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 ? |
4251 | 2^348003-20377 | 104760 | Enzo Doria | 06/2013 |
4253 | 2^347994-741 | 104757 | Lelio R Paula | 12/2016 |
4252 | (2^347993+2234)/2 | 104757 | Stefano Morozzi | 01/2021 |
4254 | (2^347990+790)/2 | 104756 | Stefano Morozzi | 09/2020 |
4255 | 25137^14702+14702^25137 | 104756 | Hans Havermann | 09/2020 |
4256 | (13*2^347972-1)/282309 | 104746 | Lelio R Paula | 03/2014 |
4257 | 2^347910-109937 | 104732 | Enzo Doria | 04/2014 |
4258 | (88*10^104709-637)/9 | 104710 | Robert Price | 11/2020 |
4259 | 5^149794+28 | 104702 | Lelio R Paula | 11/2016 |
4260 | 105^51800-2 | 104698 | Lelio R Paula | 10/2017 |
4261 | (16^86939+3^86939)/19 | 104684 | Robert Price | 06/2013 |
4262 | (15^89009-1)/14 | 104682 | Paul Bourdelais | 03/2010 |
4263 | 2^347731-18609 | 104678 | Enzo Doria | 05/2013 |
4264 | 4*6^134519+7 | 104677 | Lelio R Paula | 11/2014 |
4265 | 25321^13588+13588^25321 | 104656 | Hans Havermann | 09/2020 |
4266 | 2^347612-18545 | 104642 | Enzo Doria | 05/2013 |
4268 | 6^134465+95 | 104635 | Lelio R Paula | 11/2016 |
4267 | (2^347588+3718)/2 | 104635 | Stefano Morozzi | 09/2020 |
4269 | (166*10^104633-7)/3 | 104635 | Robert Price | 11/2020 |
4270 | 2^347517-17301 | 104614 | Enzo Doria | 05/2013 |
4271 | (2^347493+3186)/2 | 104606 | Stefano Morozzi | 09/2020 |
4272 | (2^347490+2574)/2 | 104605 | Stefano Morozzi | 09/2020 |
4273 | 3^219222-310 | 104596 | Lelio R Paula | 11/2016 |
4274 | 25993^10554+10554^25993 | 104581 | Hans Havermann | 09/2020 |
4275 | (15*2^347399-1)/13 | 104578 | Lelio R Paula | 03/2014 |
4276 | 2^347372-17853 | 104570 | Enzo Doria | 05/2013 |
4277 | 2^347314+99 | 104552 | Lelio R Paula | 09/2011 |
4278 | 2^347300-22233 | 104548 | Enzo Doria | 06/2013 |
4279 | 2^347287-19845 | 104544 | Enzo Doria | 06/2013 |
4280 | (31*2^347254-1)/891 | 104533 | Lelio R Paula | 03/2014 |
4281 | 5*F(250091)*(F(250091)-1)+1 | 104532 | Serge Batalov | 04/2015 |
4282 | 7*2^347239-3 | 104531 | Lelio R Paula | 09/2014 |
4283 | (3^219098-8)/514147439969 | 104525 | Lelio R Paula | 12/2013 |
4284 | 5*3^219047+8 | 104513 | Lelio R Paula | 09/2014 |
4285 | 2^347148-109073 | 104502 | Enzo Doria | 04/2014 |
4286 | 2^347129-20145 | 104497 | Enzo Doria | 06/2013 |
4287 | 4*10^104461-57 | 104462 | Robert Price | 10/2018 |
4288 | (2^346989+590)/2 | 104454 | Stefano Morozzi | 09/2020 |
4289 | 2^346923-19585 | 104435 | Enzo Doria | 05/2013 |
4290 | (11*2^346864-1)/175 | 104416 | Lelio R Paula | 03/2014 |
4291 | 2^346854+346854^2+1 | 104414 | Robert Price | 06/2014 |
