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 ? |
7251 | 2^360706-458903 | 108584 | Enzo Doria | 04/2015 |
7252 | 31112^3087+3087^31112 | 108567 | Hans Havermann | 11/2020 |
7253 | 2^360641-113185 | 108564 | Enzo Doria | 07/2014 |
7254 | (29*10^108559-71)/3 | 108560 | Robert Price | 07/2021 |
7256 | 2^360618+75019 | 108557 | Enzo Doria | 04/2016 |
7255 | 2^360618+2025 | 108557 | Henri Lifchitz | 01/2025 |
7257 | 2^360607-216367 | 108554 | Harrison Mesh | 06/2022 |
7258 | 29^74228+556 | 108551 | Eric Chen | 06/2025 |
7259 | (14*10^108545-221)/9 | 108546 | Robert Price | 04/2018 |
7260 | (44*10^108544-233)/9 | 108545 | Robert Price | 01/2019 |
7261 | 3*5^155286+2 | 108541 | Lelio R Paula | 11/2014 |
7262 | 7^128422+4 | 108530 | Lelio R Paula | 11/2014 |
7263 | 7^128421-80 | 108529 | Lelio R Paula | 03/2017 |
7264 | 2^360497+53807 | 108521 | Yves Minet | 07/2010 |
7265 | 2^360468-4559 | 108512 | Yves Minet | 03/2019 |
7266 | (2^360451+578)/2 | 108507 | Stefano Morozzi | 09/2020 |
7267 | 2^360366+1459 | 108481 | Lelio R Paula | 09/2011 |
7268 | 2^360364-1499 | 108481 | Yves Minet | 05/2013 |
7269 | 25662^16871+16871^25662 | 108477 | Hans Havermann | 10/2020 |
7270 | 25009^21748+21748^25009 | 108475 | Hans Havermann | 10/2020 |
7271 | 14^94642-23 | 108472 | Lelio R Paula | 05/2017 |
7272 | (197*10^108446-11)/3 | 108448 | Robert Price | 03/2020 |
7273 | (2^360229+1290)/2 | 108440 | Stefano Morozzi | 01/2021 |
7274 | 4*10^108412+9 | 108413 | Robert Price | 05/2015 |
7275 | (259*10^108409-1)/3 | 108411 | Robert Price | 12/2020 |
7276 | (2^360122+1194)/2 | 108408 | Stefano Morozzi | 09/2020 |
7277 | 133^51042-132 | 108406 | Norbert Schneider | 02/2019 |
7278 | 2^360093-351103 | 108399 | Enzo Doria | 06/2015 |
7279 | 2^360042+723 | 108384 | Lelio R Paula | 01/2012 |
7280 | 85^56164-2 | 108364 | Lelio R Paula | 10/2013 |
7281 | 2^359937-29323 | 108352 | Enzo Doria | 07/2015 |
7282 | 22^80712-23 | 108350 | Anonymous | 09/2024 |
7283 | (2^359922+1762)/2 | 108348 | Stefano Morozzi | 09/2020 |
7284 | (2^359903+758)/2 | 108342 | Stefano Morozzi | 09/2020 |
7285 | 2^359898-19997 | 108341 | Enzo Doria | 06/2013 |
7286 | (2^359886+2122)/2 | 108337 | Stefano Morozzi | 09/2020 |
7287 | 2*5^154948-7 | 108305 | Lelio R Paula | 11/2014 |
7288 | 10^108303+13 | 108304 | Lelio R Paula | 02/2017 |
7289 | 10^108303-207959 | 108303 | Patrick De Geest | 03/2017 |
7290 | 7*10^108300+69 | 108301 | Robert Price | 07/2019 |
7291 | 198^47155-47155^198 | 108300 | Norman Luhn | 09/2024 |
7292 | 2^359730+753 | 108290 | Lelio R Paula | 12/2011 |
7293 | (8*10^108273+13)/3 | 108274 | Robert Price | 06/2018 |
7294 | 2^359650-19797 | 108266 | Enzo Doria | 06/2013 |
7295 | (47*10^108261+259)/9 | 108262 | Robert Price | 03/2022 |
7296 | 91^55259-90^55259 | 108255 | Robert Price | 07/2015 |
7297 | 10^108251+85749 | 108252 | Patrick De Geest | 02/2021 |
7298 | 10^108251-50000000001 | 108251 | Robert Price | 12/2020 |
7299 | 10^108251-252081 | 108251 | Patrick De Geest | 02/2021 |
7300 | 4*10^108238-71 | 108239 | Robert Price | 12/2018 |
