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 | (77*10^93249-257)/9 | 93250 | Robert Price | 10/2017 |
9252 | 21822^18749+18749^21822 | 93245 | Hans Havermann | 09/2020 |
9253 | 3^195430-2 | 93244 | Theodore Burton | 02/2007 |
9254 | (13*10^93236+47)/3 | 93237 | Robert Price | 01/2017 |
9255 | (67*10^93226-7)/3 | 93228 | Robert Price | 10/2017 |
9256 | (86*10^93217-221)/9 | 93218 | Robert Price | 06/2017 |
9257 | 2^309637-16329 | 93211 | Enzo Doria | 04/2013 |
9258 | (41*10^93207+49)/9 | 93208 | Robert Price | 04/2016 |
9259 | 14^81312-57 | 93194 | Lelio R Paula | 06/2016 |
9260 | 41^57783-40 | 93192 | Theodore Burton | 02/2007 |
9261 | 3*10^93179-97 | 93180 | Robert Price | 08/2017 |
9262 | 5*7^110252-4 | 93175 | Lelio R Paula | 11/2014 |
9263 | 22022^16959+16959^22022 | 93140 | Hans Havermann | 08/2020 |
9264 | 83061!4-512 | 93137 | Robert Price | 11/2019 |
9265 | 25169^5016+5016^25169 | 93135 | Hans Havermann | 08/2020 |
9266 | (483641^16384+1)/2 | 93135 | Richard Fischer | 02/2021 |
9267 | 2^309380-27 | 93133 | Lelio R Paula | 11/2012 |
9268 | 25653^4268+4268^25653 | 93127 | Hans Havermann | 08/2020 |
9269 | (2^309355+1826)/2 | 93125 | Stefano Morozzi | 02/2021 |
9270 | (2^309334+2710)/2 | 93119 | Stefano Morozzi | 09/2020 |
9271 | 23*10^93116-7 | 93118 | Robert Price | 03/2016 |
9272 | (2^309270+3214)/2 | 93100 | Stefano Morozzi | 09/2020 |
9273 | (29*10^93082+1)/3 | 93083 | Robert Price | 11/2015 |
9274 | (2^309198+3738)/2 | 93078 | Stefano Morozzi | 09/2020 |
9275 | (58*10^93062-13)/9 | 93063 | Robert Price | 09/2015 |
9276 | (2^309138+1794)/2 | 93060 | Stefano Morozzi | 01/2021 |
9277 | 2^309131+1869383 | 93058 | Enzo Doria | 05/2016 |
9278 | (2^309114+3610)/2 | 93053 | Stefano Morozzi | 09/2020 |
9279 | 2^309066-93 | 93039 | Lelio R Paula | 11/2012 |
9280 | 2^309063-113269 | 93038 | Enzo Doria | 07/2014 |
9281 | 3^194965+16 | 93022 | Robert Price | 10/2013 |
9282 | (89*10^93019-737)/9 | 93020 | Robert Price | 01/2022 |
9283 | 2^308979+1869143 | 93012 | Enzo Doria | 05/2016 |
9284 | 5^133064+54 | 93008 | Lelio R Paula | 04/2016 |
9285 | (2^308947+2750)/2 | 93003 | Stefano Morozzi | 01/2021 |
9286 | (2^308905-509203)/119 | 92988 | Lelio R Paula | 06/2013 |
9287 | (94*10^92981-49)/99 | 92981 | Ray Chandler | 07/2011 |
9288 | (2*10^92976-17)/3 | 92976 | Robert Price | 08/2014 |
9289 | 24783^5642+5642^24783 | 92972 | Hans Havermann | 08/2020 |
9290 | (472489^16384+1)/2 | 92969 | Richard Fischer | 01/2021 |
9291 | 2^308809+37967 | 92961 | Phil Moore | 03/2008 |