4292 | 2^346834+11715 | 104408 | Lelio R Paula | 04/2015 |
4293 | 24276^19961+19961^24276 | 104392 | Hans Havermann | 09/2020 |
4294 | 79^55009+78 | 104387 | Norbert Schneider | 02/2015 |
4295 | 3^218744+58 | 104368 | Lelio R Paula | 11/2016 |
4296 | 9*10^104361+89 | 104362 | Robert Price | 11/2019 |
4297 | 25048^14665+14665^25048 | 104358 | Hans Havermann | 09/2020 |
4298 | 2^346649+65 | 104352 | Lelio R Paula | 09/2011 |
4299 | 2^346638-17265 | 104349 | Enzo Doria | 05/2013 |
4300 | (2^346606+2082)/2 | 104339 | Stefano Morozzi | 09/2020 |
4301 | (2^346607+2070)/2 | 104339 | Stefano Morozzi | 09/2020 |
4302 | (2^346582+3202)/2 | 104332 | Stefano Morozzi | 09/2020 |
4303 | 26092^9967+9967^26092 | 104331 | Hans Havermann | 09/2020 |
4304 | 5*10^104326+77 | 104327 | Robert Price | 12/2018 |
4305 | (2^346564+2026)/2 | 104326 | Stefano Morozzi | 09/2020 |
4306 | 2^346527-111817 | 104316 | Enzo Doria | 05/2014 |
4307 | 6^134055+101 | 104316 | Lelio R Paula | 11/2016 |
4308 | 92105!4+8 | 104312 | Robert Price | 09/2019 |
4309 | 6^134013+163 | 104283 | Lelio R Paula | 11/2016 |
4310 | 18^83049-89 | 104250 | Lelio R Paula | 01/2017 |
4311 | (2^346311+606)/2 | 104250 | Stefano Morozzi | 09/2020 |
4312 | 2^346213+484049 | 104221 | Enzo Doria | 04/2017 |
4313 | 18^83021-91 | 104214 | Lelio R Paula | 01/2017 |
4315 | 10^104212+128779 | 104213 | Patrick De Geest | 05/2016 |
4316 | 10^104212+237447 | 104213 | Patrick De Geest | 05/2016 |
4317 | 10^104212+242139 | 104213 | Patrick De Geest | 05/2016 |
4318 | 10^104212+269199 | 104213 | Patrick De Geest | 05/2016 |
4314 | 24254^19803+19803^24254 | 104213 | Hans Havermann | 09/2020 |
4319 | (10^104212-71)/1 | 104212 | Serge Batalov | 12/2014 |
4320 | 23678^23821-23821^23678 | 104202 | Norbert Schneider | 01/2020 |
4321 | 2^346078+7695 | 104180 | Lelio R Paula | 12/2016 |
4322 | 2^346055-19537 | 104173 | Enzo Doria | 05/2013 |
4323 | (3*2^346057+1)/77 | 104173 | Lelio R Paula | 12/2013 |
4324 | 2^346037-20125 | 104168 | Enzo Doria | 06/2013 |
4325 | (34*10^104168-7)/9/13/551179 | 104162 | Makoto Morimoto | 05/2016 |
4326 | 6^133854-17993 | 104159 | Enzo Doria | 03/2013 |
4327 | 2^345990+1874239 | 104154 | Enzo Doria | 06/2016 |
4328 | 2^345988-20373 | 104153 | Enzo Doria | 06/2013 |
4329 | (2^345965+2214)/2 | 104146 | Stefano Morozzi | 09/2020 |
4330 | (26*10^104132-503)/9 | 104133 | Robert Price | 07/2018 |
4331 | 3*5^148968+8 | 104125 | Lelio R Paula | 11/2014 |
4332 | (7*10^104113+53)/3 | 104114 | Robert Price | 05/2018 |
4333 | p(21253)#/2+8 | 104113 | Robert Price | 01/2017 |
4334 | 33582^1259+1259^33582 | 104106 | Hans Havermann | 09/2020 |
