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 | (509*10^162422+31)/9 | 162424 | Robert Price | 08/2022 |
2252 | (91*10^162410+11)/3 | 162412 | Robert Price | 02/2020 |
2253 | 2^539346+81049 | 162360 | Enzo Doria | 07/2016 |
2254 | 2^539220-115499 | 162322 | Enzo Doria | 07/2015 |
2255 | 2^539217-113713 | 162321 | Enzo Doria | 09/2014 |
2256 | 9^170099-8^170099 | 162316 | Jean-Louis Charton | 05/2005 |
2257 | 4*10^162263-39 | 162264 | Robert Price | 09/2020 |
2258 | (115*10^162153-1)/3 | 162155 | Serge Batalov | 12/2014 |
2259 | (10^162138+11)/3 | 162138 | Serge Batalov | 07/2015 |
2260 | (61*10^162100-403)/9 | 162101 | Robert Price | 10/2021 |
2261 | 2^538473-112723 | 162097 | Enzo Doria | 06/2014 |
2262 | 2^538157+79337 | 162002 | Enzo Doria | 06/2016 |
2263 | 2^538136-114749 | 161996 | Enzo Doria | 05/2015 |
2264 | 2^538104+1531 | 161986 | Lelio R Paula | 01/2013 |
2265 | (9*29^110727-1199)/14 | 161927 | Curtis Bright | 05/2015 |
2266 | ((2^179294-1)^3+2)/11 | 161918 | Serge Batalov | 03/2023 |
2267 | 2^537815-22737 | 161899 | Enzo Doria | 07/2013 |
2268 | 2^537804-39839 | 161896 | Enzo Doria | 09/2015 |
2269 | (754*10^161816-7)/9 | 161818 | Robert Price | 05/2020 |
2270 | 2^537458-59201 | 161791 | Enzo Doria | 10/2015 |
2271 | 2*10^161766+51 | 161767 | Robert Price | 05/2018 |
2272 | (7*10^161760+131)/3 | 161761 | Robert Price | 06/2020 |
2273 | 2^537209-49849 | 161717 | Enzo Doria | 10/2015 |
2274 | (58*10^161691+329)/9 | 161692 | Robert Price | 05/2019 |
2275 | 23*10^161668-3 | 161670 | Robert Price | 10/2021 |
2276 | 2^536937+1879397 | 161635 | Enzo Doria | 09/2016 |
2277 | 34*10^161593-9 | 161595 | Robert Price | 06/2020 |
2278 | (31*10^161582-1)/3 | 161584 | Robert Price | 02/2020 |
2279 | 2^536569-22881 | 161524 | Enzo Doria | 07/2013 |
2280 | (46*10^161505+593)/9 | 161506 | Robert Price | 04/2022 |
2281 | 10^161253+573493 | 161254 | Patrick De Geest | 07/2018 |
2282 | 10^161253-8001 | 161253 | Robert Price | 03/2018 |
2283 | 97*10^161247-3 | 161249 | Robert Price | 05/2020 |
2284 | 7*10^161192-9 | 161193 | Robert Price | 10/2015 |
2285 | 2^535415-19 | 161176 | Stefano Morozzi | 07/2022 |
2286 | 2^535269-113803 | 161133 | Enzo Doria | 09/2014 |
2287 | 2^535205-23295 | 161113 | Enzo Doria | 08/2013 |
2288 | (49*10^161105+437)/9 | 161106 | Robert Price | 09/2021 |
2289 | (89^82609+1)/90 | 161036 | Paul Bourdelais | 10/2017 |
2290 | 2^534429-19 | 160880 | Stefano Morozzi | 07/2022 |
2291 | 2^534359-23229 | 160859 | Enzo Doria | 08/2013 |
2292 | 10^160836-1001 | 160836 | Robert Price | 10/2022 |
2293 | 457^60454+60454^457 | 160803 | Norbert Schneider | 07/2021 |
2294 | 8*10^160782+53 | 160783 | Robert Price | 08/2020 |
2295 | 67^88037-66^88037 | 160763 | Robert Price | 10/2013 |
2296 | (88*10^160761-637)/9 | 160762 | Robert Price | 11/2020 |
2297 | 2^533948-1852559 | 160735 | Enzo Doria | 08/2015 |
2298 | 2^533665-23335 | 160650 | Enzo Doria | 08/2013 |
2299 | (19*10^160619+77)/3 | 160620 | Robert Price | 05/2019 |
2300 | 2^533350-5 | 160555 | Stefano Morozzi | 07/2022 |
