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) !!!
Submit your PRPs
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| Rank |
Probable prime |
Digits (> 50000) |
Who ? |
When ? |
| 3251 | (10^180683-1)*9/(10^643-1)/(10^281-1) | 179760 | Robert Price | 07/2016 |
| 3252 | (3^376657-2^376657)/22301107657 | 179701 | Mike Oakes | 04/2017 |
| 3253 | (29*10^179697+223)/9 | 179698 | Robert Price | 09/2018 |
| 3254 | U(1289697) | 179687 | Serge Batalov | 07/2024 |
| 3256 | 2^596884+73 | 179680 | Stefano Morozzi | 04/2021 |
| 3255 | (11*10^179679+13)/3 | 179680 | Robert Price | 03/2022 |
| 3257 | (68*10^179656+7)/3 | 179658 | Robert Price | 01/2020 |
| 3258 | (10^179643+173)/3 | 179643 | Robert Price | 05/2021 |
| 3259 | 2^596690-461261 | 179622 | Enzo Doria | 07/2015 |
| 3260 | 2^596684-113939 | 179620 | Enzo Doria | 10/2014 |
| 3261 | 2^596642+39 | 179608 | Jon Grantham | 08/2023 |
| 3262 | 2*10^179602+63 | 179603 | Robert Price | 05/2018 |
| 3263 | 32262^39815-39815^32262 | 179514 | Frank Doornink | 02/2024 |
| 3264 | (25*10^179488-403)/9 | 179489 | Robert Price | 01/2021 |
| 3265 | 7*10^179460+99 | 179461 | Robert Price | 05/2021 |
| 3266 | 2^596127+74333 | 179453 | Enzo Doria | 07/2023 |
| 3267 | 3^375887-112696 | 179344 | Enzo Doria | 07/2014 |
| 3268 | 4*10^179265-71 | 179266 | Robert Price | 12/2018 |
| 3269 | 2*10^179193+83 | 179194 | Robert Price | 12/2021 |
| 3270 | 2^595144-1850429 | 179157 | Enzo Doria | 08/2015 |
| 3271 | 2^595122-5 | 179150 | Stefano Morozzi | 07/2022 |
| 3272 | 3*10^179138+89 | 179139 | Robert Price | 10/2018 |
| 3273 | (52*10^179062-133)/9 | 179063 | Robert Price | 09/2021 |
| 3274 | 2^594531+82253 | 178972 | Enzo Doria | 09/2016 |
| 3275 | (664*10^178939-61)/9 | 178941 | Robert Price | 09/2021 |
| 3276 | 2^594260-2787899 | 178891 | Enzo Doria | 11/2015 |
| 3277 | (14*10^178822+67)/9 | 178823 | Robert Price | 05/2022 |
| 3278 | (29*10^178732-119)/9 | 178733 | Robert Price | 12/2021 |
| 3279 | 2^593631+101381 | 178701 | Jorge Menendez | 04/2018 |
| 3280 | 10^178695+136719 | 178696 | Patrick De Geest | 05/2016 |
| 3281 | 10^178695-71 | 178695 | Serge Batalov | 12/2014 |
| 3282 | (2*10^178649-161)/3 | 178649 | Robert Price | 12/2020 |
| 3283 | (691^62903-1)/690 | 178609 | Michael Stocker | 05/2020 |
| 3284 | (2^148330+1)^4-2 | 178608 | Cletus Emmanuel | 12/2003 |
| 3285 | 2^593242+28309 | 178584 | Enzo Doria | 01/2024 |
| 3286 | 2^593222-14145 | 178578 | Jorge Menendez | 10/2017 |
| 3287 | 7*10^178566+33 | 178567 | Robert Price | 06/2020 |
| 3288 | 2^593178+1867219 | 178565 | Enzo Doria | 03/2016 |
| 3289 | 2^593152+22441 | 178557 | Enzo Doria | 09/2023 |
| 3290 | (38*10^178456+151)/9 | 178457 | Robert Price | 03/2022 |
| 3291 | (302*10^178369+13)/9 | 178371 | Robert Price | 02/2020 |
| 3292 | (26*10^178296-113)/3 | 178297 | Robert Price | 06/2020 |
| 3293 | (18^142031-1)/17 | 178287 | Paul Bourdelais | 03/2010 |
| 3294 | (13*10^178233+137)/3 | 178234 | Robert Price | 06/2021 |
| 3295 | (5*10^178206+91)/3 | 178207 | Robert Price | 04/2018 |
