+2951^33802 | 117292 | Hans Havermann | 02/2021 |
5311 | 26789^23866+23866^26789 | 117277 | Hans Havermann | 02/2021 |
5312 | 29^80192-1*80192^29 | 117273 | Dylan Delgado | 12/2019 |
5313 | (101*10^117260+7)/9 | 117262 | Robert Price | 02/2022 |
5314 | (13*10^117249+161)/3 | 117250 | Robert Price | 11/2018 |
5315 | 2^389478+43 | 117245 | Lelio R Paula | 03/2013 |
5316 | (298*10^117229-7)/3 | 117231 | Robert Price | 06/2020 |
5317 | (2^389416+342)/2 | 117226 | Stefano Morozzi | 01/2021 |
5318 | (29*2^389351-1)/9 | 117207 | Lelio R Paula | 04/2014 |
5319 | 5*10^117205+23 | 117206 | Robert Price | 03/2019 |
5320 | (2^389341-7)/3865 | 117200 | Lelio R Paula | 03/2014 |
5321 | 27272^19827+19827^27272 | 117195 | Hans Havermann | 01/2021 |
5322 | (2^389276+2650)/2 | 117184 | Stefano Morozzi | 09/2020 |
5323 | (844*10^117175-7)/9 | 117177 | Robert Price | 05/2021 |
5324 | 10^117142+209223 | 117143 | Patrick De Geest | 04/2018 |
5326 | 10^117142-1000000001 | 117142 | Robert Price | 03/2018 |
5325 | 10^117142-966057 | 117142 | Patrick De Geest | 04/2018 |
5327 | 10^117135+360577 | 117136 | Patrick De Geest | 07/2018 |
5328 | 10^117135-401 | 117135 | Robert Price | 01/2018 |
5329 | 26415^26488-26488^26415 | 117126 | Norbert Schneider | 05/2021 |
5330 | 4*10^117123+79 | 117124 | Robert Price | 08/2018 |
5331 | 2^389033-12797413 | 117111 | Enzo Doria | 04/2016 |
5332 | (2^388987+2634)/2 | 117097 | Stefano Morozzi | 09/2020 |
5333 | 2^388960-109505 | 117089 | Enzo Doria | 04/2014 |
5334 | 7*10^117082-57 | 117083 | Robert Price | 06/2020 |
5335 | (2^388928-5)/569 | 117077 | Lelio R Paula | 03/2014 |
5336 | 2^388888-1529 | 117067 | Yves Minet | 01/2013 |
5337 | 10^117059+28879 | 117060 | Patrick De Geest | 05/2016 |
5338 | (10^117059-71)/1 | 117059 | Serge Batalov | 12/2014 |
5339 | (2^388854+3630)/2 | 117057 | Stefano Morozzi | 09/2020 |
5340 | (2^388842+2098)/2 | 117053 | Stefano Morozzi | 09/2020 |
5341 | (2^388789+3150)/2 | 117037 | Stefano Morozzi | 09/2020 |
5342 | (2^388748+378)/2 | 117025 | Stefano Morozzi | 02/2021 |
5343 | (693^41189-1)/692 | 117005 | Michael Stocker | 05/2020 |
5344 | (26*10^116999-503)/9 | 117000 | Robert Price | 07/2018 |
5345 | (2^388609+1554)/2 | 116983 | Stefano Morozzi | 09/2020 |
5346 | 38496^1093+1093^38496 | 116975 | Hans Havermann | 02/2021 |
5347 | 3^245154-6256 | 116969 | Enzo Doria | 05/2014 |
5348 | 5*10^116966-43 | 116967 | Robert Price | 01/2019 |
5349 | 2^388536-113495 | 116961 | Enzo Doria | 08/2014 |
5350 | (2^388512+382)/2 | 116954 | Stefano Morozzi | 02/2021 |
5351 | (8*32^77701+271)/31 | 116952 | Eric Chen | 11/2023 |
5352 | 2^388469-43 | 116941 | Lelio R Paula | 03/2013 |
5353 | 27417^18416+18416^27417 | 116939 | Hans Havermann | 02/2021 |
5354 | (2^388438+1842)/2 | 116932 | Stefano Morozzi | 02/2021 |
5355 | (7*2^388372-1)/111 | 116911 | Lelio R Paula | 04/2014 |
5356 | (7*2^388335-1)/25 | 116900 | Lelio R Paula | 04/2014 |
5357 | 2^388180+93 | 116854 | Donovan Johnson | 07/2008 |
