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 ? |
2251 | (10^139398+191)/3 | 139398 | Robert Price | 02/2018 |
2252 | 2^463019+79463 | 139383 | Enzo Doria | 06/2016 |
2253 | 3^292129-112756 | 139381 | Enzo Doria | 07/2014 |
2254 | (15*2^462951+1)/527 | 139361 | Lelio R Paula | 08/2015 |
2255 | 2^462855-1865707 | 139334 | Enzo Doria | 02/2016 |
2256 | (86*10^139315+121)/9 | 139316 | Robert Price | 08/2019 |
2257 | 2^462786-292043 | 139313 | Enzo Doria | 08/2015 |
2258 | (26*10^139289+1)/9 | 139290 | Serge Batalov | 05/2010 |
2259 | (17*10^139239+67)/3 | 139240 | Robert Price | 01/2019 |
2260 | 2^462238-935 | 139148 | Yves Minet | 02/2014 |
2261 | 2^461228+1870441 | 138844 | Enzo Doria | 05/2016 |
2262 | 2^461204-113675 | 138837 | Enzo Doria | 08/2014 |
2263 | (2^461139-7)/47 | 138815 | Lelio R Paula | 08/2015 |
2264 | (17*10^138812-521)/9 | 138813 | Robert Price | 04/2018 |
2265 | 3*10^138811-11 | 138812 | Robert Price | 07/2018 |
2266 | (28*10^138768+173)/3 | 138769 | Robert Price | 12/2019 |
2267 | (17*2^460963+1)/124051613 | 138757 | Lelio R Paula | 08/2015 |
2268 | (379*10^138737-1)/9 | 138739 | Robert Price | 02/2020 |
2269 | (43*10^138648+101)/9 | 138649 | Robert Price | 11/2018 |
2270 | 2^460567-113541 | 138645 | Enzo Doria | 08/2014 |
2271 | 7*10^138604+3 | 138605 | Robert Price | 08/2015 |
2272 | 2^460330+69 | 138574 | Lelio R Paula | 09/2017 |
2273 | (17*10^138569-77)/3 | 138570 | Robert Price | 08/2019 |
2274 | 2^460078-69581 | 138498 | Enzo Doria | 12/2015 |
2275 | 2^459768+1531 | 138404 | Lelio R Paula | 06/2012 |
2276 | (266*10^138328-17)/3 | 138330 | Robert Price | 07/2020 |
2277 | 2^459343-113611 | 138277 | Enzo Doria | 08/2014 |
2278 | (28*10^138240+113)/3 | 138241 | Robert Price | 03/2020 |
2279 | 2^459199+1874393 | 138233 | Enzo Doria | 06/2016 |
2280 | 2^459063-345307 | 138192 | Enzo Doria | 06/2015 |
2281 | 2*10^138162-43 | 138163 | Robert Price | 04/2018 |
2282 | 2^458927-112747 | 138151 | Enzo Doria | 06/2014 |
2283 | 2^458927-69217 | 138151 | Enzo Doria | 11/2015 |
2284 | 555555*2^458524+1513 | 138036 | Darren Bedwell | 08/2009 |
2285 | 2^458465+2^32+1 | 138012 | T. D. Noe | 06/2007 |
2286 | 2^458321-113053 | 137969 | Enzo Doria | 07/2014 |
2287 | 2^458298+79459 | 137962 | Enzo Doria | 06/2016 |
2288 | 2^458248-21815 | 137947 | Enzo Doria | 07/2013 |
2289 | 2^458194+1876549 | 137931 | Enzo Doria | 07/2016 |
2290 | (318259#+325270)/(2^2*5*11*2957) | 137907 | Makoto Morimoto | 10/2017 |
2291 | (318259#+2625)/(3^2*5*7^5*22965193) | 137899 | Makoto Morimoto | 06/2019 |
2292 | 2^457937+57 | 137853 | Lelio R Paula | 09/2017 |
