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 | 2^587560-113699 | 176874 | Enzo Doria | 09/2014 |
2252 | 20596^40995+40995^20596 | 176844 | Norbert Schneider | 07/2022 |
2253 | (101*10^176815-17)/3 | 176817 | Robert Price | 02/2020 |
2254 | (25*10^176715-37)/3 | 176716 | Robert Price | 10/2019 |
2255 | 5^252763-12862 | 176674 | Enzo Doria | 05/2014 |
2256 | 38442^38531+38531^38442 | 176658 | Norbert Schneider | 09/2022 |
2257 | 2^586829+23961 | 176654 | Enzo Doria | 10/2023 |
2258 | 2^586804+81721 | 176646 | Enzo Doria | 08/2016 |
2259 | (10^176584+77)/3 | 176584 | Robert Price | 01/2018 |
2260 | 2^586570+1874419 | 176576 | Enzo Doria | 06/2016 |
2261 | (157*10^176482-1)/3 | 176484 | Robert Price | 01/2022 |
2262 | 5*2^586202+3 | 176466 | Jon Grantham | 07/2023 |
2263 | 2^585994-3311 | 176402 | Yves Minet | 12/2014 |
2264 | 8*10^176383-81 | 176384 | Robert Price | 02/2021 |
2265 | (23*10^176357+61)/3 | 176358 | Robert Price | 01/2022 |
2266 | 2^585832-112799 | 176354 | Enzo Doria | 07/2014 |
2267 | fibonacci(843383)+4 | 176257 | Jon Grantham | 08/2023 |
2268 | (29*10^176240+49)/3 | 176241 | Robert Price | 09/2021 |
2269 | (2^585387-11)/3 | 176219 | Stefano Morozzi | 08/2022 |
2270 | 2^585336-461039 | 176204 | Enzo Doria | 06/2015 |
2271 | (25*10^176164+143)/3 | 176165 | Robert Price | 08/2020 |
2272 | 2^585095-1848127 | 176132 | Enzo Doria | 08/2015 |
2273 | (5*10^176101-311)/9 | 176101 | Robert Price | 12/2021 |
2274 | 6*10^176092+77 | 176093 | Robert Price | 06/2020 |
2275 | 2^584950-115451 | 176088 | Enzo Doria | 07/2015 |
2276 | (646*10^176044+11)/9 | 176046 | Robert Price | 10/2021 |
2277 | (7^208283-5^208283)/2 | 176020 | Jon Grantham | 08/2023 |
2278 | 2^584671-3697 | 176004 | Yves Minet | 03/2017 |
2279 | (13*10^175965-37)/3 | 175966 | Robert Price | 01/2019 |
2280 | 2*10^175860-87 | 175861 | Robert Price | 04/2018 |
2281 | (86*10^175774-221)/9 | 175775 | Robert Price | 09/2019 |
2282 | 2^583614+216037 | 175686 | Enzo Doria | 11/2023 |
2283 | 9*10^175654+47 | 175655 | Robert Price | 01/2022 |
2284 | 10^175608+2971 | 175609 | Mike Oakes | 05/2010 |
2285 | 10^175608-55829 | 175608 | Patrick De Geest | 09/2013 |
2286 | 2^583051+1879793 | 175516 | Enzo Doria | 09/2016 |
2287 | (76*10^175514+491)/9 | 175515 | Robert Price | 10/2021 |
2288 | 2^582860-23829 | 175459 | Enzo Doria | 08/2013 |
2289 | (8*10^175428-53)/9 | 175428 | Serge Batalov | 11/2022 |
2290 | (28*10^175417-67)/3 | 175418 | Robert Price | 10/2020 |
2291 | 2^582670-23913 | 175402 | Enzo Doria | 08/2013 |
2292 | 2^582573-29563 | 175372 | Enzo Doria | 08/2015 |
2293 | 2^582435-1862767 | 175331 | Enzo Doria | 12/2015 |
2294 | 10^175295-40000001 | 175295 | Robert Price | 07/2022 |
2295 | (11*10^175250-149)/3 | 175251 | Robert Price | 09/2018 |
2296 | (29*10^175236+31)/3 | 175237 | Robert Price | 04/2021 |
2297 | 2^582072-1848599 | 175222 | Enzo Doria | 08/2015 |
2298 | (404267#+465)/(3^4*5^2*31) | 175208 | Makoto Morimoto | 06/2019 |
2299 | Phi(262767,10)/259613797 | 175168 | Makoto Kamada | 12/2022 |
