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 | 10^113933+145897 | 113934 | Patrick De Geest | 02/2021 |
6254 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
6252 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
6253 | 10^113933-66167 | 113933 | Patrick De Geest | 02/2021 |
6255 | 2339662057597*10^113903+9 | 113916 | Robert Price | 03/2021 |
6256 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
6257 | 10^113894+76609 | 113895 | Patrick De Geest | 02/2021 |
6259 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
6258 | 10^113894-373947 | 113894 | Patrick De Geest | 02/2021 |
6260 | 33933^2270+2270^33933 | 113881 | Hans Havermann | 02/2021 |
6261 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
6262 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
6263 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
6264 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
6265 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
6266 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
6267 | 2^378137+1334025 | 113831 | Henri Lifchitz | 10/2024 |
6268 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
6269 | 30030^6173+6173^30030 | 113829 | Hans Havermann | 01/2021 |
6270 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
6271 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
6272 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
6273 | 34596^1949+1949^34596 | 113815 | Hans Havermann | 02/2021 |
6274 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
6275 | (28*10^113810+449)/9 | 113811 | Robert Price | 11/2021 |
6276 | 26273^21448+21448^26273 | 113799 | Hans Havermann | 02/2021 |
6277 | 2^377966+255255 | 113780 | Paul Bourdelais | 10/2021 |
6278 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
6279 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
6280 | (7*10^113757+11)/9 | 113757 | Tyler Busby | 01/2023 |
6281 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
6282 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
6283 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
6284 | (2^377791+3258)/2 | 113727 | Stefano Morozzi | 02/2021 |
6285 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
6286 | 28206^10739+10739^28206 | 113698 | Hans Havermann | 01/2021 |
6287 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
6288 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
6289 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
6290 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
6291 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |
6292 | (10^113656+7)/40231/6439001 | 113645 | Alfred Reich | 06/2021 |
6293 | (2*10^113631+529)/9 | 113631 | Robert Price | 01/2018 |
6294 | 2^377408-2798279 | 113612 | Enzo Doria | 04/2016 |
6295 | (292^46336+1)/(292^256+1) | 113605 | Martin Hopf | 12/2023 |
6296 | 13706^27459+27459^13706 | 113596 | Norbert Schneider | 12/2020 |
6297 | 5*10^113586+57 | 113587 | Robert Price | 12/2018 |
6298 | (2^377242+2794)/2 | 113561 | Stefano Morozzi | 02/2021 |
6299 | 2^377222+93679 | 113556 | Yves Minet | 02/2011 |
6301 | 262147#+316847 | 113552 | Makoto Morimoto | 02/2016 |
