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 ? |
3251 | (2^383544+5)/3 | 115458 | Lelio R Paula | 03/2014 |
3252 | 89^59221-88^59221 | 115445 | Robert Price | 06/2015 |
3253 | 2^383347+1869413 | 115399 | Enzo Doria | 05/2016 |
3254 | (103*10^115366+17)/3 | 115368 | Robert Price | 08/2020 |
3255 | (2^383232+750)/2 | 115365 | Stefano Morozzi | 09/2020 |
3256 | (2^383130-7)/131793 | 115329 | Lelio R Paula | 03/2014 |
3257 | (12^106859+1)/13 | 115320 | Paul Bourdelais | 02/2010 |
3258 | 2^383039+1869509 | 115307 | Enzo Doria | 05/2016 |
3259 | (13*2^382968+1)/7 | 115286 | Lelio R Paula | 04/2014 |
3260 | (2^382945+2046)/2 | 115278 | Stefano Morozzi | 09/2020 |
3261 | (2^382789+3182)/2 | 115231 | Stefano Morozzi | 09/2020 |
3262 | 6*10^115211+79 | 115212 | Robert Price | 09/2020 |
3263 | (2^382722+3598)/2 | 115211 | Stefano Morozzi | 09/2020 |
3264 | (2^382699+2090)/2 | 115204 | Stefano Morozzi | 09/2020 |
3265 | 2^382633+73697 | 115185 | Enzo Doria | 04/2016 |
3266 | 2^382509-113593 | 115147 | Enzo Doria | 08/2014 |
3267 | 2^382502-2791721 | 115145 | Enzo Doria | 03/2016 |
3268 | (2^382495+1274)/2 | 115143 | Stefano Morozzi | 01/2021 |
3269 | 9^120590-120590 | 115073 | Robert Price | 03/2014 |
3270 | 2^382198+9 | 115054 | Lelio R Paula | 03/2013 |
3271 | (89*10^115030-179)/9 | 115031 | Robert Price | 12/2019 |
3272 | 2^381883-59197 | 114959 | Enzo Doria | 10/2015 |
3273 | (2^381820+1626)/2 | 114939 | Stefano Morozzi | 09/2020 |
3275 | 2^381806-4481 | 114936 | Yves Minet | 07/2011 |
3274 | 2^381808-1851507 | 114936 | Enzo Doria | 08/2015 |
3276 | (2^381792+2502)/2 | 114931 | Stefano Morozzi | 09/2020 |
3277 | 2^381774-112373 | 114926 | Enzo Doria | 06/2014 |
3278 | 2^381555-109051 | 114860 | Enzo Doria | 04/2014 |
3279 | 2^381546-113501 | 114857 | Enzo Doria | 08/2014 |
3280 | (7*10^114845-19)/3 | 114846 | Robert Price | 10/2020 |
3281 | (2^381436+1182)/2 | 114824 | Stefano Morozzi | 09/2020 |
3282 | 3^240644+16 | 114817 | Robert Price | 04/2020 |
3283 | 2^381360-110429 | 114801 | Enzo Doria | 05/2014 |
3284 | (17*10^114795+79)/3 | 114796 | Robert Price | 04/2019 |
3285 | (66^63079+1)/67 | 114774 | Paul Bourdelais | 09/2017 |
3286 | 2^381252-115079 | 114769 | Enzo Doria | 06/2015 |
3287 | (2^381251+3186)/2 | 114768 | Stefano Morozzi | 09/2020 |
3288 | 2^381209-29323 | 114756 | Enzo Doria | 07/2015 |
3289 | 569^41651-568^41651 | 114754 | Jean-Louis Charton | 11/2009 |
3290 | 2^381201+1873205 | 114753 | Enzo Doria | 06/2016 |
3291 | (85*10^114737+293)/9 | 114738 | Robert Price | 12/2019 |