2300 | (2^581871-15)/7 | 175160 | Stefano Morozzi | 08/2022 |
2301 | (5*10^175117-311)/9 | 175117 | Robert Price | 12/2021 |
2302 | 10^175093-89 | 175093 | Riley Fisher | 10/2023 |
2303 | (220221^32768+1)/2 | 175075 | Serge Batalov | 03/2018 |
2304 | 2^581560+1874911 | 175068 | Enzo Doria | 06/2016 |
2305 | (25*10^174987-97)/9 | 174988 | Robert Price | 01/2022 |
2306 | (75^93307-1)/74 | 174955 | Paul Bourdelais | 09/2017 |
2307 | (27398^39601-1)/(27398^199-1) | 174855 | Ryan Propper | 11/2023 |
2308 | (861^59557-1)/860 | 174798 | Jinyuan Wang | 11/2023 |
2309 | (47*10^174790+493)/9 | 174791 | Robert Price | 03/2019 |
2310 | 2^580528-114035 | 174757 | Enzo Doria | 11/2014 |
2311 | (19*10^174754+737)/9 | 174755 | Robert Price | 07/2018 |
2312 | (35*10^174704-377)/9 | 174705 | Robert Price | 07/2018 |
2313 | (49*10^174681+293)/9 | 174682 | Robert Price | 03/2021 |
2314 | 6*10^174681+11 | 174682 | Robert Price | 07/2022 |
2315 | 3*10^174665-97 | 174666 | Robert Price | 07/2018 |
2316 | 2^580046-459161 | 174612 | Enzo Doria | 05/2015 |
2317 | 2^579992-39779 | 174595 | Enzo Doria | 09/2015 |
2318 | (29*10^174481-149)/3 | 174482 | Robert Price | 08/2022 |
2319 | (4*10^174423+101)/3 | 174424 | Robert Price | 11/2021 |
2320 | (2^579371-17)/3 | 174408 | Stefano Morozzi | 08/2022 |
2321 | (2*10^174301-71)/3 | 174301 | Robert Price | 12/2017 |
2322 | 14^152048+3 | 174267 | Paul Bourdelais | 12/2020 |
2323 | (2^578696+1)/257 | 174203 | Lelio R Paula | 09/2015 |
2324 | 2^578656-1851809 | 174193 | Enzo Doria | 08/2015 |
2325 | 2^578583-113917 | 174171 | Enzo Doria | 10/2014 |
2326 | 2^578434-83 | 174126 | Yves Minet | 07/2017 |
2327 | 15^148048-2 | 174118 | Robert Price | 06/2015 |
2328 | 5*10^174101+81 | 174102 | Robert Price | 03/2019 |
2329 | 2^578346+2794249 | 174100 | Enzo Doria | 03/2016 |
2330 | 10^173947+1372429 | 173948 | Patrick De Geest | 08/2021 |
2332 | 10^173947-500000001 | 173947 | Robert Price | 12/2020 |
2331 | 10^173947-286391 | 173947 | Patrick De Geest | 08/2021 |
2333 | 5*10^173904-13 | 173905 | Robert Price | 01/2019 |
2334 | (8*10^173862-59)/3 | 173863 | Robert Price | 08/2021 |
2335 | (71*10^173854-287)/9 | 173855 | Robert Price | 09/2019 |
2336 | (43*10^173826+101)/9 | 173827 | Robert Price | 11/2018 |
2337 | 2*10^173809-87 | 173810 | Robert Price | 04/2018 |
2338 | (2^577372-23)/7 | 173806 | Stefano Morozzi | 08/2022 |
2339 | (L!41532)/2 | 173772 | Serge Batalov | 07/2017 |
2340 | 2^577146-49877 | 173739 | Enzo Doria | 10/2015 |
2341 | 2^576889+49727 | 173661 | Enzo Doria | 09/2016 |
2342 | (167*10^173571-17)/3 | 173573 | Robert Price | 03/2020 |
2343 | 37803^37916+37916^37803 | 173562 | Norbert Schneider | 06/2022 |
2344 | 5*10^173534+29 | 173535 | Robert Price | 12/2020 |
2345 | (2^576551-1)/4612409/64758208321/242584327930759 | 173528 | Dario Alpern | 06/2014 |
2346 | 4*10^173509+99 | 173510 | Robert Price | 12/2020 |
2347 | (13*10^173467-37)/3 | 173468 | Robert Price | 01/2019 |
2348 | 2^576016-113919 | 173399 | Enzo Doria | 11/2014 |
2349 | 2^575911-1849387 | 173367 | Enzo Doria | 08/2015 |