6300 | 27172^15101+15101^27172 | 113552 | Hans Havermann | 01/2021 |
6302 | (2^377131+2018)/2 | 113528 | Stefano Morozzi | 09/2020 |
6303 | 2^377026-458903 | 113497 | Enzo Doria | 04/2015 |
6304 | (2^377014+2202)/2 | 113493 | Stefano Morozzi | 09/2020 |
6305 | 34791^1828+1828^34791 | 113488 | Hans Havermann | 02/2021 |
6306 | (2^376967+258)/2 | 113479 | Stefano Morozzi | 09/2020 |
6307 | (29*10^113468-17)/3 | 113469 | Robert Price | 09/2019 |
6308 | 35^73477-34^73477 | 113454 | Robert Price | 05/2012 |
6309 | (14*10^113451-89)/3 | 113452 | Robert Price | 05/2020 |
6310 | 2^376790+27 | 113426 | Lelio R Paula | 03/2013 |
6311 | 2^376786-112701 | 113424 | Enzo Doria | 06/2014 |
6312 | 2^376731+21 | 113408 | Lelio R Paula | 03/2013 |
6313 | 9*10^113394-73 | 113395 | Robert Price | 01/2021 |
6314 | 29660^6651+6651^29660 | 113387 | Hans Havermann | 01/2021 |
6316 | 2^376634+1871245 | 113379 | Enzo Doria | 05/2016 |
6315 | (4^188318+17)/3 | 113379 | Robert Price | 02/2017 |
6317 | E(29532)/492842728006750534663978950979381235 | 113367 | Thomas Ritschel | 01/2021 |
6318 | 2^376576-344369 | 113361 | Enzo Doria | 07/2015 |
6319 | 2^376549-4543 | 113353 | Yves Minet | 03/2019 |
6320 | (17*2^376526-1)/223 | 113345 | Lelio R Paula | 03/2014 |
6321 | (27*2^376441-1)/811 | 113319 | Lelio R Paula | 03/2014 |
6322 | (2^376350+3538)/2 | 113293 | Stefano Morozzi | 09/2020 |
6323 | 27265^14272+14272^27265 | 113273 | Hans Havermann | 02/2021 |
6324 | 32174^3315+3315^32174 | 113269 | Hans Havermann | 02/2021 |
6325 | (2*10^113248+73)/3 | 113248 | Robert Price | 09/2020 |
6326 | 2^376128-21237 | 113226 | Enzo Doria | 06/2013 |
6327 | 5*10^113224-81 | 113225 | Robert Price | 01/2019 |
6328 | 10^113220+51409 | 113221 | Patrick De Geest | 02/2021 |
6330 | 10^113220-1000001 | 113220 | Robert Price | 08/2020 |
6329 | 10^113220-61779 | 113220 | Patrick De Geest | 02/2021 |
6331 | (2^376102+1578)/2 | 113218 | Stefano Morozzi | 09/2020 |
6332 | (2^375996+3102)/2 | 113186 | Stefano Morozzi | 09/2020 |
6333 | 37258^1091+1091^37258 | 113184 | Hans Havermann | 02/2021 |
6334 | (94*10^113176+11)/3 | 113178 | Robert Price | 04/2020 |
6335 | (46*10^113162+521)/9 | 113163 | Robert Price | 02/2019 |
6336 | (2^375885+3614)/2 | 113153 | Stefano Morozzi | 09/2020 |
6337 | (2^375873+2006)/2 | 113149 | Stefano Morozzi | 09/2020 |
6338 | 3*10^113137+23 | 113138 | Robert Price | 03/2022 |
6339 | (8*10^113135-269)/9 | 113135 | Robert Price | 09/2021 |
6340 | (2^375726+694)/2 | 113105 | Stefano Morozzi | 09/2020 |
6341 | 26050^21959+21959^26050 | 113100 | Hans Havermann | 02/2021 |
6342 | 2^375638-1787171 | 113079 | Enzo Doria | 03/2016 |
6343 | (2^375563+2178)/2 | 113056 | Stefano Morozzi | 09/2020 |
6344 | 2^375537+4847 | 113048 | Yves Minet | 02/2011 |
6345 | (17^91837+16^91837)/33 | 113000 | Robert Price | 08/2013 |
6346 | 5*F(270343)*(F(270343)+1)+1 | 112997 | Serge Batalov | 04/2015 |
6347 | 2^375350-3221 | 112992 | Yves Minet | 07/2011 |
6348 | (13*2^375306+1)/7 | 112979 | Lelio R Paula | 03/2014 |
6349 | 2^375305+76757 | 112979 | Enzo Doria | 05/2016 |
6350 | 2^375258+1874989 | 112964 | Enzo Doria | 07/2016 |