5358 | (2^388167+2510)/2 | 116850 | Stefano Morozzi | 09/2020 |
5359 | (85*10^116842+293)/9 | 116843 | Robert Price | 12/2019 |
5360 | 35296^2043+2043^35296 | 116840 | Hans Havermann | 02/2021 |
5361 | (2^388081+7)/1179 | 116821 | Lelio R Paula | 03/2014 |
5362 | 38633^1056+1056^38633 | 116814 | Hans Havermann | 02/2021 |
5363 | 2^387946-113261 | 116784 | Enzo Doria | 07/2014 |
5364 | 28*36^75009+305 | 116739 | Eric Chen | 12/2022 |
5365 | 28363^13056+13056^28363 | 116737 | Hans Havermann | 02/2021 |
5366 | 2^387745+75557 | 116723 | Enzo Doria | 04/2016 |
5367 | (29*10^116719+313)/9 | 116720 | Robert Price | 04/2021 |
5368 | 29622^8699+8699^29622 | 116695 | Hans Havermann | 02/2021 |
5369 | (206*10^116687-17)/9 | 116689 | Robert Price | 01/2020 |
5370 | (7*10^116684+179)/3 | 116685 | Robert Price | 09/2018 |
5371 | 2^387554+71455 | 116666 | Enzo Doria | 04/2016 |
5372 | (602*10^116647-71)/9 | 116649 | Robert Price | 12/2021 |
5373 | 3^244461-10 | 116638 | Tyler NeSmith | 09/2022 |
5374 | (2^387447+78557)/365 | 116631 | Lelio R Paula | 12/2014 |
5375 | (2*10^116620+43)/9 | 116620 | Serge Batalov | 11/2022 |
5376 | (19*10^116617+287)/9/12893 | 116614 | Alfred Reich | 11/2022 |
5377 | (145*10^116610-1)/9 | 116612 | Serge Batalov | 12/2014 |
5378 | (25*10^116601-331)/9 | 116602 | Robert Price | 08/2018 |
5379 | (17*2^387284-1)/59 | 116584 | Lelio R Paula | 04/2014 |
5380 | (2^387284+3726)/2 | 116584 | Stefano Morozzi | 09/2020 |
5381 | 2^387211+111546435 | 116563 | Paul Bourdelais | 10/2021 |
5382 | 10^116557+987654321 | 116558 | Robert Price | 12/2019 |
5383 | 10^116557+9141 | 116558 | Patrick De Geest | 02/2020 |
5384 | 10^116557-609431 | 116557 | Patrick De Geest | 02/2020 |
5385 | (28*10^116546+41)/3 | 116547 | Robert Price | 07/2021 |
5386 | (2^387123+1814)/2 | 116536 | Stefano Morozzi | 02/2021 |
5387 | (2^387076+2046)/2 | 116522 | Stefano Morozzi | 09/2020 |
5388 | (2^387073+2570)/2 | 116521 | Stefano Morozzi | 09/2020 |
5389 | (7*10^116514-1)/3 | 116515 | Robert Price | 08/2022 |
5390 | (26*32^77406-390071011)/31 | 116508 | Eric Chen | 11/2023 |
5391 | 28846^10931+10931^28846 | 116500 | Hans Havermann | 02/2021 |
5392 | 10^116496+3593037 | 116497 | Paul Bourdelais | 10/2021 |
5393 | (31*2^386968+1)/7 | 116490 | Lelio R Paula | 03/2014 |
5394 | 2^386947+1869143 | 116483 | Enzo Doria | 05/2016 |
5395 | (30^78857-1)/29 | 116480 | Robert Price | 12/2011 |
5396 | (2^386903-3)/5 | 116469 | Lelio R Paula | 03/2014 |
5397 | 28107^13892+13892^28107 | 116441 | Hans Havermann | 02/2021 |
5398 | 29110^9957+9957^29110 | 116386 | Hans Havermann | 02/2021 |
5399 | 26469^24838+24838^26469 | 116335 | Hans Havermann | 01/2021 |
5400 | 2^386389-1864783 | 116315 | Enzo Doria | 03/2016 |
5401 | (8*32^77250-163)/31 | 116273 | Eric Chen | 11/2023 |
5402 | (31*2^386231-1)/1711 | 116266 | Lelio R Paula | 03/2014 |
5403 | 297^47012+47012^297 | 116250 | Norbert Schneider | 07/2018 |
5404 | (35*10^116223-341)/9 | 116224 | Robert Price | 08/2021 |