2293 | (8*10^137845+13)/3 | 137846 | Robert Price | 06/2018 |
2294 | 2^457822+105 | 137819 | Donovan Johnson | 06/2006 |
2295 | 2^457754-113711 | 137798 | Enzo Doria | 09/2014 |
2296 | 11^132315-360 | 137792 | Enzo Doria | 09/2012 |
2297 | 2^457347-37 | 137676 | Lelio R Paula | 09/2017 |
2298 | 2^457322-2786393 | 137668 | Enzo Doria | 10/2015 |
2299 | gcd(Phi(699340,10),(10^34967+1)^3*(10^34967+1+10^17484)-(10^69934+10^34967)*(10^17484-1)+10^104901) | 137664 | Serge Batalov | 04/2015 |
2300 | 2^457240-59939 | 137643 | Enzo Doria | 10/2015 |
2301 | (856*10^137633-1)/9 | 137635 | Robert Price | 03/2020 |
2302 | 2^457120-21575 | 137607 | Enzo Doria | 07/2013 |
2303 | (11*2^456978+1)/15 | 137564 | Lelio R Paula | 07/2015 |
2304 | (7*25^98399+10613)/12 | 137556 | Curtis Bright | 05/2015 |
2305 | 419^52446+52446^419 | 137525 | Norbert Schneider | 11/2020 |
2306 | 2^456796-113513 | 137510 | Enzo Doria | 08/2014 |
2307 | (7*2^456655+1)/39 | 137467 | Lelio R Paula | 07/2015 |
2308 | 2^456503-114361 | 137422 | Enzo Doria | 01/2015 |
2309 | (11*2^456387+1)/19 | 137386 | Lelio R Paula | 07/2015 |
2310 | (71*10^137250+1)/9 | 137251 | Serge Batalov | 05/2010 |
2311 | 2^455922+76249 | 137247 | Enzo Doria | 05/2016 |
2312 | 14^119741-119741^14 | 137239 | Serge Batalov | 04/2016 |
2313 | (19*2^455760+1)/3455 | 137196 | Lelio R Paula | 07/2015 |
2315 | 124^(2^16)+57^(2^16) | 137195 | Valeryi Kuryshev | 02/2011 |
2314 | 124^65536+57^65536 | 137195 | Serge Batalov | 03/2018 |
2316 | 2^455689+72677 | 137177 | Enzo Doria | 04/2016 |
2317 | (17948!)^2+Prime(17948) | 137118 | Robert Price | 08/2014 |
2318 | (37^87421-1)/36 | 137093 | Robert Price | 03/2012 |
2319 | 2^455323-57 | 137066 | Lelio R Paula | 09/2017 |
2320 | (73*17^111333-9)/16 | 136991 | Curtis Bright | 05/2015 |
2321 | 2^455025-112723 | 136977 | Enzo Doria | 06/2014 |
2322 | 66*10^136894+7 | 136896 | Robert Price | 07/2020 |
2323 | 2^454675-1777 | 136871 | Yves Minet | 10/2011 |
2324 | 3*10^136829+29 | 136830 | Robert Price | 05/2018 |
2325 | 4*10^136813-57 | 136814 | Robert Price | 10/2018 |
2326 | (6092*28^94536-143)/9 | 136812 | Curtis Bright | 05/2015 |
2327 | 2^454348-53 | 136773 | Lelio R Paula | 09/2017 |
2328 | 31^91680-2 | 136729 | Lelio R Paula | 05/2017 |
2329 | 2^454199-113697 | 136728 | Enzo Doria | 09/2014 |
2330 | 7*10^136716+69 | 136717 | Robert Price | 07/2019 |
2331 | 6*10^136692+53 | 136693 | Robert Price | 05/2020 |
2332 | (13*10^136687+47)/3 | 136688 | Robert Price | 10/2018 |
2333 | 2^453935-2791357 | 136649 | Enzo Doria | 01/2016 |
2334 | 2^453906-69131 | 136640 | Enzo Doria | 11/2015 |