9292 | (3^194825+5)/11656 | 92952 | Lelio R Paula | 04/2013 |
9293 | norm(Mod(x,x^3-x-1)^760951-1) | 92930 | Paul Vanderveen | 08/2022 |
9294 | (2^308679+5)/1573 | 92919 | Lelio R Paula | 06/2013 |
9295 | 2^308650+513 | 92913 | Lelio R Paula | 02/2016 |
9296 | 5*7^109936+6 | 92908 | Lelio R Paula | 10/2014 |
9297 | 2*10^92897+89 | 92898 | Robert Price | 07/2016 |
9298 | (2^308585+3182)/2 | 92894 | Stefano Morozzi | 09/2020 |
9299 | (467307^16384+1)/2 | 92891 | Richard Fischer | 01/2021 |
9300 | 12^86073-29 | 92889 | Lelio R Paula | 06/2016 |
9301 | 2^308566-117 | 92888 | Lelio R Paula | 11/2012 |
9302 | 11594^22851-22851^11594 | 92872 | Norbert Schneider | 03/2022 |
9303 | (17*10^92869-11)/3 | 92870 | Robert Price | 07/2015 |
9304 | 2^308459+75431 | 92856 | Enzo Doria | 04/2016 |
9305 | 2^308426-1866377 | 92846 | Enzo Doria | 03/2016 |
9306 | 24810^5521+5521^24810 | 92840 | Hans Havermann | 09/2020 |
9307 | (96^46831-1)/95 | 92830 | Paul Bourdelais | 09/2017 |
9308 | 2^308349-112111 | 92823 | Enzo Doria | 06/2014 |
9309 | 9*10^92821+19 | 92822 | Robert Price | 05/2016 |
9310 | 2^308337-75 | 92819 | Lelio R Paula | 11/2012 |
9311 | 4*3^194508+7 | 92805 | Lelio R Paula | 08/2014 |
9312 | 22*25^66379-113 | 92796 | Curtis Bright | 05/2015 |
9313 | 24538^6047+6047^24538 | 92792 | Hans Havermann | 09/2020 |
9314 | 11^89101-1080 | 92790 | Enzo Doria | 12/2012 |
9315 | 2^308236-1867859 | 92789 | Enzo Doria | 04/2016 |
9316 | 12^85963+25 | 92770 | Lelio R Paula | 06/2016 |
9317 | 23692^8221+8221^23692 | 92753 | Hans Havermann | 08/2020 |
9318 | 7^109750-66 | 92750 | Lelio R Paula | 04/2016 |
9319 | 9*10^92742+11 | 92743 | Robert Kinser | 01/2017 |
9320 | (2^308054+778)/2 | 92734 | Stefano Morozzi | 09/2020 |
9321 | (21*2^308046-1)/8857 | 92729 | Lelio R Paula | 02/2014 |
9322 | 2^307997-110655 | 92717 | Enzo Doria | 05/2014 |
9323 | L(443609) | 92709 | Renaud Lifchitz | 11/2003 |
9324 | 10^92691+9 | 92692 | Robert Price | 03/2011 |
9325 | 10^92691-252291 | 92691 | Patrick De Geest | 08/2013 |
9326 | (2^307883+3506)/2 | 92682 | Stefano Morozzi | 09/2020 |
9327 | 2^307879-2521 | 92681 | Yves Minet | 07/2014 |
9328 | 21269^21412+21412^21269 | 92666 | Norbert Schneider | 04/2020 |
9329 | (2^307780-7)/19959669 | 92644 | Lelio R Paula | 06/2013 |
9330 | 14^80831+47 | 92643 | Lelio R Paula | 06/2016 |
9331 | 15^78744+92 | 92611 | Lelio R Paula | 07/2016 |
9332 | 65^51083-64 | 92610 | Norbert Schneider | 11/2014 |
9333 | (2^307639+506)/2 | 92609 | Stefano Morozzi | 09/2020 |