7301 | 3^226845+68 | 108233 | Lelio R Paula | 03/2017 |
7302 | 31836^2509+2509^31836 | 108227 | Hans Havermann | 10/2020 |
7303 | 5^154822-12866 | 108216 | Enzo Doria | 05/2014 |
7304 | (9*2^359480+1)/5 | 108215 | Lelio R Paula | 03/2014 |
7305 | 2^359454-177 | 108207 | Lelio R Paula | 12/2011 |
7306 | (19*10^108184+191)/3 | 108185 | Robert Price | 05/2019 |
7307 | 6^139022-59 | 108181 | Lelio R Paula | 02/2017 |
7308 | (19*10^108176-43)/3 | 108177 | Robert Price | 05/2019 |
7309 | 2^359352-179 | 108176 | Lelio R Paula | 12/2011 |
7310 | 2^359338-1858781 | 108172 | Enzo Doria | 04/2016 |
7311 | 12^100230-97 | 108167 | Lelio R Paula | 03/2017 |
7312 | 27028^10043+10043^27028 | 108163 | Hans Havermann | 11/2020 |
7313 | (25*2^359297-1)/137 | 108159 | Lelio R Paula | 03/2014 |
7314 | (2^359262-509203)/9 | 108148 | Lelio R Paula | 12/2013 |
7315 | 26583^11702+11702^26583 | 108147 | Hans Havermann | 11/2020 |
7316 | (3^226648-8)/352715633 | 108131 | Lelio R Paula | 12/2013 |
7317 | 6^138956-59 | 108129 | Lelio R Paula | 02/2017 |
7318 | 2^359184+30285 | 108126 | Lelio R Paula | 04/2017 |
7319 | 13^97059+10 | 108119 | Lelio R Paula | 03/2017 |
7320 | 2^359137-81 | 108112 | Lelio R Paula | 12/2011 |
7322 | (15185*2^359105-1)/567 | 108103 | Lelio R Paula | 05/2015 |
7321 | 2^359110-11655 | 108103 | Lelio R Paula | 04/2017 |
7323 | 2^359069-933075 | 108091 | Paul Bourdelais | 06/2021 |
7324 | 26227^13222+13222^26227 | 108090 | Hans Havermann | 10/2020 |
7325 | 8*6^138875+5 | 108067 | Lelio R Paula | 11/2014 |
7326 | 2^358987-216919 | 108066 | Enzo Doria | 05/2014 |
7327 | (2^358977+2714)/2 | 108063 | Stefano Morozzi | 09/2020 |
7328 | 39^67918-67918^39 | 108062 | Norbert Schneider | 02/2017 |
7329 | 24^78283-1*78283^24 | 108048 | Dylan Delgado | 12/2019 |
7330 | 25282^18767+18767^25282 | 108040 | Hans Havermann | 11/2020 |
7331 | 2^358785-16933 | 108006 | Enzo Doria | 05/2013 |
7332 | (2^358783+5)/33383 | 108000 | Lelio R Paula | 12/2013 |
7333 | 6^138770-77 | 107985 | Lelio R Paula | 02/2017 |
7334 | (2^358727-1017)/94178479 | 107980 | Lelio R Paula | 12/2013 |
7335 | 10^107976+19 | 107977 | Lelio R Paula | 02/2017 |
7336 | 10^107976-202103 | 107976 | Patrick De Geest | 03/2017 |
7337 | 2^358684+75781 | 107975 | Enzo Doria | 04/2016 |
7338 | 24928^21411+21411^24928 | 107955 | Hans Havermann | 10/2020 |
7339 | 99^54094+2 | 107952 | Lelio R Paula | 06/2013 |
7340 | (2^358583-9)/391 | 107942 | Lelio R Paula | 12/2013 |
7341 | (2^358562+838)/2 | 107938 | Stefano Morozzi | 02/2021 |
7342 | 2^358538+55 | 107931 | Lelio R Paula | 12/2011 |
7343 | 2^358517-109009 | 107925 | Enzo Doria | 03/2014 |
7344 | 26395^12258+12258^26395 | 107914 | Hans Havermann | 10/2020 |
7345 | 3^226159+104 | 107906 | Lelio R Paula | 03/2017 |
7346 | 7*10^107870+39 | 107871 | Robert Price | 01/2023 |
7347 | 2^358310-11655 | 107863 | Lelio R Paula | 03/2017 |
7348 | (68*10^107852-329)/9 | 107853 | Robert Price | 03/2021 |
7349 | 2^358275-112987 | 107852 | Enzo Doria | 07/2014 |
7350 | 3^226038+4 | 107848 | Tyler NeSmith | 01/2021 |