2301 | (38*10^160508+151)/9 | 160509 | Robert Price | 03/2022 |
2302 | 3^336353-2^336353 | 160482 | Mike Oakes | 10/2007 |
2303 | 2*10^160339-63 | 160340 | Robert Price | 01/2022 |
2304 | (91*10^160249+11)/3 | 160251 | Robert Price | 02/2020 |
2305 | (772*10^160239-7)/9 | 160241 | Robert Price | 06/2022 |
2306 | (2*10^160226+187)/9 | 160226 | Robert Price | 06/2022 |
2307 | 2^532253+1868417 | 160225 | Enzo Doria | 04/2016 |
2308 | 2^532108-291989 | 160181 | Enzo Doria | 07/2015 |
2309 | (28*10^160171-79)/3 | 160172 | Robert Price | 02/2021 |
2310 | (26*10^160166-131)/3 | 160167 | Robert Price | 10/2019 |
2311 | 55*10^160145+9 | 160147 | Robert Price | 03/2022 |
2312 | (17*10^160143+67)/3 | 160144 | Robert Price | 01/2019 |
2313 | (394^61637-1)/393 | 159977 | Jinyuan Wang | 04/2023 |
2314 | (19*10^159921-49)/3 | 159922 | Robert Price | 05/2019 |
2315 | 2^531110-1853771 | 159881 | Enzo Doria | 09/2015 |
2316 | (275^65536+53^65536)/2 | 159864 | Kellen Shenton | 11/2020 |
2317 | 2^530992-1845299 | 159845 | Enzo Doria | 07/2015 |
2318 | 2^530810+25 | 159790 | Stefano Morozzi | 04/2021 |
2319 | (136*10^159736-1)/9 | 159738 | Serge Batalov | 12/2014 |
2320 | 2^530417-113833 | 159672 | Enzo Doria | 09/2014 |
2321 | 2^530406-69701 | 159669 | Enzo Doria | 02/2016 |
2322 | 5*10^159656+41 | 159657 | Robert Price | 03/2019 |
2323 | 8*10^159614+29 | 159615 | Robert Price | 11/2021 |
2324 | 7^188858-188858^7 | 159604 | Norbert Schneider | 05/2018 |
2325 | 7*10^159553-27 | 159554 | Robert Price | 11/2022 |
2326 | (24^115597-1)/23 | 159547 | Paul Bourdelais | 09/2017 |
2327 | (17*10^159438-377)/9 | 159439 | Robert Price | 03/2018 |
2328 | (203*10^159381+1)/3 | 159383 | Robert Price | 03/2020 |
2329 | 89*10^159258-9 | 159260 | Robert Price | 05/2020 |
2330 | 2^528944-461315 | 159229 | Enzo Doria | 07/2015 |
2331 | (13*10^159147+161)/3 | 159148 | Robert Price | 11/2018 |
2332 | 2^528620-113879 | 159131 | Enzo Doria | 10/2014 |
2333 | (47*10^159083+259)/9 | 159084 | Robert Price | 03/2022 |
2334 | 2^528000-23307 | 158944 | Enzo Doria | 08/2013 |
2335 | (29*10^158909+61)/9 | 158910 | Robert Price | 08/2018 |
2336 | 2^527817+1876817 | 158889 | Enzo Doria | 07/2016 |
2337 | 2^527774-49451 | 158876 | Enzo Doria | 10/2015 |
2338 | 2^527745-291553 | 158868 | Enzo Doria | 07/2015 |
2339 | (91*10^158862+11)/3 | 158864 | Robert Price | 02/2020 |
2340 | 2^527644-1856999 | 158837 | Enzo Doria | 10/2015 |
2341 | (37200^37199-37198^37199)/(2^37199) | 158822 | Henri Lifchitz | 06/2013 |
2342 | 2^527573-1863253 | 158816 | Enzo Doria | 12/2015 |
2343 | 2^527553-23401 | 158810 | Enzo Doria | 08/2013 |
2344 | (49*10^158783+437)/9 | 158784 | Robert Price | 09/2021 |
2345 | 10^158755+310207 | 158756 | Patrick De Geest | 08/2021 |
2346 | 10^158755-50001 | 158755 | Robert Price | 08/2020 |
2347 | (61*10^158748-619)/9 | 158749 | Robert Price | 03/2022 |
2348 | 11^152420-210 | 158730 | Enzo Doria | 09/2012 |
2349 | (13*10^158694+401)/9 | 158695 | Robert Price | 01/2018 |
2350 | 2^527052-69059 | 158659 | Enzo Doria | 11/2015 |