3292 | (265*10^114736+17)/3 | 114738 | Robert Price | 03/2020 |
3293 | (2^381062+2170)/2 | 114711 | Stefano Morozzi | 09/2020 |
3294 | 2^380974-112991 | 114685 | Enzo Doria | 07/2014 |
3295 | 88^58949-87 | 114626 | Norbert Schneider | 11/2015 |
3296 | (2^380773+662)/2 | 114624 | Stefano Morozzi | 09/2020 |
3297 | 2^380700+91 | 114603 | Lelio R Paula | 03/2013 |
3298 | (2^380632+2578)/2 | 114582 | Stefano Morozzi | 09/2020 |
3299 | 7*10^114577+27 | 114578 | Robert Price | 11/2020 |
3301 | 10^114575+3^114576 | 114576 | Mike Oakes | 09/2010 |
3300 | 10^114575+66561 | 114576 | Patrick De Geest | 02/2017 |
3302 | 10^114575-581081 | 114575 | Patrick De Geest | 02/2017 |
3303 | 2^380587-115315 | 114569 | Enzo Doria | 06/2015 |
3304 | 2^380513-216925 | 114546 | Enzo Doria | 05/2014 |
3305 | 2^380451-1840087 | 114528 | Enzo Doria | 06/2015 |
3306 | (2^380426+1510)/2 | 114520 | Stefano Morozzi | 09/2020 |
3307 | (15*2^380388-1)/539 | 114507 | Lelio R Paula | 04/2014 |
3308 | 2^380279-461347 | 114476 | Enzo Doria | 07/2015 |
3309 | 2^380021+1871147 | 114398 | Enzo Doria | 05/2016 |
3310 | 2^380005+1869707 | 114393 | Enzo Doria | 05/2016 |
3311 | 2^379958-113297 | 114379 | Enzo Doria | 07/2014 |
3312 | (2^379897+1274)/2 | 114361 | Stefano Morozzi | 01/2021 |
3313 | 2^379881-109105 | 114356 | Enzo Doria | 04/2014 |
3314 | (21*2^379857-1)/13 | 114349 | Lelio R Paula | 03/2014 |
3315 | (4*10^114340+83)/3 | 114341 | Robert Price | 02/2018 |
3316 | (26*10^114317-77)/3 | 114318 | Robert Price | 11/2019 |
3317 | (4*10^114296+149)/9 | 114296 | Robert Price | 04/2018 |
3318 | 88^58773-87 | 114284 | Norbert Schneider | 11/2015 |
3319 | 2^379529-2790283 | 114250 | Enzo Doria | 03/2016 |
3320 | 2^379522+1869115 | 114248 | Enzo Doria | 05/2016 |
3321 | (145*10^114226-1)/9 | 114228 | Serge Batalov | 12/2014 |
3322 | 2^379317-115033 | 114186 | Enzo Doria | 06/2015 |
3323 | (17*2^379246-1)/17407 | 114162 | Lelio R Paula | 03/2014 |
3324 | (2^379192+226)/2 | 114148 | Stefano Morozzi | 09/2020 |
3325 | 6*10^114091+23 | 114092 | Robert Price | 09/2020 |
3326 | (4*10^114076+143)/3 | 114077 | Robert Price | 03/2018 |
3327 | (2^378820+3226)/2 | 114036 | Stefano Morozzi | 01/2021 |
3328 | (2^378754+2058)/2 | 114017 | Stefano Morozzi | 09/2020 |
3329 | (46*10^113969+683)/9 | 113970 | Robert Price | 03/2019 |
3330 | 78*10^113954-7 | 113956 | Robert Price | 10/2020 |
3331 | 2*10^113941+9 | 113942 | Dmitry Domanov | 11/2010 |
3333 | (2^378478+1642)/2 | 113933 | Stefano Morozzi | 09/2020 |
3332 | 10^113933-8000000001 | 113933 | Robert Price | 11/2020 |
