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 | (2^395825+582)/2 | 119155 | Stefano Morozzi | 09/2020 |
5252 | (2^395733+1130)/2 | 119128 | Stefano Morozzi | 09/2020 |
5253 | 29548^10725+10725^29548 | 119091 | Hans Havermann | 02/2021 |
5254 | (11*2^395548+1)/3 | 119073 | Lelio R Paula | 04/2014 |
5255 | (30^80599+1)/31 | 119054 | Robert Price | 07/2013 |
5256 | (232*10^119047+17)/3 | 119049 | Robert Price | 04/2021 |
5257 | (43*10^119024-403)/9 | 119025 | Robert Price | 01/2019 |
5258 | (4*10^119021+47)/3 | 119022 | Robert Price | 01/2022 |
5259 | fibonacci(569433)+3 | 119005 | Jon Grantham | 08/2023 |
5260 | 26603^26888-26888^26603 | 118978 | Norbert Schneider | 07/2021 |
5261 | 6*10^118964+43 | 118965 | Robert Price | 07/2020 |
5262 | (8*10^118960+49)/3 | 118961 | Robert Price | 05/2020 |
5263 | 31253^6402+6402^31253 | 118959 | Hans Havermann | 02/2021 |
5264 | (194*10^118925-17)/3 | 118927 | Robert Price | 09/2020 |
5265 | 2^395062-109121 | 118926 | Enzo Doria | 04/2014 |
5266 | 2^395055-4627 | 118924 | Yves Minet | 05/2019 |
5267 | (2^395054+702)/2 | 118923 | Stefano Morozzi | 09/2020 |
5268 | 2^395018-224141 | 118913 | Harrison Mesh | 06/2022 |
5269 | (2^395011+2526)/2 | 118910 | Stefano Morozzi | 09/2020 |
5270 | (2^394918+78557)/3 | 118882 | Lelio R Paula | 01/2015 |
5271 | 25771^26948-26948^25771 | 118872 | Frank Doornink | 02/2024 |
5272 | (2^394858+1018)/2 | 118864 | Stefano Morozzi | 09/2020 |
5273 | (77*10^118862-149)/9 | 118863 | Robert Price | 03/2022 |
5274 | (16*10^118852+77)/3 | 118853 | Robert Price | 03/2019 |
5275 | fibonacci(568658)-2 | 118843 | Jon Grantham | 08/2023 |
5276 | (2^394739+3)/1883 | 118826 | Lelio R Paula | 04/2014 |
5277 | 10^118772+163599 | 118773 | Patrick De Geest | 02/2021 |
5279 | 10^118772-40000000001 | 118772 | Robert Price | 09/2020 |
5278 | 10^118772-207603 | 118772 | Patrick De Geest | 02/2021 |
5280 | 26842^26577+26577^26842 | 118763 | Hans Havermann | 02/2021 |
5281 | (2^394513+2034)/2 | 118760 | Stefano Morozzi | 09/2020 |
5282 | (2^394473+2774)/2 | 118748 | Stefano Morozzi | 02/2021 |
5283 | 27949^17710+17710^27949 | 118734 | Hans Havermann | 02/2021 |
5284 | 2^394266+71389 | 118686 | Enzo Doria | 04/2016 |
5285 | (10^118681-3^118681)/7 | 118681 | Jon Grantham | 08/2023 |
5286 | 2^394215-51261 | 118671 | Tony Prest / Never Odd Or Even | 04/2018 |
5287 | (2^394215-48793)/5/5 | 118670 | Tony Prest / Never Odd Or Even | 04/2018 |
5288 | (2^394215-17063)/3/3/5 | 118669 | Tony Prest / Never Odd Or Even | 04/2018 |
5289 | (2^394215-24393)/5/5/5/5/5 | 118668 | Tony Prest / Never Odd Or Even | 04/2018 |
5290 | (2^394215-35885)/3/71/173 | 118666 | Tony Prest / Never Odd Or Even | 04/2018 |