2351 | 2^526849-113233 | 158598 | Enzo Doria | 07/2014 |
2352 | (2^526763-lucasV(1,2,526763)+1)/2 | 158572 | Mike Oakes | 11/2016 |
2353 | (299*10^158544-17)/3 | 158546 | Robert Price | 04/2020 |
2354 | 2^526658-3551 | 158540 | Yves Minet | 03/2017 |
2355 | 7*10^158465-11 | 158466 | Robert Price | 06/2019 |
2356 | 2^526375-23115 | 158455 | Enzo Doria | 08/2013 |
2357 | (77*10^158407-293)/9 | 158408 | Robert Price | 09/2019 |
2358 | 2^526028-112793 | 158351 | Enzo Doria | 07/2014 |
2359 | (229*10^158339-1)/3 | 158341 | Robert Price | 07/2020 |
2360 | 2^525690-9221 | 158249 | Lelio R Paula | 01/2013 |
2361 | 2*10^158242-7 | 158243 | Robert Price | 10/2015 |
2362 | 2^525667-525667^2 | 158242 | Henri Lifchitz | 12/2017 |
2363 | (56*10^158230+187)/9 | 158231 | Robert Price | 10/2021 |
2364 | 10^158209-7000000001 | 158209 | Robert Price | 12/2021 |
2365 | 2^525186-1857431 | 158097 | Enzo Doria | 10/2015 |
2366 | 2^524515-23245 | 157895 | Enzo Doria | 08/2013 |
2367 | 2^524499-109087 | 157890 | Enzo Doria | 04/2014 |
2368 | 2^18067*(2^506221+1)-1 | 157827 | Makoto Morimoto | 05/2014 |
2369 | 2^524288+189093 | 157827 | Makoto Morimoto | 01/2016 |
2370 | 2^524288-164073 | 157827 | Matthias Baur | 03/2020 |
2371 | (2^(2^19)-9333)/(19*23*463*1213*31973) | 157814 | Makoto Morimoto | 09/2017 |
2372 | (10^157792+359)/9 | 157792 | Serge Batalov | 12/2014 |
2373 | 2^524154+25 | 157787 | Stefano Morozzi | 04/2021 |
2374 | 2^523924+1876591 | 157717 | Enzo Doria | 07/2016 |
2375 | (17*10^157705+67)/3 | 157706 | Robert Price | 01/2019 |
2376 | (46^94841+1)/47 | 157696 | Robert Price | 01/2014 |
2377 | 2^523774+77839 | 157672 | Enzo Doria | 05/2016 |
2378 | 2^523587+2799143 | 157616 | Enzo Doria | 04/2016 |
2379 | 2^523371+1872113 | 157551 | Enzo Doria | 05/2016 |
2380 | 8*10^157477+27 | 157478 | Robert Price | 09/2021 |
2381 | (8*10^157368+49)/3 | 157369 | Robert Price | 05/2020 |
2382 | (82*10^157319+449)/9 | 157320 | Robert Price | 11/2019 |
2383 | 2^522535-351457 | 157299 | Enzo Doria | 07/2015 |
2384 | 2^522226+2799079 | 157206 | Enzo Doria | 04/2016 |
2385 | (134*10^157191-11)/3 | 157193 | Robert Price | 01/2022 |
2386 | 2^522134-1849451 | 157178 | Enzo Doria | 08/2015 |
2387 | 9*10^157152+71 | 157153 | Robert Price | 08/2020 |
2388 | 4*10^157150+93 | 157151 | Robert Price | 08/2018 |
2389 | 2^521401-29593 | 156958 | Enzo Doria | 08/2015 |
2390 | 7*10^156956+87 | 156957 | Robert Price | 07/2019 |
2391 | 2^521348-23397 | 156942 | Enzo Doria | 08/2013 |
2392 | 10^156902+34509 | 156903 | Patrick De Geest | 08/2021 |
2394 | 10^156902-80000001 | 156902 | Robert Price | 04/2021 |
2393 | 10^156902-69789 | 156902 | Patrick De Geest | 08/2021 |
2395 | (98^78797+1)/99 | 156901 | Paul Bourdelais | 12/2017 |
2396 | 65^86522+64 | 156857 | Lelio R Paula | 04/2016 |
2397 | 8^173687-7^173687 | 156855 | Jean-Louis Charton | 09/2004 |
2398 | 2^520977-1858183 | 156830 | Enzo Doria | 10/2015 |
2399 | (8*10^156718-101)/3 | 156719 | Robert Price | 08/2018 |
2400 | (10^156640+7)/53 | 156639 | Alfred Reich | 07/2021 |