5405 | (2^386087-5)/40959 | 116220 | Lelio R Paula | 03/2014 |
5406 | (2^386054+714)/2 | 116214 | Stefano Morozzi | 09/2020 |
5407 | (10^116201+7)/39623 | 116197 | Alfred Reich | 06/2021 |
5408 | 2^385969-823 | 116189 | Yves Minet | 05/2012 |
5409 | 57^66162+2 | 116173 | Lelio R Paula | 08/2013 |
5410 | 2^385854+75049 | 116154 | Enzo Doria | 04/2016 |
5411 | (2^385832+2590)/2 | 116147 | Stefano Morozzi | 09/2020 |
5412 | 2^385775-109587 | 116130 | Enzo Doria | 04/2014 |
5413 | (14*10^116112-101)/3 | 116113 | Robert Price | 02/2019 |
5414 | Phi(236354,10)/24817171 | 116109 | Makoto Kamada | 08/2019 |
5415 | 3*10^116107+23 | 116108 | Robert Price | 03/2022 |
5416 | 28729^10990+10990^28729 | 116094 | Hans Havermann | 02/2021 |
5417 | 2^385622+71839 | 116084 | Enzo Doria | 04/2016 |
5419 | 2^385564+781 | 116067 | Lelio R Paula | 01/2012 |
5418 | (2^385565+1562)/2 | 116067 | Stefano Morozzi | 09/2020 |
5420 | (2^385563+378)/2 | 116066 | Stefano Morozzi | 02/2021 |
5421 | (2^385557+2214)/2 | 116064 | Stefano Morozzi | 09/2020 |
5422 | 29431^8776+8776^29431 | 116056 | Hans Havermann | 01/2021 |
5423 | (266*10^116024-17)/3 | 116026 | Robert Price | 07/2020 |
5424 | 30263^6820+6820^30263 | 116022 | Hans Havermann | 02/2021 |
5425 | 2^385410+71893 | 116020 | Enzo Doria | 04/2016 |
5426 | 38225^1084+1084^38225 | 116014 | Hans Havermann | 02/2021 |
5427 | 26548^23433+23433^26548 | 116011 | Hans Havermann | 02/2021 |
5428 | 12^107422-11 | 115928 | Riley Fisher | 12/2023 |
5429 | (25*2^385091+1)/27 | 115924 | Lelio R Paula | 04/2014 |
5430 | (11*10^115910-131)/3 | 115911 | Robert Price | 09/2018 |
5431 | 2^384953-111943 | 115883 | Paul Bourdelais | 02/2012 |
5432 | 43^70942-2 | 115882 | Lelio R Paula | 04/2014 |
5433 | (94*10^115879-7)/3 | 115881 | Robert Price | 03/2020 |
5434 | 7*10^115863+3 | 115864 | Robert Price | 08/2015 |
5435 | 2^384834+93 | 115847 | Donovan Johnson | 07/2008 |
5436 | 2^384809+71297 | 115840 | Enzo Doria | 04/2016 |
5437 | (55*10^115829+323)/9 | 115830 | Robert Price | 10/2021 |
5438 | (7*2^384759-1)/5 | 115825 | Lelio R Paula | 04/2014 |
5439 | 31348^4951+4951^31348 | 115822 | Hans Havermann | 02/2021 |
5440 | (793*10^115810+53)/9 | 115812 | Robert Price | 01/2022 |
5441 | (71*10^115810-17)/9 | 115811 | Ray Chandler | 08/2011 |
5442 | (97*10^115794-1)/3 | 115796 | Serge Batalov | 12/2014 |
5443 | 31115^5266+5266^31115 | 115794 | Hans Havermann | 02/2021 |
5444 | 2*10^115784-27 | 115785 | Robert Price | 05/2018 |
5445 | 10^115773+126739 | 115774 | Patrick De Geest | 04/2018 |
5447 | 10^115773-1000000001 | 115773 | Robert Price | 03/2018 |
5446 | 10^115773-396221 | 115773 | Patrick De Geest | 04/2018 |
5448 | (2^384582+778)/2 | 115771 | Stefano Morozzi | 09/2020 |
5449 | 30537^6178+6178^30537 | 115762 | Hans Havermann | 01/2021 |
5450 | 2^384530+1874305 | 115756 | Enzo Doria | 06/2016 |
5451 | (3*2^384524-1)/7631 | 115750 | Lelio R Paula | 03/2014 |
5452 | 28275^12406+12406^28275 | 115748 | Hans Havermann | 01/2021 |