2350 | 2^575879+215843 | 173357 | Enzo Doria | 11/2023 |
2351 | 515^63898+63898^515 | 173280 | Norbert Schneider | 12/2022 |
2352 | 2^575614+9 | 173278 | Stefano Morozzi | 04/2021 |
2353 | (16*10^173251+107)/3 | 173252 | Robert Price | 03/2019 |
2354 | 2^575196-114053 | 173152 | Enzo Doria | 11/2014 |
2355 | 7*10^173135-53 | 173136 | Robert Price | 06/2020 |
2356 | (301*10^173047+23)/9 | 173049 | Robert Price | 06/2021 |
2357 | 2^574591-115387 | 172970 | Enzo Doria | 07/2015 |
2358 | (202*10^172940-13)/9 | 172942 | Robert Price | 01/2020 |
2359 | 45^104608+104608^45 | 172940 | Norbert Schneider | 06/2021 |
2360 | 2^574328+81121 | 172890 | Enzo Doria | 07/2016 |
2361 | (115*10^172862-1)/3 | 172864 | Serge Batalov | 12/2014 |
2362 | 3*10^172812+13 | 172813 | Robert Price | 07/2018 |
2363 | 4*5^247217-9 | 172798 | Lelio R Paula | 10/2015 |
2364 | 6*10^172772-31 | 172773 | Robert Price | 07/2020 |
2365 | 69061#+(467617#)/(69061#) | 172766 | Makoto Morimoto | 10/2017 |
2366 | (34*10^172693-547)/9 | 172694 | Robert Price | 03/2021 |
2367 | 5*10^172692+57 | 172693 | Robert Price | 12/2018 |
2368 | 2^573668-2782319 | 172692 | Enzo Doria | 09/2015 |
2369 | 2^573444-1855109 | 172624 | Enzo Doria | 09/2015 |
2370 | (14*10^172532-29)/3 | 172533 | Robert Price | 09/2021 |
2371 | 3^361608+2 | 172531 | Robert Price | 08/2014 |
2372 | (54^99563+1)/55 | 172481 | Paul Bourdelais | 11/2017 |
2373 | 2^572592-114893 | 172368 | Enzo Doria | 05/2015 |
2374 | 4*10^172354-39 | 172355 | Robert Price | 09/2020 |
2375 | 37535^37674+37674^37535 | 172338 | Norbert Schneider | 05/2022 |
2376 | (11*10^172322-119)/9 | 172323 | Robert Price | 11/2021 |
2377 | 5^246497-4^246497 | 172295 | Jean-Louis Charton | 11/2008 |
2378 | (38^109037-1)/37 | 172254 | Paul Bourdelais | 10/2017 |
2379 | (4*10^172241+89)/3 | 172242 | Robert Price | 08/2021 |
2380 | 3*10^172219+73 | 172220 | Robert Price | 08/2018 |
2381 | A352736(15) | 172032 | Kellen Shenton | 05/2022 |
2382 | 2^571300+25861 | 171979 | Enzo Doria | 11/2023 |
2383 | (36^110503+1)/37 | 171975 | Lelio R Paula | 04/2017 |
2384 | 8*10^171934+71 | 171935 | Robert Price | 05/2022 |
2385 | (11*10^171933+601)/9 | 171934 | Robert Price | 02/2022 |
2386 | (22*10^171929-73)/3 | 171930 | Robert Price | 04/2019 |
2387 | (49*10^171892-319)/9 | 171893 | Robert Price | 10/2021 |
2388 | 4*10^171891-63 | 171892 | Robert Price | 10/2018 |
2389 | 67*10^171882-9 | 171884 | Robert Price | 03/2021 |
2390 | 2^570980-2782289 | 171883 | Enzo Doria | 09/2015 |
2391 | (25*10^171849-37)/3 | 171850 | Robert Price | 10/2019 |
2392 | 10^171832+387909 | 171833 | Patrick De Geest | 08/2021 |
2393 | 10^171832-10001 | 171832 | Robert Price | 02/2021 |
2395 | (2^(12*47563+13)-9)/7 | 171818 | Kevin Acres | 11/2013 |
2394 | (2^570769-9)/7 | 171818 | Stefano Morozzi | 08/2022 |
2396 | (74*10^171651-29)/9 | 171652 | Robert Price | 06/2022 |
2397 | (536*10^171601-23)/9 | 171603 | Robert Price | 02/2020 |
2398 | 7*10^171535-99 | 171536 | Robert Price | 06/2019 |
2399 | (28*10^171534+101)/3 | 171535 | Robert Price | 09/2020 |