2401 | 2^520314-1849481 | 156631 | Enzo Doria | 08/2015 |
2402 | (16*10^156563-637)/9 | 156564 | Robert Price | 09/2022 |
2403 | (13*10^156522+89)/3 | 156523 | Robert Price | 09/2018 |
2405 | (2^(6*86654+1)-5)/3 | 156513 | Kevin Acres | 12/2013 |
2404 | (2^519925-5)/3 | 156513 | Stefano Morozzi | 08/2022 |
2406 | (4*10^156501+119)/3 | 156502 | Robert Price | 12/2021 |
2407 | (83*10^156476-101)/9 | 156477 | Robert Price | 08/2021 |
2408 | 3*10^156435-97 | 156436 | Robert Price | 07/2018 |
2409 | (8*10^156371-77)/3 | 156372 | Robert Price | 07/2018 |
2410 | 53^90679-52^90679 | 156356 | Robert Price | 01/2013 |
2411 | (418*10^156309-31)/9 | 156311 | Robert Price | 05/2022 |
2412 | 2^519106-59141 | 156267 | Enzo Doria | 10/2015 |
2413 | 2^519053+6077 | 156251 | Enzo Doria | 02/2016 |
2414 | 2^518968-1860989 | 156225 | Enzo Doria | 11/2015 |
2415 | (25*10^156221-7)/9 | 156222 | Robert Price | 05/2018 |
2416 | 2^518931-2782297 | 156214 | Enzo Doria | 09/2015 |
2417 | 2^518608-2939 | 156117 | Yves Minet | 09/2014 |
2418 | 10*27^109005+697 | 156027 | Curtis Bright | 05/2015 |
2419 | (49*10^155933-319)/9 | 155934 | Robert Price | 10/2021 |
2420 | 2^517967-113485 | 155924 | Enzo Doria | 08/2014 |
2421 | (103*10^155904+17)/3 | 155906 | Robert Price | 08/2020 |
2422 | (8*10^155887-719)/9 | 155887 | Robert Price | 03/2021 |
2423 | 4*10^155853-39 | 155854 | Robert Price | 09/2020 |
2424 | 2^517698+4429 | 155843 | Lelio R Paula | 01/2013 |
2425 | 2^517543-113257 | 155796 | Enzo Doria | 07/2014 |
2426 | (14*10^155740-143)/3 | 155741 | Robert Price | 11/2018 |
2427 | 31^104309-30^104309 | 155563 | Henri Lifchitz | 08/2013 |
2428 | 2^516742-39851 | 155555 | Enzo Doria | 09/2015 |
2429 | (22*10^155543+47)/3 | 155544 | Robert Price | 06/2021 |
2430 | (19*10^155504+131)/3 | 155505 | Robert Price | 06/2020 |
2431 | 7*10^155496+61 | 155497 | Robert Price | 08/2019 |
2432 | 2^516440+1878421 | 155464 | Enzo Doria | 08/2016 |
2433 | 2^515863-1867237 | 155291 | Enzo Doria | 03/2016 |
2434 | 2^515786-461861 | 155268 | Enzo Doria | 07/2015 |
2435 | 2^515644-113247 | 155225 | Enzo Doria | 07/2014 |
2436 | (7*10^155203+107)/3 | 155204 | Robert Price | 02/2018 |
2437 | 10^155163+206871 | 155164 | Patrick De Geest | 05/2021 |
2438 | 10^155163-40001 | 155163 | Robert Price | 01/2021 |
2439 | (17*10^155133-89)/9 | 155134 | Robert Price | 07/2018 |
2440 | 2^515227-114997 | 155099 | Enzo Doria | 06/2015 |
2442 | EPrime(155025) | 155025 | Eric W. Weisstein | 10/2010 |
2441 | (14*10^155024-221)/9 | 155025 | Robert Price | 04/2018 |
2443 | (4*10^155022+173)/3 | 155023 | Robert Price | 02/2018 |
2444 | (266*10^154897+1)/3 | 154899 | Robert Price | 04/2020 |
2445 | (115*10^154869-1)/3 | 154871 | Serge Batalov | 12/2014 |
2446 | (86*10^154853-221)/9 | 154854 | Robert Price | 09/2019 |
2447 | 2^514361+78167 | 154839 | Enzo Doria | 05/2016 |
2448 | 2^514281-115033 | 154815 | Enzo Doria | 06/2015 |
2449 | 2^514194-3311 | 154788 | Yves Minet | 12/2014 |
2450 | (14*10^154773+37)/3 | 154774 | Robert Price | 12/2018 |
2451 | (16*10^154748+131)/3 | 154749 | Robert Price | 03/2022 |