2335 | 2^453791+1873913 | 136605 | Enzo Doria | 06/2016 |
2336 | (21*2^453740-1)/2735 | 136588 | Lelio R Paula | 07/2015 |
2337 | 580^49413+49413^580 | 136550 | Norbert Schneider | 02/2019 |
2338 | 2^453503-2788297 | 136519 | Enzo Doria | 11/2015 |
2339 | 34989^7958+7958^34989 | 136486 | Hans Havermann | 12/2020 |
2340 | 31007^25158+25158^31007 | 136452 | Hans Havermann | 11/2020 |
2341 | 2^453267-113827 | 136447 | Enzo Doria | 09/2014 |
2342 | 32234^17085+17085^32234 | 136435 | Hans Havermann | 12/2020 |
2343 | 2^453102+1877629 | 136398 | Enzo Doria | 08/2016 |
2344 | 31016^24981+24981^31016 | 136397 | Hans Havermann | 12/2020 |
2345 | (62*10^136384-791)/9 | 136385 | Robert Price | 06/2019 |
2346 | 34171^9798+9798^34171 | 136382 | Hans Havermann | 12/2020 |
2347 | 32756^14541+14541^32756 | 136350 | Hans Havermann | 11/2020 |
2348 | 32205^17086+17086^32205 | 136313 | Hans Havermann | 12/2020 |
2349 | (25*10^136303+161)/3 | 136304 | Robert Price | 09/2019 |
2350 | 35337^7196+7196^35337 | 136299 | Hans Havermann | 12/2020 |
2351 | (182*10^136291+1)/3 | 136293 | Robert Price | 02/2020 |
2352 | 23*10^136225-7 | 136227 | Robert Price | 02/2020 |
2353 | L(651821) | 136223 | Henri Lifchitz | 09/2008 |
2354 | 2^452460+77701 | 136205 | Enzo Doria | 05/2016 |
2355 | 2^452440-114755 | 136199 | Enzo Doria | 04/2015 |
2356 | 32286^16489+16489^32286 | 136157 | Hans Havermann | 11/2020 |
2357 | (47*10^136141+493)/9 | 136142 | Robert Price | 03/2019 |
2358 | 40392^2339+2339^40392 | 136082 | Hans Havermann | 12/2020 |
2359 | 2*10^136057-43 | 136058 | Robert Price | 04/2018 |
2360 | (2^451956-5)/181 | 136051 | Lelio R Paula | 07/2015 |
2361 | 2^451947-22165 | 136050 | Enzo Doria | 07/2013 |
2362 | 2^451787-21805 | 136002 | Enzo Doria | 07/2013 |
2363 | 2^451734-112817 | 135986 | Enzo Doria | 06/2014 |
2364 | (11*2^451714+1)/75 | 135979 | Lelio R Paula | 07/2015 |
2365 | 2^451705-21879 | 135977 | Enzo Doria | 07/2013 |
2366 | 2^451560-112349 | 135934 | Enzo Doria | 06/2014 |
2367 | 2^451530+45 | 135925 | Lelio R Paula | 08/2017 |
2368 | 31622^19875+19875^31622 | 135922 | Hans Havermann | 11/2020 |
2369 | (25*10^135906-73)/3 | 135907 | Robert Price | 09/2019 |
2370 | 2^451450-21743 | 135900 | Enzo Doria | 07/2013 |
2371 | 35218^7199+7199^35218 | 135846 | Hans Havermann | 11/2020 |
2372 | 35661^6446+6446^35661 | 135844 | Hans Havermann | 11/2020 |
2373 | 31869^18248+18248^31869 | 135801 | Hans Havermann | 12/2020 |
2374 | 2^450805+65 | 135706 | Lelio R Paula | 08/2017 |
2375 | 30761^25728+25728^30761 | 135669 | Hans Havermann | 12/2020 |