5453 | (44*10^115697-413)/9 | 115698 | Robert Price | 03/2019 |
5454 | 31408^4825+4825^31408 | 115692 | Hans Havermann | 02/2021 |
5455 | (2^384317+1026)/2 | 115691 | Stefano Morozzi | 09/2020 |
5456 | 2^384312+81091 | 115690 | Yves Minet | 02/2010 |
5457 | (2^384265+2150)/2 | 115675 | Stefano Morozzi | 09/2020 |
5458 | (2^384246+2182)/2 | 115670 | Stefano Morozzi | 09/2020 |
5459 | (4*10^115668-79)/3 | 115669 | Robert Price | 02/2018 |
5460 | (2^384193+242)/2 | 115654 | Stefano Morozzi | 09/2020 |
5461 | (266947#+1267)/(7^2*181) | 115647 | Makoto Morimoto | 06/2019 |
5462 | (266947#+273808)/(2*109*157) | 115646 | Makoto Morimoto | 10/2017 |
5463 | 18^92125-92125^18 | 115642 | Dylan Delgado | 08/2019 |
5464 | 3^242337-4042 | 115625 | Enzo Doria | 05/2014 |
5465 | (2^383931+674)/2 | 115575 | Stefano Morozzi | 09/2020 |
5466 | 7*10^115561+79 | 115562 | Robert Price | 07/2019 |
5467 | (2^383872+3046)/2 | 115557 | Stefano Morozzi | 09/2020 |
5468 | (11*2^383859-13)/15 | 115553 | Renaud Lifchitz | 03/2011 |
5469 | 2^383780-3149 | 115530 | Yves Minet | 10/2014 |
5470 | (41*10^115523-383)/9 | 115524 | Robert Price | 11/2018 |
5471 | 27316^16923+16923^27316 | 115506 | Hans Havermann | 02/2021 |
5472 | (2^383618+618)/2 | 115481 | Stefano Morozzi | 09/2020 |
5473 | 2^383560-455 | 115464 | Yves Minet | 03/2012 |
5474 | (2^383544+5)/3 | 115458 | Lelio R Paula | 03/2014 |
5475 | (22*10^115454+119)/3 | 115455 | Robert Price | 04/2021 |
5476 | 89^59221-88^59221 | 115445 | Robert Price | 06/2015 |
5477 | (7^136559-5^136559)/2 | 115406 | Jon Grantham | 08/2023 |
5478 | 2^383347+1869413 | 115399 | Enzo Doria | 05/2016 |
5479 | (103*10^115366+17)/3 | 115368 | Robert Price | 08/2020 |
5480 | (2^383232+750)/2 | 115365 | Stefano Morozzi | 09/2020 |
5481 | (2^383130-7)/131793 | 115329 | Lelio R Paula | 03/2014 |
5482 | (12^106859+1)/13 | 115320 | Paul Bourdelais | 02/2010 |
5483 | 29961^7060+7060^29961 | 115315 | Hans Havermann | 01/2021 |
5485 | 2^383039+1869509 | 115307 | Enzo Doria | 05/2016 |
5484 | 32918^3183+3183^32918 | 115307 | Hans Havermann | 01/2021 |
5486 | 36964^1315+1315^36964 | 115288 | Hans Havermann | 02/2021 |
5487 | (13*2^382968+1)/7 | 115286 | Lelio R Paula | 04/2014 |
5488 | (22*10^115282+29)/3 | 115283 | Robert Price | 02/2021 |
5489 | (2^382945+2046)/2 | 115278 | Stefano Morozzi | 09/2020 |
5490 | (2^382789+3182)/2 | 115231 | Stefano Morozzi | 09/2020 |
5491 | 30416^6141+6141^30416 | 115224 | Hans Havermann | 02/2021 |
5492 | 6*10^115211+79 | 115212 | Robert Price | 09/2020 |
5493 | (2^382722+3598)/2 | 115211 | Stefano Morozzi | 09/2020 |
5494 | 27987^13078+13078^27987 | 115210 | Hans Havermann | 02/2021 |
5495 | (2^382699+2090)/2 | 115204 | Stefano Morozzi | 09/2020 |
5496 | (22*10^115188-193)/9 | 115189 | Robert Price | 07/2022 |
5497 | 2^382633+73697 | 115185 | Enzo Doria | 04/2016 |
5498 | 2^382509-113593 | 115147 | Enzo Doria | 08/2014 |
5499 | 2^382502-2791721 | 115145 | Enzo Doria | 03/2016 |
5500 | (2^382495+1274)/2 | 115143 | Stefano Morozzi | 01/2021 |