2400 | 2^569563-1854607 | 171456 | Enzo Doria | 09/2015 |
2401 | 3*10^171391+59 | 171392 | Robert Price | 08/2018 |
2402 | (13*10^171306-37)/3 | 171307 | Robert Price | 01/2019 |
2403 | 8*10^171296+87 | 171297 | Robert Price | 10/2019 |
2404 | (55^98419-1)/54 | 171284 | Paul Bourdelais | 12/2017 |
2405 | (29*10^171275+67)/3 | 171276 | Robert Price | 12/2019 |
2406 | (457*10^171248-7)/9 | 171250 | Robert Price | 09/2022 |
2407 | (194*10^171193+1)/3 | 171195 | Robert Price | 01/2022 |
2408 | 2^568642+1876129 | 171179 | Enzo Doria | 07/2016 |
2409 | 4*10^171146-47 | 171147 | Robert Price | 10/2021 |
2410 | (19*10^170970+131)/3 | 170971 | Robert Price | 06/2020 |
2411 | (46*10^170932+143)/9 | 170933 | Robert Price | 01/2019 |
2412 | (79*10^170901-1)/3 | 170903 | Robert Price | 02/2020 |
2413 | (58*10^170842+221)/9 | 170843 | Robert Price | 02/2017 |
2414 | (29*10^170830+583)/9 | 170831 | Robert Price | 06/2021 |
2415 | 3*10^170800+73 | 170801 | Robert Price | 08/2018 |
2416 | (29*10^170799-119)/9 | 170800 | Robert Price | 12/2021 |
2417 | 11^164009-270 | 170798 | Enzo Doria | 10/2012 |
2418 | 2*10^170767+41 | 170768 | Robert Price | 05/2018 |
2419 | 3^357706-112358 | 170670 | Enzo Doria | 07/2014 |
2420 | TheodorusPrime(170657) | 170657 | Eric W. Weisstein | 08/2013 |
2421 | 2^566871+1876163 | 170646 | Enzo Doria | 07/2016 |
2422 | (10657#+(404267#)/(10657#))/(1675379) | 170624 | Makoto Morimoto | 10/2017 |
2423 | 2^566496+7 | 170533 | Donovan Johnson | 07/2006 |
2424 | (7^201653+1)/8 | 170416 | Paul Bourdelais | 02/2010 |
2425 | 7*10^170348+33 | 170349 | Robert Price | 06/2020 |
2426 | 2^565183-352021 | 170138 | Enzo Doria | 07/2015 |
2427 | 2^565143-114457 | 170125 | Enzo Doria | 02/2015 |
2428 | 3*2^565024-5 | 170090 | Jon Grantham | 07/2023 |
2429 | 2^564976-49869 | 170075 | Enzo Doria | 10/2015 |
2430 | (7*10^169944+23)/3 | 169945 | Robert Price | 06/2020 |
2431 | (37*10^169943+251)/9 | 169944 | Robert Price | 09/2021 |
2432 | 2^564526-351401 | 169940 | Enzo Doria | 07/2015 |
2433 | 4*10^169759+19 | 169760 | Robert Price | 10/2018 |
2434 | (2^563903-13)/5 | 169752 | Stefano Morozzi | 08/2022 |
2435 | 21^128327-20^128327 | 169677 | Jean-Louis Charton | 11/2014 |
2436 | 2^563583+89 | 169656 | Stefano Morozzi | 04/2021 |
2437 | 2^563311+215813 | 169574 | Enzo Doria | 11/2023 |
2438 | (4*10^169559+83)/3 | 169560 | Robert Price | 02/2018 |
2439 | (194*10^169461-17)/3 | 169463 | Robert Price | 09/2020 |
2440 | 2^562749+2793047 | 169405 | Enzo Doria | 02/2016 |
2441 | (154*10^169337-1)/9 | 169339 | Serge Batalov | 12/2014 |
2442 | (109*10^169337+11)/3 | 169339 | Robert Price | 06/2022 |
2443 | (56*10^169311+691)/9 | 169312 | Robert Price | 05/2019 |
2444 | 2^562369-29503 | 169290 | Enzo Doria | 08/2015 |
2445 | (17*10^169283+19)/9 | 169284 | Robert Price | 03/2021 |
2446 | 10^169074-5000000001 | 169074 | Robert Price | 09/2022 |
2447 | 8*10^169071+59 | 169072 | Robert Price | 11/2021 |
2448 | (10^169044+173)/3 | 169044 | Robert Price | 05/2021 |
2449 | (22*10^169019+383)/9 | 169020 | Robert Price | 05/2018 |