2376 | 2*10^135663+69 | 135664 | Robert Price | 06/2020 |
2377 | 33962^9867+9867^33962 | 135651 | Hans Havermann | 11/2020 |
2378 | 34474^8605+8605^34474 | 135647 | Hans Havermann | 11/2020 |
2379 | 34501^8508+8508^34501 | 135583 | Hans Havermann | 12/2020 |
2380 | 35487^6614+6614^35487 | 135577 | Hans Havermann | 12/2020 |
2381 | (13725^32768+1)/131074 | 135573 | Lelio R Paula | 03/2012 |
2382 | (313007#+121876)/(2*30469*424867) | 135559 | Makoto Morimoto | 08/2020 |
2383 | (5^193939+1)/6 | 135557 | Paul Bourdelais | 12/2008 |
2384 | 40387^2260+2260^40387 | 135463 | Hans Havermann | 12/2020 |
2385 | 2^449944-113247 | 135447 | Enzo Doria | 07/2014 |
2386 | 2^449770-113247 | 135395 | Enzo Doria | 07/2014 |
2387 | 2^449637-21471 | 135355 | Enzo Doria | 07/2013 |
2388 | 2^449571-75 | 135335 | Lelio R Paula | 08/2017 |
2389 | (14*10^135301-53)/3 | 135302 | Robert Price | 12/2018 |
2390 | (3*2^449494+1)/61340334013 | 135301 | Lelio R Paula | 07/2015 |
2391 | (13*10^135279+437)/9 | 135280 | Robert Price | 04/2018 |
2392 | 2^449330-35 | 135262 | Lelio R Paula | 08/2017 |
2393 | (19*2^449325+1)/9 | 135261 | Lelio R Paula | 07/2015 |
2394 | (13*10^135184-43)/3 | 135185 | Robert Price | 11/2018 |
2395 | 2^449049-115309 | 135178 | Enzo Doria | 07/2015 |
2396 | (10^135174+191)/3 | 135174 | Robert Price | 02/2018 |
2397 | 2^449030-15 | 135172 | Lelio R Paula | 08/2017 |
2398 | (27*2^448936-1)/11 | 135144 | Lelio R Paula | 07/2015 |
2399 | 32221^15586+15586^32221 | 135095 | Hans Havermann | 12/2020 |
2400 | 2^448762-351401 | 135091 | Enzo Doria | 07/2015 |
2401 | 2*10^135083-43 | 135084 | Robert Price | 04/2018 |
2402 | 31357^20278+20278^31357 | 135056 | Hans Havermann | 12/2020 |
2403 | 32242^15445+15445^32242 | 135055 | Hans Havermann | 12/2020 |
2404 | k(198476) | 135054 | Robert Price | 07/2020 |
2405 | 32461^14470+14470^32461 | 135053 | Hans Havermann | 12/2020 |
2406 | 31353^20288+20288^31353 | 135045 | Hans Havermann | 12/2020 |
2407 | 5*10^135019+77 | 135020 | Robert Price | 12/2018 |
2408 | (10^135011+719)/9 | 135011 | Serge Batalov | 12/2014 |
2409 | (197*10^134978+7)/3 | 134980 | Robert Price | 12/2020 |
2410 | 2^448386-2786261 | 134978 | Enzo Doria | 03/2016 |
2411 | (73*10^134939+143)/9 | 134940 | Robert Price | 07/2019 |
2412 | 33957^9392+9392^33957 | 134903 | Hans Havermann | 12/2020 |
2413 | 2^448113-113949 | 134896 | Enzo Doria | 11/2014 |
2414 | 30958^22755+22755^30958 | 134887 | Hans Havermann | 12/2020 |
2415 | 2^447888-2786639 | 134828 | Enzo Doria | 03/2016 |
2416 | 2^447870-396765 | 134823 | Enzo Doria | 07/2013 |
2417 | 2^447766-21791 | 134791 | Enzo Doria | 07/2013 |