4335 | 32958^1441+1441^32958 | 104104 | Hans Havermann | 09/2020 |
4336 | 29127^3748+3748^29127 | 104095 | Hans Havermann | 09/2020 |
4337 | 322^41507+41507^322 | 104094 | Norbert Schneider | 12/2014 |
4338 | 6^133768-17423 | 104092 | Enzo Doria | 03/2013 |
4339 | 2^345786-19557 | 104092 | Enzo Doria | 05/2013 |
4340 | 171^46606-2 | 104072 | Lelio R Paula | 10/2017 |
4341 | (29*10^104071+91)/3 | 104072 | Robert Price | 04/2020 |
4342 | 2^345704-149 | 104068 | Lelio R Paula | 09/2011 |
4343 | 25821^10712+10712^25821 | 104056 | Hans Havermann | 09/2020 |
4344 | (2^345662+2038)/2 | 104055 | Stefano Morozzi | 09/2020 |
4345 | 6^133704-17273 | 104042 | Enzo Doria | 02/2013 |
4346 | (3*2^345590-1)/23 | 104033 | Lelio R Paula | 03/2014 |
4347 | (19*2^345590+1)/33859 | 104030 | Lelio R Paula | 12/2013 |
4348 | 2^345547+11 | 104021 | Lelio R Paula | 09/2011 |
4349 | (19*10^104014-49)/3 | 104015 | Robert Price | 05/2019 |
4350 | (2^345519+1794)/2 | 104012 | Stefano Morozzi | 01/2021 |
4351 | 2^345504+723 | 104008 | Lelio R Paula | 09/2011 |
4352 | 2^345498-20037 | 104006 | Enzo Doria | 06/2013 |
4354 | 18^82854-37 | 104005 | Lelio R Paula | 01/2017 |
4353 | (2^345497+2070)/2 | 104005 | Stefano Morozzi | 09/2020 |
4355 | 2^345486-109065 | 104002 | Enzo Doria | 04/2014 |
4356 | 2^345487-1851507 | 104002 | Enzo Doria | 08/2015 |
4358 | 2^345476+777 | 103999 | Lelio R Paula | 09/2011 |
4357 | (2^345477+1554)/2 | 103999 | Stefano Morozzi | 09/2020 |
4359 | 4*10^103981-83 | 103982 | Robert Price | 10/2018 |
4360 | (2^345327+3206)/2 | 103954 | Stefano Morozzi | 09/2020 |
4361 | (5*2^345321-1)/3 | 103953 | Lelio R Paula | 03/2014 |
4362 | (2^345267-9)/539049127 | 103927 | Lelio R Paula | 12/2013 |
4363 | (19*10^103907+77)/3 | 103908 | Robert Price | 05/2019 |
4364 | (78557*2^345163+1)/78589 | 103905 | Lelio R Paula | 07/2014 |
4365 | 2^345106-112181 | 103888 | Enzo Doria | 06/2014 |
4366 | 2^345079+99 | 103880 | Lelio R Paula | 09/2011 |
4367 | (2^345034+1198)/2 | 103866 | Stefano Morozzi | 09/2020 |
4368 | 11^99736-1500 | 103865 | Enzo Doria | 12/2012 |
4369 | (2^345010+2082)/2 | 103859 | Stefano Morozzi | 09/2020 |
4370 | 23263^23784-23784^23263 | 103857 | Norbert Schneider | 01/2020 |
4371 | 23927^21814+21814^23927 | 103813 | Hans Havermann | 09/2020 |
4372 | 2^344842-20157 | 103808 | Enzo Doria | 06/2013 |
4373 | (47*2^344822+1)/1071 | 103801 | Lelio R Paula | 05/2015 |
4374 | 2^344802-20765 | 103796 | Enzo Doria | 06/2013 |
4375 | 2^344794-3821 | 103794 | Yves Minet | 05/2011 |