7351 | 2^358233-20005 | 107839 | Enzo Doria | 06/2013 |
7352 | 2^358194+175 | 107828 | Lelio R Paula | 12/2011 |
7353 | 107^53125-106 | 107812 | Norbert Schneider | 12/2016 |
7354 | (25*10^107804+161)/3 | 107805 | Robert Price | 09/2019 |
7355 | 3^225922-2 | 107793 | Lelio R Paula | 10/2013 |
7356 | 12^99874-95 | 107783 | Lelio R Paula | 03/2017 |
7357 | (2^358042+3042)/2 | 107782 | Stefano Morozzi | 09/2020 |
7358 | 2^358030-171 | 107778 | Lelio R Paula | 12/2011 |
7359 | 11^103488-4170 | 107772 | Enzo Doria | 12/2012 |
7360 | 2^357930-4547 | 107748 | Yves Minet | 03/2019 |
7361 | 2^357911-20089 | 107742 | Enzo Doria | 06/2013 |
7362 | 2^357885+12857 | 107735 | Yves Minet | 07/2010 |
7363 | (248587#+9289)/(7*1327*111512899) | 107733 | Makoto Morimoto | 06/2019 |
7364 | 7^127473+40 | 107728 | Lelio R Paula | 03/2017 |
7365 | 5*2^357850+9 | 107725 | Lelio R Paula | 09/2014 |
7366 | 2^357786-291191 | 107705 | Enzo Doria | 06/2015 |
7367 | 28907^5320+5320^28907 | 107705 | Hans Havermann | 10/2020 |
7368 | 18^85798-23 | 107700 | Lelio R Paula | 04/2017 |
7369 | 3*10^107697+61 | 107698 | Robert Price | 09/2020 |
7370 | (14*10^107694-83)/3 | 107695 | Robert Price | 12/2018 |
7371 | 2^357688-1854269 | 107675 | Enzo Doria | 09/2015 |
7372 | 2^357671-112225 | 107670 | Enzo Doria | 06/2014 |
7373 | 34323^1370+1370^34323 | 107662 | Hans Havermann | 11/2020 |
7374 | 2^357618-21005 | 107654 | Enzo Doria | 06/2013 |
7375 | 2^357615-17505 | 107653 | Enzo Doria | 05/2013 |
7376 | 26350^12171+12171^26350 | 107649 | Hans Havermann | 11/2020 |
7377 | (30^72871-1)/29 | 107638 | Robert Price | 12/2011 |
7378 | 25021^20026+20026^25021 | 107631 | Hans Havermann | 11/2020 |
7379 | 2^357521+1870217 | 107625 | Enzo Doria | 05/2016 |
7380 | 2^357477-1203 | 107612 | Lelio R Paula | 03/2017 |
7381 | 26008^13693+13693^26008 | 107583 | Hans Havermann | 10/2020 |
7382 | 25432^16991+16991^25432 | 107583 | Hans Havermann | 10/2020 |
7383 | (2^357371+3726)/2 | 107580 | Stefano Morozzi | 09/2020 |
7384 | 5^153894-18 | 107568 | Lelio R Paula | 02/2017 |
7385 | 2^357324-69059 | 107566 | Enzo Doria | 11/2015 |
7386 | 467^40296+40296^467 | 107563 | Norbert Schneider | 03/2019 |
7387 | 29446^4495+4495^29446 | 107559 | Hans Havermann | 10/2020 |
7388 | (8*10^107556-149)/3 | 107557 | Robert Price | 12/2021 |
7389 | 2^357283-869688105 | 107553 | Lelio R Paula | 01/2017 |
7390 | 2^357280-16805 | 107552 | Enzo Doria | 05/2013 |
7391 | 10^107547+481719 | 107548 | Patrick De Geest | 03/2017 |
7392 | 10^107547-47 | 107547 | Lelio R Paula | 02/2017 |
7393 | (23*10^107540+37)/3 | 107541 | Robert Price | 02/2022 |
7394 | 17^87396-32 | 107537 | Lelio R Paula | 04/2017 |
7395 | (746*10^107505+1)/9 | 107507 | Robert Price | 03/2020 |
7396 | (2^357125+1730)/2 | 107506 | Stefano Morozzi | 09/2020 |
7397 | 2^357081+107 | 107493 | Lelio R Paula | 12/2011 |
7398 | 2^357058-3911 | 107486 | Yves Minet | 04/2017 |
7399 | 3^225274+64 | 107484 | Lelio R Paula | 03/2017 |
7400 | 5*10^107482-57 | 107483 | Robert Price | 03/2021 |
7401 | 18^85619-95 | 107476 | Lelio R Paula | 04/2017 |