3334 | 4*10^113905+91 | 113906 | Robert Price | 08/2018 |
3335 | 10^113894-10000001 | 113894 | Robert Price | 09/2020 |
3336 | 4*10^113874-21 | 113875 | Robert Price | 09/2018 |
3337 | 2^378272-16565 | 113872 | Enzo Doria | 04/2013 |
3338 | 2^378217-109075 | 113855 | Enzo Doria | 04/2014 |
3339 | 2^378198+67 | 113849 | Lelio R Paula | 03/2013 |
3340 | 2^378173-3249 | 113842 | Yves Minet | 12/2014 |
3341 | (2^378155-5)/6597 | 113833 | Lelio R Paula | 03/2014 |
3342 | 92941#+(355049#)/(92941#) | 113830 | Makoto Morimoto | 10/2017 |
3343 | 2^378128-262977 | 113828 | Enzo Doria | 05/2014 |
3344 | 2^378123-113215 | 113827 | Enzo Doria | 07/2014 |
3345 | 6^146274-18893 | 113824 | Enzo Doria | 03/2013 |
3346 | 2^378078+75 | 113813 | Lelio R Paula | 03/2013 |
3347 | 2^377940+63 | 113772 | Lelio R Paula | 03/2013 |
3348 | 2^377941-2785843 | 113772 | Enzo Doria | 04/2016 |
3349 | 2^377810+1875109 | 113733 | Enzo Doria | 09/2016 |
3350 | 2^377803-1838635 | 113731 | Enzo Doria | 05/2015 |
3351 | 2^377799+52223 | 113729 | Yves Minet | 08/2010 |
3352 | 11^109190-270 | 113710 | Enzo Doria | 12/2012 |
3353 | 7*10^113693+9 | 113694 | Robert Price | 09/2015 |
3354 | 13899^27442+27442^13899 | 113692 | Norbert Schneider | 12/2020 |
3355 | (82*10^113681+161)/9 | 113682 | Robert Price | 10/2019 |
3356 | 26121^22478+22478^26121 | 113673 | Hans Havermann | 01/2021 |
3357 | 4*10^113658+49 | 113659 | Robert Price | 01/2019 |
3358 | (2*10^113631+529)/9 | 113631 | Robert Price | 01/2018 |
3359 | 2^377408-2798279 | 113612 | Enzo Doria | 04/2016 |
3360 | 13706^27459+27459^13706 | 113596 | Norbert Schneider | 12/2020 |
3361 | 5*10^113586+57 | 113587 | Robert Price | 12/2018 |
3362 | 2^377222+93679 | 113556 | Yves Minet | 02/2011 |
3363 | 262147#+316847 | 113552 | Makoto Morimoto | 02/2016 |
3364 | (2^377131+2018)/2 | 113528 | Stefano Morozzi | 09/2020 |
3365 | 2^377026-458903 | 113497 | Enzo Doria | 04/2015 |
3366 | (2^377014+2202)/2 | 113493 | Stefano Morozzi | 09/2020 |
3367 | (2^376967+258)/2 | 113479 | Stefano Morozzi | 09/2020 |
3368 | (29*10^113468-17)/3 | 113469 | Robert Price | 09/2019 |
3369 | 35^73477-34^73477 | 113454 | Robert Price | 05/2012 |
3370 | (14*10^113451-89)/3 | 113452 | Robert Price | 05/2020 |
3371 | 2^376790+27 | 113426 | Lelio R Paula | 03/2013 |
3372 | 2^376786-112701 | 113424 | Enzo Doria | 06/2014 |
3373 | 2^376731+21 | 113408 | Lelio R Paula | 03/2013 |
3374 | 9*10^113394-73 | 113395 | Robert Price | 01/2021 |
3376 | 2^376634+1871245 | 113379 | Enzo Doria | 05/2016 |