4376 | 2^344782-109103 | 103790 | Enzo Doria | 04/2014 |
4377 | 3^217521-112292 | 103784 | Enzo Doria | 06/2014 |
4378 | 2^344759-3505 | 103783 | Lelio R Paula | 11/2016 |
4379 | 2^344735-147 | 103776 | Lelio R Paula | 09/2011 |
4380 | L(496533)-p(496533) | 103770 | Robert Price | 02/2018 |
4381 | 2*10^103749-69 | 103750 | Robert Price | 03/2018 |
4382 | (55459*2^344617+1)/3 | 103745 | Lelio R Paula | 07/2014 |
4383 | 2^344607-108891 | 103738 | Enzo Doria | 03/2014 |
4384 | (148*10^103718-1)/3 | 103720 | Robert Price | 03/2020 |
4385 | 2^344541-19561 | 103718 | Enzo Doria | 05/2013 |
4386 | (1017*2^344529-1)/569 | 103714 | Lelio R Paula | 05/2015 |
4387 | (2^344503+1758)/2 | 103706 | Stefano Morozzi | 09/2020 |
4388 | (4*10^103703-1)/3 | 103704 | Serge Batalov | 08/2009 |
4389 | 2^344444-455 | 103688 | Yves Minet | 03/2012 |
4390 | (13*53^238539*617^9)/10^307661 | 103673 | Hans Havermann | 06/2017 |
4391 | (2^344367+2738)/2 | 103665 | Stefano Morozzi | 01/2021 |
4393 | 2^344317-109933 | 103650 | Enzo Doria | 04/2014 |
4392 | 26055^9508+9508^26055 | 103650 | Hans Havermann | 09/2020 |
4394 | 2^344309-1852033 | 103648 | Enzo Doria | 08/2015 |
4395 | 25729^10672+10672^25729 | 103643 | Hans Havermann | 09/2020 |
4396 | (2^344289+1166)/2 | 103642 | Stefano Morozzi | 09/2020 |
4397 | 3*10^103637+61 | 103638 | Robert Price | 09/2020 |
4398 | (2^344246+1614)/2 | 103629 | Stefano Morozzi | 09/2020 |
4399 | 2^344213-21253 | 103619 | Enzo Doria | 06/2013 |
4400 | 2^344196-465 | 103614 | Lelio R Paula | 11/2016 |
4401 | 2^344187-20001 | 103611 | Enzo Doria | 06/2013 |
4402 | (117191#+(355573#)/(117191#))/(200233679) | 103602 | Makoto Morimoto | 10/2017 |
4403 | 20^79621-19^79621 | 103590 | Jean-Louis Charton | 03/2011 |
4404 | Phi(155391,10)/1243129 | 103586 | Maksym Voznyy | 09/2015 |
4405 | 2^344085-20365 | 103580 | Enzo Doria | 06/2013 |
4406 | 5^148169+54 | 103566 | Lelio R Paula | 11/2016 |
4407 | 2^344017-108795 | 103560 | Enzo Doria | 03/2014 |
4408 | (2^343974+2670)/2 | 103547 | Stefano Morozzi | 09/2020 |
4409 | 2^343971-5077 | 103546 | Lelio R Paula | 09/2011 |
4410 | (7*10^103531+71)/3 | 103532 | Robert Price | 05/2018 |
4411 | 2^343899+141 | 103524 | Lelio R Paula | 08/2011 |
4412 | (2^343933+1)/(3*98294339989393) | 103520 | Gord Palameta | 08/2018 |
4413 | 6^133029-67 | 103517 | Lelio R Paula | 11/2016 |
4414 | (17*2^343877+1)/13625 | 103515 | Lelio R Paula | 12/2013 |
4415 | 66*10^103507+7 | 103509 | Robert Price | 07/2020 |
4416 | 26297^8614+8614^26297 | 103485 | Hans Havermann | 09/2020 |
4417 | (2^343764+2658)/2 | 103483 | Stefano Morozzi | 09/2020 |