2452 | 6*10^154742-31 | 154743 | Robert Price | 07/2020 |
2453 | (14*10^154645+139)/9 | 154646 | Robert Price | 11/2021 |
2454 | 5*10^154618-79 | 154619 | Robert Price | 03/2021 |
2455 | (167*10^154595+1)/3 | 154597 | Robert Price | 04/2021 |
2456 | 2^513437+1879337 | 154560 | Enzo Doria | 09/2016 |
2457 | 2^513306-113201 | 154521 | Enzo Doria | 07/2014 |
2458 | (10^154494-13)/3 | 154494 | Robert Price | 12/2017 |
2459 | 23462^35343+35343^23462 | 154462 | Norbert Schneider | 04/2023 |
2460 | 2^513042-113787 | 154442 | Enzo Doria | 09/2014 |
2461 | (151*10^154434-7)/9 | 154436 | Robert Price | 01/2020 |
2462 | (118*10^154428+11)/3 | 154430 | Robert Price | 03/2022 |
2463 | (14*10^154413+139)/9 | 154414 | Robert Price | 11/2021 |
2464 | (77*10^154389-149)/9 | 154390 | Robert Price | 03/2022 |
2465 | 10^154359+49 | 154360 | Robert Price | 02/2023 |
2466 | 2^512628-114053 | 154317 | Enzo Doria | 11/2014 |
2467 | 2^512529-49783 | 154287 | Enzo Doria | 10/2015 |
2468 | (355573#+6991)/(6991*2031599) | 154285 | Makoto Morimoto | 06/2019 |
2469 | 2^512315-115297 | 154223 | Enzo Doria | 06/2015 |
2470 | (47*10^154221+241)/9 | 154222 | Robert Price | 06/2022 |
2471 | (7*10^154211+173)/9 | 154211 | Robert Price | 01/2022 |
2472 | 2^512168+79561 | 154178 | Enzo Doria | 06/2016 |
2473 | (77*10^154175+67)/9 | 154176 | Robert Price | 12/2021 |
2474 | (5*10^154158-311)/9 | 154158 | Robert Price | 12/2021 |
2475 | (89*10^154103-737)/9 | 154104 | Robert Price | 01/2022 |
2476 | 4*10^154078-57 | 154079 | Robert Price | 10/2018 |
2477 | (355049#+1859)/(11*13*10148891*62587751) | 154051 | Makoto Morimoto | 06/2019 |
2478 | 7*10^154038-57 | 154039 | Robert Price | 06/2020 |
2479 | 2^511658-114371 | 154025 | Enzo Doria | 02/2015 |
2480 | (57^87719+1)/58 | 154022 | Paul Bourdelais | 11/2017 |
2481 | 2^511580-1846679 | 154001 | Enzo Doria | 07/2015 |
2482 | (259*10^153998-1)/3 | 154000 | Robert Price | 12/2020 |
2483 | 2^511541-113713 | 153990 | Enzo Doria | 09/2014 |
2484 | 49958^32768+49957^32768 | 153965 | Kellen Shenton | 11/2020 |
2485 | 10^153935+970801 | 153936 | Patrick De Geest | 05/2021 |
2486 | 10^153935-1000001 | 153935 | Robert Price | 08/2020 |
2487 | (16*10^153868-73)/3 | 153869 | Robert Price | 07/2020 |
2488 | 9*10^153851-67 | 153852 | Robert Price | 11/2020 |
2489 | 2^511077-112723 | 153850 | Enzo Doria | 06/2014 |
2490 | 69^83653-68^83653 | 153826 | Robert Price | 12/2013 |
2491 | 2^510963-113707 | 153816 | Enzo Doria | 09/2014 |
2492 | (154*10^153810-1)/9 | 153812 | Serge Batalov | 12/2014 |
2493 | (181*25^109991-75077)/8 | 153763 | Curtis Bright | 05/2015 |
2494 | 2*10^153746-3 | 153747 | Robert Price | 07/2015 |
2495 | 2^510626+1179379 | 153714 | Enzo Doria | 05/2016 |
2496 | (7*10^153664+113)/3 | 153665 | Robert Price | 02/2018 |
2497 | 2^510420+81121 | 153652 | Enzo Doria | 07/2016 |
2498 | (197*7^181761-1)/2 | 153608 | Karsten Bonath | 05/2017 |
2499 | 2^509894-459161 | 153494 | Enzo Doria | 05/2015 |
2500 | (74*10^153489+673)/9 | 153490 | Robert Price | 02/2022 |