2418 | (19*10^134786-49)/3 | 134787 | Robert Price | 05/2019 |
2419 | Phi(202173,10)/10064306514895399941409 | 134758 | Makoto Kamada | 05/2019 |
2420 | (23*2^447647+1)/155 | 134755 | Lelio R Paula | 07/2015 |
2421 | 2^447552-1019 | 134727 | Yves Minet | 02/2014 |
2422 | (2*10^134720-41)/3 | 134720 | Robert Price | 02/2018 |
2423 | 2^447437-113569 | 134692 | Enzo Doria | 08/2014 |
2424 | 2^447426-65 | 134689 | Lelio R Paula | 08/2017 |
2425 | 2^447357-2797573 | 134668 | Enzo Doria | 04/2016 |
2426 | 2^447278+93 | 134645 | Lelio R Paula | 08/2017 |
2427 | 33193^11370+11370^33193 | 134623 | Hans Havermann | 12/2020 |
2428 | 30736^23979+23979^30736 | 134619 | Hans Havermann | 11/2020 |
2429 | 2^447094-113193 | 134589 | Enzo Doria | 07/2014 |
2430 | 3^282041-9934 | 134568 | Enzo Doria | 05/2014 |
2431 | 2^447019-109039 | 134567 | Enzo Doria | 04/2014 |
2432 | 31376^19437+19437^31376 | 134561 | Hans Havermann | 12/2020 |
2433 | 2^446990-22041 | 134558 | Enzo Doria | 07/2013 |
2434 | 123^64370+122 | 134528 | Karsten Bonath | 04/2018 |
2435 | 30100^29461+29461^30100 | 134525 | Hans Havermann | 11/2020 |
2436 | 31251^20158+20158^31251 | 134519 | Hans Havermann | 11/2020 |
2437 | 2^446782-39701 | 134495 | Enzo Doria | 09/2015 |
2438 | 31817^16858+16858^31817 | 134485 | Hans Havermann | 12/2020 |
2439 | 32448^13937+13937^32448 | 134471 | Hans Havermann | 12/2020 |
2440 | 2^446494-16485 | 134409 | Enzo Doria | 04/2013 |
2441 | 30042^29785+29785^30042 | 134408 | Hans Havermann | 12/2020 |
2442 | 29504^30069-30069^29504 | 134405 | Norbert Schneider | 12/2014 |
2443 | 2^446441+65 | 134393 | Lelio R Paula | 08/2017 |
2444 | (10^134385+719)/9 | 134385 | Serge Batalov | 12/2014 |
2445 | 30043^29688+29688^30043 | 134370 | Hans Havermann | 12/2020 |
2446 | (2*10^134364+529)/9 | 134364 | Robert Price | 01/2018 |
2447 | 34657^7512+7512^34657 | 134323 | Hans Havermann | 11/2020 |
2448 | (13*2^446187-1)/661 | 134314 | Lelio R Paula | 07/2015 |
2449 | 5^192103+3^192103+2^192103+1 | 134275 | Matthias Baur | 12/2019 |
2450 | 42035^1564+1564^42035 | 134270 | Hans Havermann | 12/2020 |
2452 | (2^446022-7)/339 | 134264 | Lelio R Paula | 07/2015 |
2451 | 30621^24248+24248^30621 | 134264 | Hans Havermann | 12/2020 |
2453 | 30341^26562+26562^30341 | 134237 | Hans Havermann | 11/2020 |
2454 | 29976^29953+29953^29976 | 134186 | Hans Havermann | 12/2020 |
2455 | 35038^6755+6755^35038 | 134183 | Hans Havermann | 12/2020 |
2456 | 63^74568+2 | 134174 | Lelio R Paula | 06/2017 |
2457 | 2^445671-22071 | 134161 | Enzo Doria | 07/2013 |