5291 | (2^394215-47925)/353/2269 | 118665 | Tony Prest / Never Odd Or Even | 04/2018 |
5292 | 29201^11568+11568^29201 | 118652 | Hans Havermann | 02/2021 |
5293 | (2^394215-62819)/3/3/3/3/3/3/3/7/21737/1082153/1953673 | 118650 | Tony Prest / Never Odd Or Even | 04/2018 |
5294 | (2^394131+2214)/2 | 118645 | Stefano Morozzi | 09/2020 |
5295 | (17*10^118635+13)/3 | 118636 | Robert Price | 01/2019 |
5296 | 5^169727+2 | 118635 | Lelio R Paula | 08/2013 |
5297 | 2^394034-3 | 118617 | M. Frind & P. Underwood | 09/2002 |
5298 | (7*10^118607+179)/3 | 118608 | Robert Price | 09/2018 |
5299 | 3^248572-6242 | 118599 | Enzo Doria | 05/2014 |
5300 | (3*2^393964-1)/181 | 118594 | Lelio R Paula | 04/2014 |
5301 | (49*10^118576-319)/9 | 118577 | Robert Price | 10/2021 |
5302 | 7^140303-6 | 118570 | Robert Price | 11/2014 |
5303 | (2^393864+2686)/2 | 118565 | Stefano Morozzi | 09/2020 |
5304 | (2^393862+3150)/2 | 118564 | Stefano Morozzi | 09/2020 |
5305 | (2^393849+3762)/2 | 118561 | Stefano Morozzi | 01/2021 |
5306 | (529*10^118554-7)/9 | 118556 | Robert Price | 08/2021 |
5307 | 26712^26777-26777^26712 | 118534 | Norbert Schneider | 07/2021 |
5308 | (2^393661+2666)/2 | 118504 | Stefano Morozzi | 09/2020 |
5309 | (2^393650+1614)/2 | 118501 | Stefano Morozzi | 09/2020 |
5310 | 27655^19264+19264^27655 | 118495 | Hans Havermann | 02/2021 |
5311 | 10^118485+469413 | 118486 | Patrick De Geest | 05/2021 |
5312 | 10^118485-50001 | 118485 | Robert Price | 08/2020 |
5313 | (2^393528+1030)/2 | 118464 | Stefano Morozzi | 09/2020 |
5314 | 2^393427+75683 | 118434 | Enzo Doria | 04/2016 |
5315 | 2^393376-114749 | 118418 | Enzo Doria | 04/2015 |
5316 | (65*10^118417+493)/9 | 118418 | Robert Price | 08/2021 |
5317 | 30960^6671+6671^30960 | 118397 | Hans Havermann | 02/2021 |
5318 | (2^393234+2010)/2 | 118375 | Stefano Morozzi | 09/2020 |
5319 | (2^393205+3030)/2 | 118367 | Stefano Morozzi | 09/2020 |
5320 | (2^393191+2174)/2 | 118362 | Stefano Morozzi | 09/2020 |
5321 | 2^393180+73 | 118359 | Lelio R Paula | 04/2013 |
5322 | 3*2^393125+5 | 118343 | Jon Grantham | 07/2023 |
5323 | 27546^19771+19771^27546 | 118339 | Hans Havermann | 02/2021 |
5324 | p(23881)#/2+256 | 118328 | Robert Price | 05/2017 |
5325 | (10^118324+359)/9 | 118324 | Serge Batalov | 12/2014 |
5326 | (2^393048+570)/2 | 118319 | Stefano Morozzi | 09/2020 |
5327 | p(23879)#/2+64 | 118317 | Robert Price | 03/2017 |
5328 | 2^393033+20147 | 118315 | Enzo Doria | 08/2023 |
5329 | 8*10^118308+59 | 118309 | Robert Price | 11/2021 |
5330 | 9*10^118282+29 | 118283 | Robert Price | 03/2022 |
5331 | (238*10^118265-1)/3 | 118267 | Robert Price | 04/2020 |