6351 | (29^77237-1)/28 | 112950 | Robert Price | 04/2012 |
6352 | 26620^17483+17483^26620 | 112939 | Hans Havermann | 01/2021 |
6353 | 2^375154+74959 | 112933 | Enzo Doria | 04/2016 |
6354 | (266*10^112930-17)/3 | 112932 | Robert Price | 07/2020 |
6355 | (2^375134+778)/2 | 112927 | Stefano Morozzi | 09/2020 |
6356 | (85*10^112919+293)/9 | 112920 | Robert Price | 12/2019 |
6357 | 29412^6899+6899^29412 | 112907 | Hans Havermann | 02/2021 |
6358 | (191*10^112903-17)/3 | 112905 | Robert Price | 11/2021 |
6359 | (2^375022+1210)/2 | 112893 | Stefano Morozzi | 09/2020 |
6360 | 25992^22039+22039^25992 | 112889 | Hans Havermann | 02/2021 |
6361 | 2^374993-2787163 | 112885 | Enzo Doria | 11/2015 |
6362 | (697*10^112869+23)/9 | 112871 | Robert Price | 04/2021 |
6363 | (19*10^112851+377)/9 | 112852 | Alfred Reich | 11/2022 |
6364 | 30420^5111+5111^30420 | 112813 | Hans Havermann | 02/2021 |
6365 | 13699^27268+27268^13699 | 112800 | Norbert Schneider | 12/2020 |
6366 | 123^53970+2 | 112793 | Lelio R Paula | 06/2013 |
6367 | 12308*27^78796+23 | 112790 | Eric Chen | 07/2023 |
6368 | 28942^7887+7887^28942 | 112785 | Hans Havermann | 01/2021 |
6369 | 2^374640-4685 | 112778 | Yves Minet | 05/2019 |
6370 | 12^104501-11 | 112776 | Riley Fisher | 12/2023 |
6371 | 57290^93+93^57290 | 112775 | Hans Havermann | 01/2021 |
6372 | 115^54726-54726^115 | 112774 | Norman Luhn | 09/2024 |
6373 | 13608^27277-27277^13608 | 112758 | Norbert Schneider | 12/2022 |
6374 | (2^374556+682)/2 | 112753 | Stefano Morozzi | 09/2020 |
6375 | p(22852)#/2+512 | 112745 | Robert Price | 05/2017 |
6376 | (11*10^112742-131)/3 | 112743 | Robert Price | 09/2018 |
6377 | 2^374441-21049 | 112718 | Enzo Doria | 06/2013 |
6378 | 4*10^112706-93 | 112707 | Robert Price | 07/2020 |
6379 | (2^374321+1)/3 | 112682 | H.&R. Lifchitz | 12/2000 |
6380 | (2^374213+1766)/2 | 112650 | Stefano Morozzi | 09/2020 |
6381 | (2^374094+2674)/2 | 112614 | Stefano Morozzi | 09/2020 |
6382 | (13^101089-1)/12 | 112607 | Paul Bourdelais | 03/2010 |
6383 | 3^236011-8828 | 112606 | Enzo Doria | 05/2014 |
6384 | 31854^3427+3427^31854 | 112602 | Hans Havermann | 02/2021 |
6385 | (85*10^112530-373)/9 | 112531 | Robert Price | 05/2021 |
6386 | (2^373748+1090)/2 | 112510 | Stefano Morozzi | 09/2020 |
6387 | (2^373727+1806)/2 | 112503 | Stefano Morozzi | 01/2021 |
6388 | 27618^11821+11821^27618 | 112479 | Hans Havermann | 01/2021 |
6389 | U(602921)/4823369 | 112479 | Serge Batalov | 07/2024 |
6390 | 26343^18580+18580^26343 | 112460 | Hans Havermann | 01/2021 |
6391 | 39^70666+2 | 112435 | Lelio R Paula | 11/2013 |
6392 | (134*10^112400+7)/3 | 112402 | Robert Price | 02/2020 |
6393 | (2^373365+326)/2 | 112394 | Stefano Morozzi | 09/2020 |
6394 | (2^373350-7)/3 | 112390 | Lelio R Paula | 03/2014 |
6395 | 2^373306-1840931 | 112377 | Enzo Doria | 06/2015 |
6396 | 8*10^112371-37 | 112372 | Robert Price | 03/2022 |
6397 | (2^373262+270)/2 | 112363 | Stefano Morozzi | 09/2020 |
6398 | 6*10^112359-79 | 112360 | Robert Price | 03/2022 |
6399 | 34409^1842+1842^34409 | 112356 | Hans Havermann | 02/2021 |
6400 | (142*10^112352-7)/9 | 112354 | Robert Price | 12/2019 |