9334 | 131^43737-130 | 92604 | Norbert Schneider | 02/2019 |
9335 | 6^118996-16943 | 92597 | Enzo Doria | 02/2013 |
9336 | (16*10^92591+41)/3 | 92592 | Robert Price | 10/2017 |
9337 | (2^307566+3094)/2 | 92587 | Stefano Morozzi | 09/2020 |
9338 | (2^307528+1578)/2 | 92575 | Stefano Morozzi | 09/2020 |
9339 | 9*10^92573+83 | 92574 | Robert Price | 11/2020 |
9340 | (3^194014-7)/2 | 92568 | Lelio R Paula | 04/2013 |
9341 | (2^307498+570)/2 | 92566 | Stefano Morozzi | 09/2020 |
9342 | (63^51439+1)/64 | 92555 | Paul Bourdelais | 08/2017 |
9343 | (26*10^92546-77)/3 | 92547 | Robert Price | 02/2017 |
9344 | 3^193954+340 | 92540 | Lelio R Paula | 04/2016 |
9345 | 2^307407-307407^2 | 92539 | Henri Lifchitz | 05/2005 |
9346 | 24299^6430+6430^24299 | 92536 | Hans Havermann | 08/2020 |
9348 | (17*10^92513-11)/3 | 92514 | Robert Price | 07/2015 |
9347 | 2^307322-399071 | 92514 | Harrison Mesh | 06/2022 |
9349 | 2^307318-4211 | 92512 | Yves Minet | 06/2011 |
9350 | 2^307280+193 | 92501 | Lelio R Paula | 11/2012 |
9351 | 20^71075-33 | 92471 | Lelio R Paula | 05/2016 |
9352 | 2^307126-78795 | 92455 | Lelio R Paula | 02/2016 |
9353 | 2^307125-49789 | 92454 | Henri Lifchitz | 08/2005 |
9354 | 2^307099+23 | 92447 | Lelio R Paula | 11/2012 |
9355 | 129^43798+128 | 92440 | Norbert Schneider | 01/2019 |
9356 | 21826^17181+17181^21826 | 92435 | Hans Havermann | 09/2020 |
9357 | Phi(277314,10)/3327769 | 92430 | Makoto Kamada | 05/2019 |
9358 | (2^307022+1038)/2 | 92423 | Stefano Morozzi | 09/2020 |
9359 | 7*10^92414-69 | 92415 | Robert Price | 01/2017 |
9360 | (9787*2^306975-1)/2129155 | 92407 | Lelio R Paula | 04/2015 |
9361 | 22280^14037+14037^22280 | 92402 | Hans Havermann | 09/2020 |
9362 | 2^306915+1873373 | 92391 | Enzo Doria | 06/2016 |
9363 | 2^306908+12915 | 92389 | Lelio R Paula | 02/2016 |
9364 | 21620^18759+18759^21620 | 92387 | Hans Havermann | 09/2020 |
9365 | (2^306897+1142)/2 | 92385 | Stefano Morozzi | 09/2020 |
9366 | 19^72244-80 | 92383 | Lelio R Paula | 06/2016 |
9368 | 2^306849+75 | 92371 | Lelio R Paula | 11/2012 |
9367 | (397*10^92369+53)/9 | 92371 | Robert Price | 11/2017 |
9369 | 2^306813-869688105 | 92360 | Lelio R Paula | 08/2016 |
9370 | (2^306797+2694)/2 | 92355 | Stefano Morozzi | 09/2020 |
9371 | 21713^17914+17914^21713 | 92350 | Hans Havermann | 09/2020 |
9372 | 2^306672-109059 | 92318 | Enzo Doria | 04/2014 |
9373 | (430463^16384+1)/2 | 92307 | Richard Fischer | 09/2020 |
9374 | 6^118604-17063 | 92292 | Enzo Doria | 02/2013 |
9375 | 21969^15878+15878^21969 | 92288 | Hans Havermann | 08/2020 |