2458 | 532^49205+49205^532 | 134129 | Norbert Schneider | 02/2019 |
2459 | (28*10^134122-1)/9 | 134123 | Serge Batalov | 10/2014 |
2460 | 2^445515-113859 | 134114 | Enzo Doria | 10/2014 |
2461 | (22*10^134111-37)/3 | 134112 | Robert Price | 04/2019 |
2462 | 2^445328-216917 | 134058 | Enzo Doria | 05/2014 |
2463 | 2^445202+1874899 | 134020 | Enzo Doria | 06/2016 |
2464 | (13*2^445102+1)/17 | 133989 | Lelio R Paula | 07/2015 |
2465 | (35*10^133979-593)/9 | 133980 | Robert Price | 07/2018 |
2466 | 2*10^133955+51 | 133956 | Robert Price | 05/2018 |
2467 | 2^444964-113639 | 133948 | Enzo Doria | 08/2014 |
2468 | 28118^30097-30097^28118 | 133902 | Norbert Schneider | 01/2015 |
2469 | 10^133902-100001 | 133902 | Robert Price | 07/2020 |
2470 | 2^444784-113859 | 133894 | Enzo Doria | 10/2014 |
2471 | (266*10^133603+1)/3 | 133605 | Robert Price | 04/2020 |
2472 | (89*10^133580+1)/9 | 133581 | Serge Batalov | 05/2010 |
2473 | 62^74521-61^74521 | 133571 | Robert Price | 05/2013 |
2474 | (182*10^133543+1)/3 | 133545 | Robert Price | 02/2020 |
2475 | (103*10^133535+11)/3 | 133537 | Robert Price | 01/2021 |
2476 | 2^443198-1867301 | 133416 | Enzo Doria | 03/2016 |
2477 | (19*10^133390+101)/3 | 133391 | Robert Price | 09/2019 |
2478 | (71^72043-1)/70 | 133369 | Paul Bourdelais | 09/2017 |
2479 | 2^442799+1874633 | 133296 | Enzo Doria | 06/2016 |
2480 | L(637751) | 133283 | Henri Lifchitz | 09/2008 |
2481 | 109^65410-2 | 133269 | Lelio R Paula | 10/2017 |
2482 | 2^442688-113489 | 133263 | Enzo Doria | 08/2014 |
2483 | (7*2^442640-1)/783 | 133246 | Lelio R Paula | 07/2015 |
2484 | (9*2^442463+1)/17441 | 133192 | Lelio R Paula | 07/2015 |
2485 | 2^442437-1849993 | 133187 | Enzo Doria | 08/2015 |
2486 | 2^442430-2791721 | 133185 | Enzo Doria | 03/2016 |
2487 | (13*2^442428-1)/309 | 133183 | Lelio R Paula | 07/2015 |
2488 | (25*2^442379-1)/71 | 133169 | Lelio R Paula | 07/2015 |
2489 | 452^50145+50145^452 | 133142 | Norbert Schneider | 09/2020 |
2490 | (2^441799+78557)/5 | 132995 | Lelio R Paula | 07/2015 |
2491 | 2^441781-2787883 | 132990 | Enzo Doria | 03/2016 |
2492 | (3*2^441685+1)/749 | 132959 | Lelio R Paula | 07/2015 |
2493 | 2^441618+37 | 132941 | Lelio R Paula | 08/2017 |
2494 | 2^441521-113575 | 132912 | Enzo Doria | 08/2014 |
2495 | 7*10^132884-99 | 132885 | Robert Price | 06/2019 |
2496 | (103*10^132879+17)/3 | 132881 | Robert Price | 08/2020 |
2497 | 2^441396-65 | 132874 | Lelio R Paula | 08/2017 |
2498 | 2^441330+1871959 | 132854 | Enzo Doria | 05/2016 |
2499 | 2^441063+383 | 132774 | Lelio R Paula | 06/2012 |
2500 | (22*10^132746+53)/3 | 132747 | Robert Price | 07/2019 |