6401 | 25580^24633+24633^25580 | 112336 | Hans Havermann | 01/2021 |
6402 | 2^373117+1872107 | 112320 | Enzo Doria | 05/2016 |
6403 | (2^373083+2226)/2 | 112309 | Stefano Morozzi | 01/2021 |
6404 | (35*10^112308+37)/9 | 112309 | Robert Price | 05/2022 |
6405 | 27963^10376+10376^27963 | 112301 | Hans Havermann | 01/2021 |
6406 | 2^373040+2792761 | 112297 | Enzo Doria | 03/2016 |
6407 | (156^51203-1)/155 | 112293 | Anonymous | 06/2024 |
6408 | EPrime(112280) | 112280 | Eric W. Weisstein | 10/2010 |
6409 | 2^372950+7 | 112270 | Lelio R Paula | 03/2013 |
6410 | 2^372945+1872167 | 112268 | Enzo Doria | 05/2016 |
6411 | 36274^1243+1243^36274 | 112249 | Hans Havermann | 02/2021 |
6412 | (55*10^112246+629)/9 | 112247 | Robert Price | 01/2022 |
6413 | (2^372872+3798)/2 | 112246 | Stefano Morozzi | 02/2021 |
6414 | 25513^25060+25060^25513 | 112232 | Hans Havermann | 02/2021 |
6415 | (53*10^112229-611)/9 | 112230 | Robert Price | 02/2022 |
6416 | (2^372754+3094)/2 | 112210 | Stefano Morozzi | 09/2020 |
6417 | 6*10^112208+43 | 112209 | Robert Price | 07/2020 |
6418 | (2^372728+3058)/2 | 112203 | Stefano Morozzi | 09/2020 |
6419 | 32061^3158+3158^32061 | 112195 | Hans Havermann | 02/2021 |
6420 | 2^372695+72113 | 112193 | Enzo Doria | 04/2016 |
6421 | 2^372689-2563 | 112191 | Yves Minet | 07/2014 |
6422 | 2^372638-1859441 | 112176 | Enzo Doria | 03/2016 |
6423 | 3^235074-113056 | 112159 | Enzo Doria | 07/2014 |
6424 | -E(29242)/489848365125303139750087 | 112140 | Thomas Ritschel | 01/2021 |
6425 | (2^372514+1822)/2 | 112138 | Stefano Morozzi | 02/2021 |
6426 | 29468^6387+6387^29468 | 112135 | Hans Havermann | 02/2021 |
6427 | (2^372428+3742)/2 | 112112 | Stefano Morozzi | 01/2021 |
6428 | 25375^25454-25454^25375 | 112110 | Norbert Schneider | 01/2021 |
6429 | 131^52949-130 | 112108 | Norbert Schneider | 02/2019 |
6430 | (91*10^112098-1)/9 | 112100 | Robert Price | 04/2015 |
6431 | (68*10^112087-149)/9 | 112088 | Robert Price | 07/2022 |
6432 | 2^372314+212419 | 112078 | Enzo Doria | 08/2023 |
6433 | 2^372310-85061 | 112077 | Paul Bourdelais | 02/2012 |
6434 | (10^112067+53)/9 | 112067 | Serge Batalov | 07/2015 |
6435 | (2^372237+2082)/2 | 112055 | Stefano Morozzi | 09/2020 |
6436 | 75^59760+2 | 112054 | Lelio R Paula | 08/2013 |
6437 | (2^372200+1710)/2 | 112044 | Stefano Morozzi | 09/2020 |
6438 | (10^112048+7)/8747/373157 | 112039 | Alfred Reich | 06/2021 |
6439 | (2^372181+2250)/2 | 112038 | Stefano Morozzi | 01/2021 |
6440 | F(536233)/825798821/1102495049/5641839306317 | 112036 | Serge Batalov | 07/2024 |
6441 | (2^372161+2574)/2 | 112032 | Stefano Morozzi | 09/2020 |
6442 | 8*10^112018-49 | 112019 | Robert Price | 08/2019 |
6444 | 2^372108+74221 | 112016 | Enzo Doria | 04/2016 |
6443 | (2^372109+2046)/2 | 112016 | Stefano Morozzi | 09/2020 |
6445 | 8*10^112011-73 | 112012 | Robert Price | 01/2022 |
6446 | (184*10^112007-1)/3 | 112009 | Robert Price | 02/2020 |
6447 | 5*10^112003+81 | 112004 | Robert Price | 03/2019 |
6448 | 215^48018+48018^215 | 112000 | Norbert Schneider | 10/2018 |
6449 | 25549^24148+24148^25549 | 111979 | Hans Havermann | 02/2021 |