9376 | 21690^17959+17959^21690 | 92276 | Hans Havermann | 08/2020 |
9377 | (7613^23773-1)/7612 | 92273 | Serge Batalov | 08/2020 |
9378 | (31*2^306487-1)/2291 | 92260 | Lelio R Paula | 02/2014 |
9379 | 2^306467-1835611 | 92256 | Enzo Doria | 04/2015 |
9380 | 2^306433+76427 | 92246 | Enzo Doria | 05/2016 |
9381 | 2^306430-113313 | 92245 | Enzo Doria | 07/2014 |
9382 | (23*10^92244-59)/9 | 92245 | Robert Price | 03/2017 |
9383 | (47*2^306398+1)/1384803 | 92231 | Lelio R Paula | 04/2015 |
9384 | 21436^20043+20043^21436 | 92217 | Hans Havermann | 09/2020 |
9385 | (25*10^92186+59)/3 | 92187 | Robert Price | 05/2016 |
9386 | 2^306213-286183 | 92180 | Enzo Doria | 06/2015 |
9387 | 10^92178+61 | 92179 | Robert Price | 04/2011 |
9388 | 10^92178-175509 | 92178 | Patrick De Geest | 07/2013 |
9389 | 2^306200+1185 | 92176 | Lelio R Paula | 02/2016 |
9390 | (2^306171+1514)/2 | 92167 | Stefano Morozzi | 09/2020 |
9391 | 2^306153+11 | 92162 | Lelio R Paula | 11/2012 |
9392 | (2^306141-7)/5 | 92157 | Lelio R Paula | 06/2013 |
9393 | 21725^17454+17454^21725 | 92156 | Hans Havermann | 09/2020 |
9394 | (26*10^92152-23)/3 | 92153 | Robert Price | 08/2016 |
9395 | (2*10^92152+67)/3 | 92152 | Robert Price | 04/2017 |
9396 | 21368^20457+20457^21368 | 92115 | Hans Havermann | 08/2020 |
9397 | (2^305996+3018)/2 | 92114 | Stefano Morozzi | 09/2020 |
9398 | 4*10^92099+3*7^92099 | 92100 | Mike Oakes | 12/2010 |
9399 | 30504^1045+1045^30504 | 92096 | Hans Havermann | 08/2020 |
9400 | 24334^6085+6085^24334 | 92087 | Hans Havermann | 08/2020 |
9401 | (2^305898+2262)/2 | 92085 | Stefano Morozzi | 01/2021 |
9402 | (4591^25147-1)/4590 | 92083 | Serge Batalov | 08/2020 |
9403 | 115^44684+114 | 92081 | Norbert Schneider | 07/2018 |
9404 | (416771^16384+1)/2 | 92077 | Richard Fischer | 08/2020 |
9405 | (2^305867-2^152934+1)/5 | 92075 | | 11/2006 |
9406 | 2^305851-109495 | 92071 | Enzo Doria | 04/2014 |
9407 | (416213^16384+1)/2 | 92067 | Richard Fischer | 08/2020 |
9409 | 12^85300+35 | 92055 | Lelio R Paula | 06/2016 |
9408 | ((10^46026-1)*1000/9+739)*10^46026+(10^46026-1)/9 | 92055 | Hans Havermann | 05/2019 |
9410 | 21732^17201+17201^21732 | 92048 | Hans Havermann | 08/2020 |
9411 | 7*10^92043-39 | 92044 | Robert Price | 07/2020 |
9412 | 3^192911+92 | 92042 | Lelio R Paula | 04/2016 |
9413 | 2^305723-276271 | 92032 | Enzo Doria | 07/2015 |
9414 | (2^305714+3738)/2 | 92029 | Stefano Morozzi | 09/2020 |
9415 | (2^305707+798)/2 | 92027 | Stefano Morozzi | 01/2021 |
9416 | (2^305682+3202)/2 | 92020 | Stefano Morozzi | 09/2020 |