4418 | 26499^8036+8036^26499 | 103480 | Hans Havermann | 09/2020 |
4419 | (136*10^103474-1)/9 | 103476 | Serge Batalov | 12/2014 |
4420 | 5^148014-24 | 103458 | Lelio R Paula | 11/2016 |
4421 | 28316^4503+4503^28316 | 103453 | Hans Havermann | 09/2020 |
4422 | (22*10^103450+53)/3 | 103451 | Robert Price | 07/2019 |
4423 | 2^343604+9931 | 103436 | Yves Minet | 01/2011 |
4424 | 2^343601-21189 | 103435 | Enzo Doria | 06/2013 |
4425 | 2^343508-115229 | 103407 | Enzo Doria | 06/2015 |
4426 | (509203*2^343491-1)/16147 | 103403 | Lelio R Paula | 05/2015 |
4427 | 3^216685-113056 | 103386 | Enzo Doria | 07/2014 |
4428 | 24829^14580+14580^24829 | 103382 | Hans Havermann | 08/2020 |
4429 | 91355!4+1024 | 103381 | Robert Price | 09/2019 |
4430 | 44^62903-43^62903 | 103379 | Jean-Louis Charton | 08/2012 |
4431 | 11^99262-20 | 103371 | Lelio R Paula | 12/2016 |
4432 | (509203*2^343357-1)/5 | 103366 | Lelio R Paula | 05/2015 |
4433 | (2^343363+3570)/2 | 103363 | Stefano Morozzi | 09/2020 |
4434 | 2^343356-2791889 | 103361 | Enzo Doria | 03/2016 |
4435 | (25*10^103360-241)/9 | 103361 | Robert Price | 05/2018 |
4436 | Phi(104989,10)/2939693 | 103350 | Maksym Voznyy | 09/2015 |
4437 | 24826^14511+14511^24826 | 103319 | Hans Havermann | 09/2020 |
4438 | 2^343182-171 | 103309 | Lelio R Paula | 08/2011 |
4439 | (23619^23619+23620^23620)/7187 | 103294 | David Broadhurst | 09/2009 |
4440 | 2^343079-1835611 | 103278 | Enzo Doria | 05/2015 |
4441 | 2^343063-1147 | 103273 | Yves Minet | 06/2013 |
4442 | 24741^14930+14930^24741 | 103271 | Hans Havermann | 09/2020 |
4443 | (16*10^103255-73)/3 | 103256 | Robert Price | 07/2020 |
4444 | 9*6^132689+7 | 103254 | Lelio R Paula | 11/2014 |
4445 | 2^342979-18085 | 103247 | Enzo Doria | 05/2013 |
4446 | (16*10^103244-31)/3 | 103245 | Robert Price | 03/2019 |
4447 | 2^342966-16995 | 103244 | Lelio R Paula | 11/2016 |
4448 | 23797^21774+21774^23797 | 103230 | Hans Havermann | 09/2020 |
4449 | 25388^11637+11637^25388 | 103224 | Hans Havermann | 09/2020 |
4450 | (2^342876+2086)/2 | 103216 | Stefano Morozzi | 09/2020 |
4451 | (16*10^103214-31)/3 | 103215 | Robert Price | 03/2019 |
4453 | (37*10^103211+611)/9 | 103212 | Robert Price | 11/2018 |
4452 | (238*10^103210-1)/3 | 103212 | Robert Price | 04/2020 |
4454 | 5^147650+16 | 103203 | Lelio R Paula | 11/2016 |
4455 | gcd(Phi(852060,10),(10^42603+1)^3*(10^42603+1-10^21302)+(10^85206+10^42603)*(10^21302+1)+10^127809) | 103201 | Serge Batalov | 04/2015 |
4456 | 2^342817+2707067 | 103199 | Enzo Doria | 03/2016 |
4457 | 43^63177-42 | 103198 | Theodore Burton | 02/2007 |