6450 | 2^371970-459491 | 111975 | Enzo Doria | 05/2015 |
6451 | 41^69427-40^69427 | 111971 | Robert Price | 07/2012 |
6452 | (2^371782+358)/2 | 111918 | Stefano Morozzi | 01/2021 |
6453 | 15*25^80056+446 | 111915 | Eric Chen | 10/2023 |
6454 | (34*10^111902-547)/9 | 111903 | Robert Price | 03/2021 |
6455 | (10^111902+71)/9 | 111902 | Serge Batalov | 07/2015 |
6456 | (2^371625+1610)/2 | 111870 | Stefano Morozzi | 09/2020 |
6457 | 29080^7027+7027^29080 | 111865 | Hans Havermann | 01/2021 |
6458 | 31307^3730+3730^31307 | 111820 | Hans Havermann | 02/2021 |
6459 | 2^371440-112865 | 111815 | Enzo Doria | 06/2014 |
6460 | 2^371426-459161 | 111811 | Enzo Doria | 05/2015 |
6461 | 25768^21771+21771^25768 | 111779 | Hans Havermann | 02/2021 |
6462 | 28022^9747+9747^28022 | 111777 | Hans Havermann | 02/2021 |
6463 | (94*10^111770+11)/3 | 111772 | Robert Price | 04/2020 |
6464 | 2^371276-461267 | 111766 | Enzo Doria | 07/2015 |
6465 | 6*10^111760+43 | 111761 | Robert Price | 07/2020 |
6466 | 98079!4-1024 | 111747 | Robert Price | 11/2019 |
6467 | (2^371186+3162)/2 | 111738 | Stefano Morozzi | 09/2020 |
6468 | 10^(2*55862+1)-4*10^55862-1 | 111725 | Robert Price | 07/2023 |
6469 | 2^371098-113261 | 111712 | Enzo Doria | 07/2014 |
6470 | (7*10^111706-61)/9 | 111706 | Erik Branger | 01/2011 |
6471 | 2^371044+1869391 | 111696 | Enzo Doria | 05/2016 |
6472 | 26310^17581+17581^26310 | 111688 | Hans Havermann | 01/2021 |
6473 | 88^57416+87 | 111645 | Norbert Schneider | 11/2015 |
6474 | 2^370829-109693 | 111631 | Enzo Doria | 04/2014 |
6475 | 15^94906-2 | 111619 | Lelio R Paula | 11/2013 |
6476 | 25835^20904+20904^25835 | 111614 | Hans Havermann | 02/2021 |
6477 | (2^370726+1198)/2 | 111600 | Stefano Morozzi | 09/2020 |
6478 | 27758^10479+10479^27758 | 111597 | Hans Havermann | 02/2021 |
6479 | 2^370695-4627 | 111591 | Yves Minet | 05/2019 |
6480 | 7*10^111579-87 | 111580 | Robert Price | 06/2020 |
6481 | 2^370643-37 | 111575 | Yves Minet | 01/2012 |
6482 | (2^370655-5)/2439 | 111575 | Lelio R Paula | 03/2014 |
6483 | 2^370610-3911 | 111565 | Yves Minet | 04/2017 |
6484 | (154*10^111549+17)/9 | 111551 | Robert Price | 06/2022 |
6485 | 26200^18097+18097^26200 | 111550 | Hans Havermann | 01/2021 |
6486 | 286^45405+45405^286 | 111532 | Norbert Schneider | 04/2015 |
6487 | 4*5^159563-9 | 111531 | Lelio R Paula | 11/2014 |
6488 | 5*10^111528-21 | 111529 | Robert Price | 12/2021 |
6489 | (4^185225+23)/3 | 111517 | Robert Price | 10/2018 |
6490 | 2^370393-2797093 | 111500 | Enzo Doria | 03/2016 |
6491 | (21*2^370382-1)/629 | 111495 | Lelio R Paula | 03/2014 |
6492 | (2^370371+554)/2 | 111493 | Stefano Morozzi | 09/2020 |
6493 | 6^143273-5 | 111489 | Paul Bourdelais | 01/2021 |
6494 | 2^370292-2791979 | 111469 | Enzo Doria | 03/2016 |
6495 | (14*10^111443-11)/3 | 111444 | Robert Price | 08/2022 |
6496 | 2^370177-461551 | 111435 | Enzo Doria | 07/2015 |
6497 | (2^370169+1706)/2 | 111432 | Stefano Morozzi | 09/2020 |
6498 | 109^54683-108 | 111413 | Norbert Schneider | 02/2017 |
6500 | (83*10^111406-173)/9 | 111407 | Robert Price | 04/2021 |
6499 | 959^37362-37362^959 | 111407 | Henri Lifchitz | 10/2024 |