7402 | (2^357013+2526)/2 | 107472 | Stefano Morozzi | 09/2020 |
7403 | 3^225248-130 | 107471 | Lelio R Paula | 03/2017 |
7404 | (5*2^356961+1)/143 | 107455 | Lelio R Paula | 03/2014 |
7405 | 27029^9448+9448^27029 | 107450 | Hans Havermann | 11/2020 |
7406 | 2^356935-18085 | 107449 | Enzo Doria | 05/2013 |
7407 | 55396^87+87^55396 | 107442 | Hans Havermann | 10/2020 |
7408 | (14*10^107439-137)/3 | 107440 | Robert Price | 04/2021 |
7409 | 164^48505-48505^164 | 107432 | Norman Luhn | 09/2024 |
7410 | 10^107409+37 | 107410 | Lelio R Paula | 02/2017 |
7411 | 10^107409-329633 | 107409 | Patrick De Geest | 03/2017 |
7412 | (2^356799+3134)/2 | 107407 | Stefano Morozzi | 09/2020 |
7413 | 2^356787-109027 | 107404 | Enzo Doria | 04/2014 |
7414 | (2^356767+78557)/5 | 107397 | Lelio R Paula | 12/2013 |
7415 | 3*10^107389+77 | 107390 | Robert Price | 09/2018 |
7416 | (11*10^107385+61)/9 | 107386 | Tyler Busby | 01/2023 |
7417 | 5^153622-42 | 107378 | Lelio R Paula | 02/2017 |
7418 | 2^356699-97305 | 107378 | Lelio R Paula | 03/2017 |
7419 | 13129^26072-26072^13129 | 107371 | Norbert Schneider | 09/2022 |
7420 | 2^356654+85 | 107364 | Lelio R Paula | 11/2011 |
7421 | 2^356641-19921 | 107360 | Enzo Doria | 06/2013 |
7422 | 5*10^107355+11 | 107356 | Robert Price | 08/2019 |
7423 | 2^356623-465 | 107355 | Lelio R Paula | 03/2017 |
7424 | 19^83945-92 | 107345 | Lelio R Paula | 02/2017 |
7425 | 25491^16246+16246^25491 | 107337 | Hans Havermann | 11/2020 |
7426 | 10^107334+28149 | 107335 | Patrick De Geest | 03/2017 |
7427 | 10^107334-83 | 107334 | Lelio R Paula | 02/2017 |
7428 | 2^356518+885 | 107323 | Lelio R Paula | 03/2017 |
7429 | (59*10^107322+391)/9 | 107323 | Robert Price | 03/2021 |
7430 | (3*2^356506-1)/467 | 107317 | Lelio R Paula | 03/2014 |
7431 | (2^356468+1162)/2 | 107308 | Stefano Morozzi | 09/2020 |
7432 | 2^356442-114713 | 107300 | Enzo Doria | 04/2015 |
7433 | 2^356425+72257 | 107295 | Enzo Doria | 04/2016 |
7434 | 2^356351-112237 | 107273 | Enzo Doria | 06/2014 |
7436 | (6^137845-11)/5 | 107264 | Paul Bourdelais | 11/2011 |
7435 | 2^356322+74539 | 107264 | Enzo Doria | 04/2016 |
7437 | 2^356274+1874479 | 107250 | Enzo Doria | 06/2016 |
7438 | 24606^22819+22819^24606 | 107241 | Hans Havermann | 10/2020 |
7439 | (2^356243+3686)/2 | 107240 | Stefano Morozzi | 09/2020 |
7440 | 20^82422-93 | 107234 | Lelio R Paula | 02/2017 |
7441 | 2^356157-20365 | 107214 | Enzo Doria | 06/2013 |
7442 | 8*10^107209+99 | 107210 | Robert Price | 03/2021 |
7443 | 2^356129-18949 | 107206 | Enzo Doria | 05/2013 |
7444 | (15^91163-1)/13873078234414574 | 107200 | Paul Vanderveen | 06/2025 |
7445 | 2^356100-109587 | 107197 | Enzo Doria | 04/2014 |
7446 | 2*10^107173-37 | 107174 | Robert Price | 11/2021 |
7447 | (2^355978+3262)/2 | 107160 | Stefano Morozzi | 02/2021 |
7448 | (31*2^355970+1)/5 | 107159 | Lelio R Paula | 03/2014 |
7449 | 2^355968-64875 | 107158 | Lelio R Paula | 01/2017 |
7450 | 2^355952+1870321 | 107153 | Enzo Doria | 05/2016 |
7451 | 27069^9088+9088^27069 | 107152 | Hans Havermann | 11/2020 |