5333 | (53*10^118216+1)/9 | 118217 | Serge Batalov | 11/2010 |
5332 | (2^392707+1710)/2 | 118217 | Stefano Morozzi | 09/2020 |
5334 | (7*2^392697+1)/15 | 118214 | Renaud Lifchitz | 03/2011 |
5335 | (157^53819-1)/156 | 118179 | Alexander Zhirkov | 06/2016 |
5336 | (2^392567+2750)/2 | 118175 | Stefano Morozzi | 01/2021 |
5337 | 55*10^118172+3 | 118174 | Robert Price | 07/2022 |
5338 | (19*2^392551+1)/3 | 118171 | Lelio R Paula | 04/2014 |
5339 | 27185^22222+22222^27185 | 118168 | Hans Havermann | 01/2021 |
5340 | 2^392516-29219 | 118160 | Enzo Doria | 07/2015 |
5341 | 66*10^118155+7 | 118157 | Robert Price | 07/2020 |
5342 | Phi(177231,10) | 118152 | Maksym Voznyy | 09/2015 |
5343 | 2^392413-112123 | 118129 | Enzo Doria | 06/2014 |
5344 | 27491^19798+19798^27491 | 118119 | Hans Havermann | 02/2021 |
5345 | (49*10^118109-319)/9 | 118110 | Robert Price | 10/2021 |
5346 | (2^392342+5)/27 | 118106 | Lelio R Paula | 04/2014 |
5347 | (2^392333+1782)/2 | 118104 | Stefano Morozzi | 01/2021 |
5348 | 2^392300-1841429 | 118095 | Enzo Doria | 06/2015 |
5349 | 3*2^392260-5 | 118083 | Jon Grantham | 07/2023 |
5350 | 2^392220+75241 | 118070 | Enzo Doria | 04/2016 |
5351 | 27836^17409+17409^27836 | 118047 | Hans Havermann | 02/2021 |
5352 | 10^118009+664447 | 118010 | Patrick De Geest | 07/2018 |
5353 | 10^118009-20001 | 118009 | Robert Price | 01/2018 |
5354 | (14*10^117991+61)/3 | 117992 | Robert Price | 12/2018 |
5355 | 2^391933+5897 | 117984 | Yves Minet | 02/2011 |
5356 | 2^391912-1849259 | 117978 | Enzo Doria | 08/2015 |
5357 | (2*34^77029+1043)/11 | 117968 | Eric Chen | 08/2023 |
5358 | 5*2^391836-3 | 117956 | Jon Grantham | 07/2023 |
5359 | 28357^14440+14440^28357 | 117953 | Hans Havermann | 02/2021 |
5360 | (2^391812+5)/14181 | 117944 | Lelio R Paula | 04/2014 |
5361 | (7*10^117930-79)/9 | 117930 | Robert Price | 01/2021 |
5362 | 27531^19168+19168^27531 | 117904 | Hans Havermann | 02/2021 |
5363 | (46*10^117902+683)/9 | 117903 | Robert Price | 03/2019 |
5364 | 26349^26668-26668^26349 | 117893 | Frank Doornink | 03/2024 |
5365 | 27670^18189+18189^27670 | 117869 | Hans Havermann | 02/2021 |
5366 | (7*2^391521+1)/285 | 117858 | Lelio R Paula | 04/2014 |
5367 | (56*10^117856+583)/9 | 117857 | Robert Price | 08/2021 |
5368 | 32169^4606+4606^32169 | 117846 | Hans Havermann | 02/2021 |
5369 | (71*10^117803-143)/9 | 117804 | Robert Price | 10/2021 |
5370 | 2^391324+90901 | 117801 | Yves Minet | 02/2011 |
5371 | 2^391301-1859923 | 117794 | Enzo Doria | 03/2016 |
5372 | 2^391261-4131 | 117782 | Yves Minet | 02/2018 |
5373 | (2^391211+554)/2 | 117766 | Stefano Morozzi | 09/2020 |