2450 | (64*10^168935+143)/9 | 168936 | Robert Price | 05/2021 |
2451 | (22*10^168906+47)/3 | 168907 | Robert Price | 06/2021 |
2452 | 5*10^168807+87 | 168808 | Robert Price | 02/2019 |
2453 | (77*10^168803-149)/9 | 168804 | Robert Price | 03/2022 |
2454 | 4*10^168788-47 | 168789 | Robert Price | 10/2021 |
2455 | (154*10^168781+17)/9 | 168783 | Robert Price | 06/2022 |
2456 | (418*10^168751-31)/9 | 168753 | Robert Price | 05/2022 |
2457 | (29*10^168711+691)/9 | 168712 | Robert Price | 08/2021 |
2458 | 2^560405-39943 | 168699 | Enzo Doria | 09/2015 |
2459 | (397*10^168634+17)/9 | 168636 | Robert Price | 03/2022 |
2460 | 89*10^168614-7 | 168616 | Robert Price | 04/2020 |
2461 | (23*10^168608-83)/3 | 168609 | Robert Price | 08/2019 |
2462 | (94*10^168541+11)/3 | 168543 | Robert Price | 04/2020 |
2463 | 2^559686-350471 | 168483 | Enzo Doria | 06/2015 |
2464 | 2^559672-1855889 | 168479 | Enzo Doria | 09/2015 |
2465 | (5*10^168379-473)/9 | 168379 | Robert Price | 01/2018 |
2466 | 2^559000-59939 | 168276 | Enzo Doria | 11/2015 |
2467 | 2^558830+25009 | 168225 | Enzo Doria | 11/2023 |
2468 | (16*10^168181+41)/3 | 168182 | Robert Price | 03/2019 |
2469 | (19*10^168173+77)/3 | 168174 | Robert Price | 05/2019 |
2470 | (74*10^168170+529)/9 | 168171 | Robert Price | 02/2022 |
2471 | 2^558640-114149 | 168168 | Enzo Doria | 12/2014 |
2472 | 2^558498-1849277 | 168125 | Enzo Doria | 08/2015 |
2473 | 2^558476-1850249 | 168119 | Enzo Doria | 08/2015 |
2474 | (7*10^168067+71)/3 | 168068 | Robert Price | 05/2018 |
2475 | 2^558150-1852211 | 168020 | Enzo Doria | 08/2015 |
2476 | 2^558125-113661 | 168013 | Enzo Doria | 09/2014 |
2477 | 393^64747-392^64747 | 167980 | Jean-Louis Charton | 08/2012 |
2478 | (8*10^167965-11)/3 | 167966 | Robert Price | 07/2023 |
2479 | (13*10^167911+179)/3 | 167912 | Robert Price | 11/2018 |
2480 | (26*10^167861-167)/3 | 167862 | Robert Price | 10/2019 |
2481 | 36^107827-107827 | 167812 | Kellen Shenton | 06/2023 |
2482 | 39^105456+2 | 167788 | Robert Price | 04/2015 |
2483 | (28^115877-1)/27 | 167691 | Paul Bourdelais | 12/2017 |
2484 | 6*10^167662-13 | 167663 | Robert Price | 07/2022 |
2485 | 10^167639+121911 | 167640 | Patrick De Geest | 07/2018 |
2486 | 10^167639-8001 | 167639 | Robert Price | 03/2018 |
2487 | 2^556504-462059 | 167525 | Enzo Doria | 07/2015 |
2488 | (88*10^167522-637)/9 | 167523 | Robert Price | 11/2020 |
2489 | 5*10^167510+9 | 167511 | Robert Price | 08/2015 |
2490 | 7*10^167437-93 | 167438 | Robert Price | 08/2020 |
2491 | 11^160663+6 | 167314 | Paul Bourdelais | 02/2021 |
2492 | (7*10^167308-493)/9 | 167308 | Robert Price | 04/2021 |
2493 | 2^555759+74333 | 167301 | Enzo Doria | 07/2023 |
2494 | 2^555343+1874333 | 167175 | Enzo Doria | 06/2016 |
2495 | 2^555210-461261 | 167135 | Enzo Doria | 07/2015 |
2496 | 9*10^166996+59 | 166997 | Robert Price | 01/2020 |
2497 | (68*10^166976-329)/9 | 166977 | Robert Price | 03/2021 |
2498 | (64*10^166901+683)/9 | 166902 | Robert Price | 06/2022 |
2499 | 2^554372+1880527 | 166883 | Enzo Doria | 11/2016 |
2500 | 2^554299+72173 | 166861 | Enzo Doria | 04/2016 |