3375 | (4^188318+17)/3 | 113379 | Robert Price | 02/2017 |
3377 | E(29532)/492842728006750534663978950979381235 | 113367 | Thomas Ritschel | 01/2021 |
3378 | 2^376576-344369 | 113361 | Enzo Doria | 07/2015 |
3379 | 2^376549-4543 | 113353 | Yves Minet | 03/2019 |
3380 | (17*2^376526-1)/223 | 113345 | Lelio R Paula | 03/2014 |
3381 | (27*2^376441-1)/811 | 113319 | Lelio R Paula | 03/2014 |
3382 | (2^376350+3538)/2 | 113293 | Stefano Morozzi | 09/2020 |
3383 | (2*10^113248+73)/3 | 113248 | Robert Price | 09/2020 |
3384 | 2^376128-21237 | 113226 | Enzo Doria | 06/2013 |
3385 | 5*10^113224-81 | 113225 | Robert Price | 01/2019 |
3386 | 10^113220-1000001 | 113220 | Robert Price | 08/2020 |
3387 | (2^376102+1578)/2 | 113218 | Stefano Morozzi | 09/2020 |
3388 | (2^375996+3102)/2 | 113186 | Stefano Morozzi | 09/2020 |
3389 | (94*10^113176+11)/3 | 113178 | Robert Price | 04/2020 |
3390 | (46*10^113162+521)/9 | 113163 | Robert Price | 02/2019 |
3391 | (2^375885+3614)/2 | 113153 | Stefano Morozzi | 09/2020 |
3392 | (2^375873+2006)/2 | 113149 | Stefano Morozzi | 09/2020 |
3393 | (2^375726+694)/2 | 113105 | Stefano Morozzi | 09/2020 |
3394 | 2^375638-1787171 | 113079 | Enzo Doria | 03/2016 |
3395 | (2^375563+2178)/2 | 113056 | Stefano Morozzi | 09/2020 |
3396 | 2^375537+4847 | 113048 | Yves Minet | 02/2011 |
3397 | (17^91837+16^91837)/33 | 113000 | Robert Price | 08/2013 |
3398 | 5*F(270343)*(F(270343)+1)+1 | 112997 | Serge Batalov | 04/2015 |
3399 | 2^375350-3221 | 112992 | Yves Minet | 07/2011 |
3400 | (13*2^375306+1)/7 | 112979 | Lelio R Paula | 03/2014 |
3401 | 2^375305+76757 | 112979 | Enzo Doria | 05/2016 |
3402 | 2^375258+1874989 | 112964 | Enzo Doria | 07/2016 |
3403 | (29^77237-1)/28 | 112950 | Robert Price | 04/2012 |
3404 | 2^375154+74959 | 112933 | Enzo Doria | 04/2016 |
3405 | (266*10^112930-17)/3 | 112932 | Robert Price | 07/2020 |
3406 | (2^375134+778)/2 | 112927 | Stefano Morozzi | 09/2020 |
3407 | (85*10^112919+293)/9 | 112920 | Robert Price | 12/2019 |
3408 | (2^375022+1210)/2 | 112893 | Stefano Morozzi | 09/2020 |
3409 | 2^374993-2787163 | 112885 | Enzo Doria | 11/2015 |
3410 | 13699^27268+27268^13699 | 112800 | Norbert Schneider | 12/2020 |
3411 | 123^53970+2 | 112793 | Lelio R Paula | 06/2013 |
3412 | 2^374640-4685 | 112778 | Yves Minet | 05/2019 |
3413 | (2^374556+682)/2 | 112753 | Stefano Morozzi | 09/2020 |
3414 | p(22852)#/2+512 | 112745 | Robert Price | 05/2017 |
3415 | (11*10^112742-131)/3 | 112743 | Robert Price | 09/2018 |
3416 | 2^374441-21049 | 112718 | Enzo Doria | 06/2013 |