9417 | 2^305676+93 | 92018 | Donovan Johnson | 06/2008 |
9418 | 3^192819+4 | 91999 | Lelio R Paula | 04/2013 |
9419 | (38*10^91997-83)/99 | 91997 | Ray Chandler | 07/2011 |
9420 | 2^305539+72911 | 91977 | Enzo Doria | 04/2016 |
9421 | Phi(212014,10) | 91960 | Makoto Kamada | 04/2019 |
9422 | (2^305423+2666)/2 | 91942 | Stefano Morozzi | 09/2020 |
9423 | (2^305427-5)/1297998213 | 91934 | Lelio R Paula | 06/2013 |
9424 | 82077!4-128 | 91928 | Robert Price | 11/2019 |
9425 | 85^47642+84 | 91922 | Norbert Schneider | 10/2015 |
9426 | 10^91920+187651 | 91921 | Patrick De Geest | 07/2013 |
9427 | 10^91920-39 | 91920 | Robert Price | 05/2011 |
9428 | (3975^25537-1)/3974 | 91913 | Serge Batalov | 08/2020 |
9429 | (22*10^91903+41)/9 | 91904 | Robert Price | 02/2015 |
9431 | 2^305275-27 | 91897 | Lelio R Paula | 11/2012 |
9430 | 3^192607-7726 | 91897 | Enzo Doria | 05/2014 |
9432 | (2^305243+2274)/2 | 91887 | Stefano Morozzi | 02/2021 |
9433 | (2^305220+1278)/2 | 91881 | Stefano Morozzi | 01/2021 |
9435 | (6958^23911-1)/6957 | 91874 | Serge Batalov | 08/2020 |
9434 | 9*10^91873-91 | 91874 | Robert Price | 06/2021 |
9436 | (7*10^91856-61)/9 | 91856 | Erik Branger | 12/2009 |
9437 | (2^305107+1274)/2 | 91847 | Stefano Morozzi | 01/2021 |
9438 | (89*10^91839-53)/9 | 91840 | Robert Price | 11/2015 |
9439 | 1678^28479+28479^1678 | 91839 | Norbert Schneider | 08/2020 |
9440 | (2^304996+406)/2 | 91813 | Stefano Morozzi | 02/2021 |
9441 | 8*5^131351+7 | 91812 | Lelio R Paula | 10/2014 |
9442 | 3^192426-68 | 91811 | Lelio R Paula | 04/2016 |
9443 | 2^304979-375607 | 91808 | Harrison Mesh | 06/2022 |
9445 | 2^304973+117 | 91807 | Lelio R Paula | 11/2012 |
9444 | (2^304974+234)/2 | 91807 | Stefano Morozzi | 09/2020 |
9446 | 2^304939-1863727 | 91796 | Enzo Doria | 03/2016 |
9447 | 2^304902+8415 | 91785 | Lelio R Paula | 02/2016 |
9448 | 18^73119+61 | 91785 | Lelio R Paula | 06/2016 |
9449 | (78557*2^304740+1)/159 | 91739 | Lelio R Paula | 06/2014 |
9450 | 2^304747+71355 | 91738 | Lelio R Paula | 02/2016 |
9451 | (2^304738+2262)/2 | 91735 | Stefano Morozzi | 01/2021 |
9452 | 8*3^192251+5 | 91728 | Lelio R Paula | 08/2014 |
9453 | (2^304693+3746)/2 | 91722 | Stefano Morozzi | 01/2021 |
9454 | 2^304674-87 | 91717 | Lelio R Paula | 11/2012 |
9455 | (28*10^91715+701)/9 | 91716 | Robert Price | 11/2017 |
9456 | (2^304698+9)/1362809953 | 91715 | Lelio R Paula | 06/2013 |
9457 | (2^304671+1046)/2 | 91715 | Stefano Morozzi | 09/2020 |
9458 | (2^304667+3614)/2 | 91714 | Stefano Morozzi | 09/2020 |