5374 | 38434^1157+1157^38434 | 117737 | Hans Havermann | 01/2021 |
5375 | (2^391115-7)/11791 | 117734 | Lelio R Paula | 04/2014 |
5376 | (85*10^117719+473)/9 | 117720 | Robert Price | 01/2020 |
5377 | (2^391049+582)/2 | 117718 | Stefano Morozzi | 09/2020 |
5378 | (21*23^86445-197)/22 | 117715 | Eric Chen | 08/2023 |
5379 | 31083^6122+6122^31083 | 117709 | Hans Havermann | 02/2021 |
5380 | 2^390960+1880881 | 117691 | Enzo Doria | 11/2016 |
5381 | (71*10^117685+91)/9 | 117686 | Robert Price | 04/2022 |
5383 | 2^390897-1860553 | 117672 | Enzo Doria | 11/2015 |
5382 | (61*10^117671+47)/9 | 117672 | Robert Price | 07/2022 |
5384 | 2^390889-115003 | 117670 | Enzo Doria | 06/2015 |
5385 | 12^109010+35 | 117642 | Riley Fisher | 08/2023 |
5386 | 2^390779-30337 | 117637 | Paul Bourdelais | 02/2012 |
5387 | (2^390735+378)/2 | 117623 | Stefano Morozzi | 02/2021 |
5388 | 11^112916-780 | 117590 | Enzo Doria | 12/2012 |
5389 | 27641^17916+17916^27641 | 117564 | Hans Havermann | 02/2021 |
5390 | 27906^16321+16321^27906 | 117561 | Hans Havermann | 02/2021 |
5391 | 27294^20275+20275^27294 | 117555 | Hans Havermann | 02/2021 |
5392 | (2^390473+1002)/2 | 117544 | Stefano Morozzi | 09/2020 |
5393 | 35912^1875+1875^35912 | 117541 | Hans Havermann | 02/2021 |
5394 | (16^97613-5^97613)/11 | 117537 | Robert Price | 07/2013 |
5395 | (2^390438+1774)/2 | 117534 | Stefano Morozzi | 01/2021 |
5396 | (2^390418+1722)/2 | 117528 | Stefano Morozzi | 09/2020 |
5397 | (2^390403+654)/2 | 117523 | Stefano Morozzi | 09/2020 |
5398 | (2^390343+2246)/2 | 117505 | Stefano Morozzi | 01/2021 |
5399 | 28043^15478+15478^28043 | 117493 | Hans Havermann | 02/2021 |
5400 | (17*2^390272+1)/1983 | 117482 | Lelio R Paula | 04/2014 |
5401 | 37077^1474+1474^37077 | 117479 | Hans Havermann | 01/2021 |
5402 | 31^78759-30 | 117459 | Theodore Burton | 02/2007 |
5403 | (2^390176+1222)/2 | 117455 | Stefano Morozzi | 09/2020 |
5404 | (2^390133+1242)/2 | 117442 | Stefano Morozzi | 09/2020 |
5405 | (2^390013+2634)/2 | 117406 | Stefano Morozzi | 09/2020 |
5406 | 3*10^117402+61 | 117403 | Robert Price | 09/2020 |
5407 | 2^389998+37 | 117402 | Lelio R Paula | 03/2013 |
5408 | (2^389977+314)/2 | 117395 | Stefano Morozzi | 09/2020 |
5409 | (2^389921+342)/2 | 117378 | Stefano Morozzi | 01/2021 |
5410 | 71830^43+43^71830 | 117333 | Hans Havermann | 01/2021 |
5411 | (2^389768+1582)/2 | 117332 | Stefano Morozzi | 09/2020 |
5412 | 33802^2951+2951^33802 | 117292 | Hans Havermann | 02/2021 |
5413 | 26789^23866+23866^26789 | 117277 | Hans Havermann | 02/2021 |
5414 | 29^80192-1*80192^29 | 117273 | Dylan Delgado | 12/2019 |
5415 | (101*10^117260+7)/9 | 117262 | Robert Price | 02/2022 |