4458 | 2^342805-21025 | 103195 | Enzo Doria | 06/2013 |
4459 | 24129^18890+18890^24129 | 103182 | Hans Havermann | 09/2020 |
4460 | 2^342714+12285 | 103168 | Lelio R Paula | 11/2016 |
4461 | 18^82178-4481 | 103156 | Enzo Doria | 03/2013 |
4462 | 2^342665-17349 | 103153 | Enzo Doria | 05/2013 |
4463 | 30408^2467+2467^30408 | 103150 | Hans Havermann | 09/2020 |
4464 | (2^342615+3090)/2 | 103138 | Stefano Morozzi | 09/2020 |
4465 | 2^342612+175 | 103137 | Lelio R Paula | 08/2011 |
4466 | 5*2^342585-3 | 103130 | Lelio R Paula | 09/2014 |
4467 | (16*10^103127+77)/3 | 103128 | Robert Price | 03/2019 |
4468 | 2^342503-19209 | 103104 | Enzo Doria | 05/2013 |
4469 | (2^342484+2586)/2 | 103098 | Stefano Morozzi | 09/2020 |
4470 | 2^342476+51 | 103096 | Lelio R Paula | 08/2011 |
4471 | 2^342460-18885 | 103091 | Enzo Doria | 05/2013 |
4472 | (194*10^103074-17)/3 | 103076 | Robert Price | 09/2020 |
4473 | 7^121968+82 | 103075 | Lelio R Paula | 10/2016 |
4474 | 2^342392-108863 | 103071 | Enzo Doria | 03/2014 |
4475 | (2^342389+806)/2 | 103070 | Stefano Morozzi | 01/2021 |
4476 | 3^216014+28 | 103065 | Lelio R Paula | 11/2016 |
4477 | 2^342353-108879 | 103059 | Enzo Doria | 03/2014 |
4478 | 10^103055+276081 | 103056 | Patrick De Geest | 12/2016 |
4479 | 10^103055-53 | 103055 | Lelio R Paula | 11/2016 |
4480 | (58*10^103046+419)/9 | 103047 | Robert Price | 05/2019 |
4481 | 7^121933+4 | 103046 | Lelio R Paula | 11/2014 |
4482 | (2^342235+3618)/2 | 103023 | Stefano Morozzi | 09/2020 |
4483 | 19^80560-68 | 103017 | Lelio R Paula | 10/2016 |
4484 | 2^342209+76607 | 103016 | Enzo Doria | 05/2016 |
4486 | 328^40945+40945^328 | 103013 | Norbert Schneider | 12/2014 |
4485 | 12^95454-91 | 103013 | Lelio R Paula | 10/2016 |
4487 | 25833^9716+9716^25833 | 103009 | Hans Havermann | 01/2019 |
4488 | 65^56809-64^56809 | 102990 | Robert Price | 08/2013 |
4489 | 5^147338-6 | 102985 | Lelio R Paula | 11/2014 |
4490 | 2^342084-112739 | 102978 | Enzo Doria | 06/2014 |
4491 | 9*6^132330+7 | 102974 | Lelio R Paula | 11/2014 |
4492 | 2^342034-16737 | 102963 | Enzo Doria | 05/2013 |
4493 | (9787*2^342011-1)/25 | 102959 | Lelio R Paula | 05/2015 |
4494 | (3^215802-5)/124343756 | 102956 | Lelio R Paula | 12/2013 |
4495 | 2^341991-17421 | 102950 | Enzo Doria | 05/2013 |
4496 | 2^341969-112225 | 102943 | Enzo Doria | 06/2014 |
4497 | (9787*2^341864-1)/3 | 102915 | Lelio R Paula | 05/2015 |
4498 | 2^341859+1869413 | 102910 | Enzo Doria | 05/2016 |
4499 | 10^102909+61 | 102910 | Lelio R Paula | 11/2016 |
4500 | 10^102909-67049 | 102909 | Patrick De Geest | 12/2016 |