9459 | (5*10^91707-119)/3 | 91708 | Robert Price | 04/2016 |
9460 | 2^304633-112771 | 91704 | Enzo Doria | 06/2014 |
9461 | (5*10^91702+19)/3 | 91703 | Robert Price | 07/2022 |
9462 | (395347^16384+1)/2 | 91701 | Richard Fischer | 06/2020 |
9463 | 9*10^91693+23 | 91694 | Robert Price | 11/2017 |
9464 | 2^304578-16713 | 91688 | Enzo Doria | 05/2013 |
9465 | 6^117824-18713 | 91685 | Enzo Doria | 03/2013 |
9466 | (17^74509-1)/16 | 91679 | Paul Bourdelais | 03/2010 |
9467 | (2^304551-7)/8912231 | 91673 | Lelio R Paula | 06/2013 |
9468 | (2^304532-3)/3013 | 91670 | Lelio R Paula | 06/2013 |
9469 | 2^304507-16645 | 91666 | Enzo Doria | 04/2013 |
9470 | (29*2^304489+1)/31 | 91661 | Lelio R Paula | 09/2013 |
9471 | 21221^20860+20860^21221 | 91661 | Hans Havermann | 08/2020 |
9472 | 2^304471-1836127 | 91655 | Enzo Doria | 04/2015 |
9473 | 2^304433+32769 | 91644 | Lelio R Paula | 02/2016 |
9474 | 2^304410-16625 | 91637 | Enzo Doria | 04/2013 |
9475 | 183^40500-2 | 91630 | Lelio R Paula | 10/2017 |
9476 | (2^304373+1850)/2 | 91626 | Stefano Morozzi | 02/2021 |
9477 | (203*10^91613+1)/3 | 91615 | Robert Price | 09/2017 |
9478 | 4*5^131052+7 | 91603 | Lelio R Paula | 10/2014 |
9480 | 21725^16458+16458^21725 | 91601 | Hans Havermann | 08/2020 |
9479 | (2^304290+3270)/2 | 91601 | Stefano Morozzi | 02/2021 |
9481 | 2^304278-4001 | 91597 | Yves Minet | 04/2017 |
9482 | 2^304272-1866377 | 91595 | Enzo Doria | 03/2016 |
9483 | 2^304268+19683 | 91594 | Lelio R Paula | 02/2016 |
9484 | 21718^16497+16497^21718 | 91594 | Hans Havermann | 08/2020 |
9485 | 22759^10576+10576^22759 | 91590 | Hans Havermann | 08/2020 |
9486 | (29*10^91584-119)/9 | 91585 | Robert Price | 12/2021 |
9487 | 2^304162-161 | 91562 | Lelio R Paula | 11/2012 |
9488 | 5*7^108331+6 | 91552 | Lelio R Paula | 10/2014 |
9489 | 2^304082-2787821 | 91538 | Enzo Doria | 03/2016 |
9490 | 2^304015+64133 | 91518 | David Broadhurst | 06/2002 |
9491 | (7*2^303963+1)/3 | 91503 | Lelio R Paula | 09/2013 |
9492 | (2^303945-9)/3617 | 91494 | Lelio R Paula | 06/2013 |
9493 | (384007^16384+1)/2 | 91494 | Richard Fischer | 05/2020 |
9494 | 2^303930+1880527 | 91493 | Enzo Doria | 10/2016 |
9495 | 24810^4859+4859^24810 | 91464 | Hans Havermann | 08/2020 |
9496 | 2*10^91457+29 | 91458 | Robert Price | 02/2021 |
9497 | 3^191684-68 | 91457 | Lelio R Paula | 04/2016 |
9498 | (21*2^303813-1)/169 | 91456 | Lelio R Paula | 02/2014 |
9499 | (298*10^91446+11)/3 | 91448 | Robert Price | 11/2020 |
9500 | 11^87797-120 | 91432 | Enzo Doria | 07/2012 |