5416 | (13*10^117249+161)/3 | 117250 | Robert Price | 11/2018 |
5417 | 2^389478+43 | 117245 | Lelio R Paula | 03/2013 |
5418 | (298*10^117229-7)/3 | 117231 | Robert Price | 06/2020 |
5419 | (2^389416+342)/2 | 117226 | Stefano Morozzi | 01/2021 |
5420 | (29*2^389351-1)/9 | 117207 | Lelio R Paula | 04/2014 |
5421 | 5*10^117205+23 | 117206 | Robert Price | 03/2019 |
5422 | (2^389341-7)/3865 | 117200 | Lelio R Paula | 03/2014 |
5423 | 27272^19827+19827^27272 | 117195 | Hans Havermann | 01/2021 |
5424 | (2^389276+2650)/2 | 117184 | Stefano Morozzi | 09/2020 |
5425 | (844*10^117175-7)/9 | 117177 | Robert Price | 05/2021 |
5426 | 10^117142+209223 | 117143 | Patrick De Geest | 04/2018 |
5428 | 10^117142-1000000001 | 117142 | Robert Price | 03/2018 |
5427 | 10^117142-966057 | 117142 | Patrick De Geest | 04/2018 |
5429 | 10^117135+360577 | 117136 | Patrick De Geest | 07/2018 |
5430 | 10^117135-401 | 117135 | Robert Price | 01/2018 |
5431 | 26415^26488-26488^26415 | 117126 | Norbert Schneider | 05/2021 |
5432 | 4*10^117123+79 | 117124 | Robert Price | 08/2018 |
5433 | 2^389033-12797413 | 117111 | Enzo Doria | 04/2016 |
5434 | (2^388987+2634)/2 | 117097 | Stefano Morozzi | 09/2020 |
5435 | 2^388960-109505 | 117089 | Enzo Doria | 04/2014 |
5436 | 7*10^117082-57 | 117083 | Robert Price | 06/2020 |
5437 | (2^388928-5)/569 | 117077 | Lelio R Paula | 03/2014 |
5438 | 2^388888-1529 | 117067 | Yves Minet | 01/2013 |
5439 | 10^117059+28879 | 117060 | Patrick De Geest | 05/2016 |
5440 | (10^117059-71)/1 | 117059 | Serge Batalov | 12/2014 |
5441 | (2^388854+3630)/2 | 117057 | Stefano Morozzi | 09/2020 |
5442 | (2^388842+2098)/2 | 117053 | Stefano Morozzi | 09/2020 |
5443 | (2^388789+3150)/2 | 117037 | Stefano Morozzi | 09/2020 |
5444 | (2^388748+378)/2 | 117025 | Stefano Morozzi | 02/2021 |
5445 | (693^41189-1)/692 | 117005 | Michael Stocker | 05/2020 |
5446 | (26*10^116999-503)/9 | 117000 | Robert Price | 07/2018 |
5447 | (2^388609+1554)/2 | 116983 | Stefano Morozzi | 09/2020 |
5448 | 38496^1093+1093^38496 | 116975 | Hans Havermann | 02/2021 |
5449 | 3^245154-6256 | 116969 | Enzo Doria | 05/2014 |
5450 | 5*10^116966-43 | 116967 | Robert Price | 01/2019 |
5451 | 2^388536-113495 | 116961 | Enzo Doria | 08/2014 |
5452 | (2^388512+382)/2 | 116954 | Stefano Morozzi | 02/2021 |
5453 | (8*32^77701+271)/31 | 116952 | Eric Chen | 11/2023 |
5454 | 2^388469-43 | 116941 | Lelio R Paula | 03/2013 |
5455 | 27417^18416+18416^27417 | 116939 | Hans Havermann | 02/2021 |
5456 | (2^388438+1842)/2 | 116932 | Stefano Morozzi | 02/2021 |
5457 | (7*2^388372-1)/111 | 116911 | Lelio R Paula | 04/2014 |
5458 | (7*2^388335-1)/25 | 116900 | Lelio R Paula | 04/2014 |