7452 | 2^355915-117 | 107142 | Lelio R Paula | 11/2011 |
7453 | (16*10^107140+77)/3 | 107141 | Robert Price | 03/2019 |
7454 | 24628^22369+22369^24628 | 107124 | Hans Havermann | 11/2020 |
7455 | (19*2^355838-1)/15 | 107119 | Lelio R Paula | 03/2014 |
7456 | 3^224495-8 | 107112 | Lelio R Paula | 12/2013 |
7457 | 3*10^107096+23 | 107097 | Robert Price | 03/2022 |
7458 | 25077^18632+18632^25077 | 107086 | Hans Havermann | 10/2020 |
7459 | 7^126691-44 | 107067 | Lelio R Paula | 03/2017 |
7460 | 4^177830-8243 | 107065 | Mike Oakes | 02/2011 |
7461 | (509*10^107057+13)/9 | 107059 | Robert Price | 05/2022 |
7462 | 2^355604+214051 | 107048 | Enzo Doria | 10/2023 |
7463 | (2^355562+1702)/2 | 107035 | Stefano Morozzi | 09/2020 |
7464 | 10^107028+3593037 | 107029 | Paul Bourdelais | 10/2021 |
7465 | 10^107028-43301 | 107028 | Alfred Reich | 08/2024 |
7466 | 2^355524-21237 | 107024 | Enzo Doria | 06/2013 |
7468 | (10223*2^355515+1)/4855 | 107022 | Lelio R Paula | 07/2014 |
7467 | 24843^20320+20320^24843 | 107022 | Hans Havermann | 10/2020 |
7469 | 2^355494-19901 | 107015 | Enzo Doria | 06/2013 |
7470 | 2^355454-869688105 | 107003 | Lelio R Paula | 01/2017 |
7471 | 4*10^106997+13 | 106998 | Robert Price | 02/2021 |
7472 | (8*10^106985-629)/9 | 106985 | Robert Price | 01/2022 |
7473 | 2^355366+54615 | 106976 | Lelio R Paula | 01/2017 |
7474 | (31*2^355346-1)/1563 | 106969 | Lelio R Paula | 03/2014 |
7475 | 2^355275-19861 | 106949 | Enzo Doria | 06/2013 |
7476 | 27503^7738+7738^27503 | 106949 | Hans Havermann | 11/2020 |
7477 | (85*10^106943+77)/9 | 106944 | Robert Price | 07/2022 |
7478 | (9179*2^355234-1)/1915 | 106937 | Lelio R Paula | 05/2015 |
7479 | 4*10^106914-93 | 106915 | Robert Price | 07/2020 |
7480 | (151^49069-1)/90910031243271150 | 106904 | Paul Vanderveen | 06/2025 |
7481 | 4*10^106892-3 | 106893 | Robert Price | 10/2015 |
7482 | (66^58741+1)/67 | 106881 | Paul Bourdelais | 08/2017 |
7483 | 2^355001+335931 | 106866 | Enzo Doria | 08/2024 |
7484 | 2^354946-3 | 106850 | M. Frind & P. Underwood | 08/2002 |
7485 | (2^354945+3134)/2 | 106849 | Stefano Morozzi | 09/2020 |
7486 | (46^64384+1)/(46^128+1) | 106843 | Martin Hopf | 12/2023 |
7487 | 2^354873-18693 | 106828 | Enzo Doria | 05/2013 |
7488 | (2^354831+3186)/2 | 106815 | Stefano Morozzi | 09/2020 |
7490 | 2^354828+4257 | 106814 | Lelio R Paula | 01/2017 |
7489 | 25216^17217+17217^25216 | 106814 | Hans Havermann | 10/2020 |
7491 | (2^354799+1218)/2 | 106805 | Stefano Morozzi | 09/2020 |
7492 | 436^40459+40459^436 | 106791 | Norbert Schneider | 03/2019 |
7493 | (21^80761+1)/22 | 106783 | Paul Bourdelais | 03/2010 |
7494 | (115*10^106772-7)/9 | 106774 | Robert Price | 02/2020 |
7495 | 2^354640-354640+1 | 106758 | Robert Price | 10/2014 |
7496 | (2^354603+2606)/2 | 106746 | Stefano Morozzi | 09/2020 |
7497 | 2^354585+95 | 106741 | Lelio R Paula | 10/2011 |
7498 | (206*10^106737+1)/9 | 106739 | Robert Price | 04/2022 |
7499 | 2^354553+189 | 106732 | Lelio R Paula | 10/2011 |
7500 | (8*10^106729-71)/9 | 106729 | Serge Batalov | 11/2022 |