3417 | 4*10^112706-93 | 112707 | Robert Price | 07/2020 |
3418 | (2^374321+1)/3 | 112682 | H.&R. Lifchitz | 12/2000 |
3419 | (2^374213+1766)/2 | 112650 | Stefano Morozzi | 09/2020 |
3420 | (2^374094+2674)/2 | 112614 | Stefano Morozzi | 09/2020 |
3421 | (13^101089-1)/12 | 112607 | Paul Bourdelais | 03/2010 |
3422 | 3^236011-8828 | 112606 | Enzo Doria | 05/2014 |
3423 | (2^373748+1090)/2 | 112510 | Stefano Morozzi | 09/2020 |
3424 | (2^373727+1806)/2 | 112503 | Stefano Morozzi | 01/2021 |
3425 | 26343^18580+18580^26343 | 112460 | Hans Havermann | 01/2021 |
3426 | 39^70666+2 | 112435 | Lelio R Paula | 11/2013 |
3427 | (134*10^112400+7)/3 | 112402 | Robert Price | 02/2020 |
3428 | (2^373365+326)/2 | 112394 | Stefano Morozzi | 09/2020 |
3429 | (2^373350-7)/3 | 112390 | Lelio R Paula | 03/2014 |
3430 | 2^373306-1840931 | 112377 | Enzo Doria | 06/2015 |
3431 | (2^373262+270)/2 | 112363 | Stefano Morozzi | 09/2020 |
3432 | (142*10^112352-7)/9 | 112354 | Robert Price | 12/2019 |
3433 | 2^373117+1872107 | 112320 | Enzo Doria | 05/2016 |
3434 | (2^373083+2226)/2 | 112309 | Stefano Morozzi | 01/2021 |
3435 | 27963^10376+10376^27963 | 112301 | Hans Havermann | 01/2021 |
3436 | 2^373040+2792761 | 112297 | Enzo Doria | 03/2016 |
3437 | EPrime(112280) | 112280 | Eric W. Weisstein | 10/2010 |
3438 | 2^372950+7 | 112270 | Lelio R Paula | 03/2013 |
3439 | 2^372945+1872167 | 112268 | Enzo Doria | 05/2016 |
3440 | (2^372754+3094)/2 | 112210 | Stefano Morozzi | 09/2020 |
3441 | 6*10^112208+43 | 112209 | Robert Price | 07/2020 |
3442 | (2^372728+3058)/2 | 112203 | Stefano Morozzi | 09/2020 |
3443 | 2^372695+72113 | 112193 | Enzo Doria | 04/2016 |
3444 | 2^372689-2563 | 112191 | Yves Minet | 07/2014 |
3445 | 2^372638-1859441 | 112176 | Enzo Doria | 03/2016 |
3446 | 3^235074-113056 | 112159 | Enzo Doria | 07/2014 |
3447 | -E(29242)/489848365125303139750087 | 112140 | Thomas Ritschel | 01/2021 |
3448 | (2^372428+3742)/2 | 112112 | Stefano Morozzi | 01/2021 |
3449 | 131^52949-130 | 112108 | Norbert Schneider | 02/2019 |
3450 | (91*10^112098-1)/9 | 112100 | Robert Price | 04/2015 |
3451 | 2^372310-85061 | 112077 | Paul Bourdelais | 02/2012 |
3452 | (10^112067+53)/9 | 112067 | Serge Batalov | 07/2015 |
3453 | (2^372237+2082)/2 | 112055 | Stefano Morozzi | 09/2020 |
3454 | 75^59760+2 | 112054 | Lelio R Paula | 08/2013 |
3455 | (2^372200+1710)/2 | 112044 | Stefano Morozzi | 09/2020 |
3456 | (2^372181+2250)/2 | 112038 | Stefano Morozzi | 01/2021 |
3457 | (2^372161+2574)/2 | 112032 | Stefano Morozzi | 09/2020 |
3458 | 8*10^112018-49 | 112019 | Robert Price | 08/2019 |