5459 | 2^388180+93 | 116854 | Donovan Johnson | 07/2008 |
5460 | (2^388167+2510)/2 | 116850 | Stefano Morozzi | 09/2020 |
5461 | (85*10^116842+293)/9 | 116843 | Robert Price | 12/2019 |
5462 | 35296^2043+2043^35296 | 116840 | Hans Havermann | 02/2021 |
5463 | (2^388081+7)/1179 | 116821 | Lelio R Paula | 03/2014 |
5464 | 38633^1056+1056^38633 | 116814 | Hans Havermann | 02/2021 |
5465 | 2^387946-113261 | 116784 | Enzo Doria | 07/2014 |
5466 | 28*36^75009+305 | 116739 | Eric Chen | 12/2022 |
5467 | 28363^13056+13056^28363 | 116737 | Hans Havermann | 02/2021 |
5468 | 2^387745+75557 | 116723 | Enzo Doria | 04/2016 |
5469 | (29*10^116719+313)/9 | 116720 | Robert Price | 04/2021 |
5470 | 29622^8699+8699^29622 | 116695 | Hans Havermann | 02/2021 |
5471 | (206*10^116687-17)/9 | 116689 | Robert Price | 01/2020 |
5472 | (7*10^116684+179)/3 | 116685 | Robert Price | 09/2018 |
5473 | 2^387554+71455 | 116666 | Enzo Doria | 04/2016 |
5474 | (602*10^116647-71)/9 | 116649 | Robert Price | 12/2021 |
5475 | 3^244461-10 | 116638 | Tyler NeSmith | 09/2022 |
5476 | (2^387447+78557)/365 | 116631 | Lelio R Paula | 12/2014 |
5477 | (2*10^116620+43)/9 | 116620 | Serge Batalov | 11/2022 |
5478 | (19*10^116617+287)/9/12893 | 116614 | Alfred Reich | 11/2022 |
5479 | (145*10^116610-1)/9 | 116612 | Serge Batalov | 12/2014 |
5480 | (25*10^116601-331)/9 | 116602 | Robert Price | 08/2018 |
5481 | 25791^26428-26428^25791 | 116587 | Frank Doornink | 03/2024 |
5483 | (17*2^387284-1)/59 | 116584 | Lelio R Paula | 04/2014 |
5482 | (2^387284+3726)/2 | 116584 | Stefano Morozzi | 09/2020 |
5484 | 2^387211+111546435 | 116563 | Paul Bourdelais | 10/2021 |
5485 | 10^116557+987654321 | 116558 | Robert Price | 12/2019 |
5486 | 10^116557+9141 | 116558 | Patrick De Geest | 02/2020 |
5487 | 10^116557-609431 | 116557 | Patrick De Geest | 02/2020 |
5488 | (28*10^116546+41)/3 | 116547 | Robert Price | 07/2021 |
5489 | (2^387123+1814)/2 | 116536 | Stefano Morozzi | 02/2021 |
5490 | (2^387076+2046)/2 | 116522 | Stefano Morozzi | 09/2020 |
5491 | (2^387073+2570)/2 | 116521 | Stefano Morozzi | 09/2020 |
5492 | (7*10^116514-1)/3 | 116515 | Robert Price | 08/2022 |
5493 | (26*32^77406-390071011)/31 | 116508 | Eric Chen | 11/2023 |
5494 | 28846^10931+10931^28846 | 116500 | Hans Havermann | 02/2021 |
5495 | 10^116496+3593037 | 116497 | Paul Bourdelais | 10/2021 |
5496 | (31*2^386968+1)/7 | 116490 | Lelio R Paula | 03/2014 |
5497 | 2^386947+1869143 | 116483 | Enzo Doria | 05/2016 |
5498 | (30^78857-1)/29 | 116480 | Robert Price | 12/2011 |
5499 | (2^386903-3)/5 | 116469 | Lelio R Paula | 03/2014 |
5500 | 28107^13892+13892^28107 | 116441 | Hans Havermann | 02/2021 |