3460 | 2^372108+74221 | 112016 | Enzo Doria | 04/2016 |
3459 | (2^372109+2046)/2 | 112016 | Stefano Morozzi | 09/2020 |
3461 | (184*10^112007-1)/3 | 112009 | Robert Price | 02/2020 |
3462 | 5*10^112003+81 | 112004 | Robert Price | 03/2019 |
3463 | 215^48018+48018^215 | 112000 | Norbert Schneider | 10/2018 |
3464 | 2^371970-459491 | 111975 | Enzo Doria | 05/2015 |
3465 | 41^69427-40^69427 | 111971 | Robert Price | 07/2012 |
3466 | (2^371782+358)/2 | 111918 | Stefano Morozzi | 01/2021 |
3467 | (10^111902+71)/9 | 111902 | Serge Batalov | 07/2015 |
3468 | (2^371625+1610)/2 | 111870 | Stefano Morozzi | 09/2020 |
3469 | 2^371440-112865 | 111815 | Enzo Doria | 06/2014 |
3470 | 2^371426-459161 | 111811 | Enzo Doria | 05/2015 |
3471 | (94*10^111770+11)/3 | 111772 | Robert Price | 04/2020 |
3472 | 2^371276-461267 | 111766 | Enzo Doria | 07/2015 |
3473 | 6*10^111760+43 | 111761 | Robert Price | 07/2020 |
3474 | 98079!4-1024 | 111747 | Robert Price | 11/2019 |
3475 | (2^371186+3162)/2 | 111738 | Stefano Morozzi | 09/2020 |
3476 | 2^371098-113261 | 111712 | Enzo Doria | 07/2014 |
3477 | (7*10^111706-61)/9 | 111706 | Erik Branger | 01/2011 |
3478 | 2^371044+1869391 | 111696 | Enzo Doria | 05/2016 |
3479 | 26310^17581+17581^26310 | 111688 | Hans Havermann | 01/2021 |
3480 | 88^57416+87 | 111645 | Norbert Schneider | 11/2015 |
3481 | 2^370829-109693 | 111631 | Enzo Doria | 04/2014 |
3482 | 15^94906-2 | 111619 | Lelio R Paula | 11/2013 |
3483 | (2^370726+1198)/2 | 111600 | Stefano Morozzi | 09/2020 |
3484 | 2^370695-4627 | 111591 | Yves Minet | 05/2019 |
3485 | 7*10^111579-87 | 111580 | Robert Price | 06/2020 |
3487 | 2^370643-37 | 111575 | Yves Minet | 01/2012 |
3486 | (2^370655-5)/2439 | 111575 | Lelio R Paula | 03/2014 |
3488 | 2^370610-3911 | 111565 | Yves Minet | 04/2017 |
3489 | 286^45405+45405^286 | 111532 | Norbert Schneider | 04/2015 |
3490 | 4*5^159563-9 | 111531 | Lelio R Paula | 11/2014 |
3491 | (4^185225+23)/3 | 111517 | Robert Price | 10/2018 |
3492 | 2^370393-2797093 | 111500 | Enzo Doria | 03/2016 |
3493 | (21*2^370382-1)/629 | 111495 | Lelio R Paula | 03/2014 |
3494 | (2^370371+554)/2 | 111493 | Stefano Morozzi | 09/2020 |
3495 | 2^370292-2791979 | 111469 | Enzo Doria | 03/2016 |
3496 | 2^370177-461551 | 111435 | Enzo Doria | 07/2015 |
3497 | (2^370169+1706)/2 | 111432 | Stefano Morozzi | 09/2020 |
3498 | 109^54683-108 | 111413 | Norbert Schneider | 02/2017 |
3499 | (2^370043+1010)/2 | 111394 | Stefano Morozzi | 09/2020 |
3500 | 32961^2396+2396^32961 | 111392 | Hans Havermann | 10/2020 |