| 3296 | 2^591872-291929 | 178172 | Enzo Doria | 07/2015 |
| 3297 | 2^591682-286871 | 178115 | Enzo Doria | 07/2015 |
| 3298 | (7^210761-5^210761)/2 | 178114 | Jon Grantham | 08/2023 |
| 3299 | 4*10^178091+9 | 178092 | Robert Price | 05/2015 |
| 3300 | 2^591435+335931 | 178040 | Enzo Doria | 09/2024 |
| 3301 | 410747#+478483 | 178007 | Makoto Morimoto | 02/2016 |
| 3302 | 3*10^177872+73 | 177873 | Robert Price | 08/2018 |
| 3303 | 2^590762-396765 | 177838 | Enzo Doria | 08/2013 |
| 3304 | (136*10^177754-1)/9 | 177756 | Serge Batalov | 12/2014 |
| 3305 | ReverseConsecutiveIntegerPrime(37765) | 177719 | Eric W. Weisstein | 04/2010 |
| 3306 | Smr(37765) | 177719 | Eric W. Weisstein | 12/2021 |
| 3307 | 598^128^209+209^128^598 | 177594 | Henri Lifchitz | 03/2025 |
| 3308 | (266*10^177577-11)/3 | 177579 | Robert Price | 04/2020 |
| 3309 | Phi(100093,2^7) | 177512 | Henri Lifchitz | 10/2025 |
| 3310 | (11*10^177466-113)/3 | 177467 | Robert Price | 10/2018 |
| 3311 | 3*10^177373-97 | 177374 | Robert Price | 07/2018 |
| 3312 | 2^589204+1881151 | 177369 | Enzo Doria | 11/2016 |
| 3313 | 2^589132-69479 | 177347 | Enzo Doria | 01/2016 |
| 3314 | (3^371666)/(371666^3) | 177314 | Henri Lifchitz | 05/2025 |
| 3315 | (101*10^177302-17)/3 | 177304 | Robert Price | 02/2020 |
| 3316 | 2^588912-113519 | 177281 | Enzo Doria | 08/2014 |
| 3317 | (682*10^177266-7)/9 | 177268 | Robert Price | 08/2021 |
| 3318 | 15^150706+2 | 177245 | Robert Price | 06/2015 |
| 3319 | 2^588403+335931 | 177127 | Enzo Doria | 09/2024 |
| 3320 | (31208^39601-1)/(31208^199-1) | 177084 | Ryan Propper | 11/2023 |
| 3321 | 2^588194-1861001 | 177065 | Enzo Doria | 11/2015 |
| 3322 | 5*10^177016-37 | 177017 | Robert Price | 11/2021 |
| 3323 | fibonacci(846926)-4 | 176997 | Jon Grantham | 08/2023 |
| 3324 | (23*10^176964-47)/3 | 176965 | Robert Price | 09/2019 |
| 3325 | 20530^41031+41031^20530 | 176942 | Norbert Schneider | 07/2022 |
| 3326 | (19*10^176932+521)/9 | 176933 | Robert Price | 05/2018 |
| 3327 | (8*10^176908+43)/3 | 176909 | Robert Price | 06/2018 |
| 3328 | Phi(146979,2^7) | 176905 | Henri Lifchitz | 10/2025 |
| 3329 | (85*10^176900+203)/9 | 176901 | Robert Price | 09/2020 |
| 3330 | 2^587560-113699 | 176874 | Enzo Doria | 09/2014 |
| 3331 | 20596^40995+40995^20596 | 176844 | Norbert Schneider | 07/2022 |
| 3332 | (101*10^176815-17)/3 | 176817 | Robert Price | 02/2020 |
| 3333 | (25*10^176715-37)/3 | 176716 | Robert Price | 10/2019 |
| 3334 | 5^252763-12862 | 176674 | Enzo Doria | 05/2014 |
| 3335 | 38442^38531+38531^38442 | 176658 | Norbert Schneider | 09/2022 |
| 3336 | 2^586829+23961 | 176654 | Enzo Doria | 10/2023 |
| 3337 | 2^586804+81721 | 176646 | Enzo Doria | 08/2016 |
| 3338 | (10^176584+77)/3 | 176584 | Robert Price | 01/2018 |
| 3339 | 2^586570+1874419 | 176576 | Enzo Doria | 06/2016 |
| 3340 | (157*10^176482-1)/3 | 176484 | Robert Price | 01/2022 |
| 3341 | 5*2^586202+3 | 176466 | Jon Grantham | 07/2023 |
| 3342 | 27122^39799-39799^27122 | 176442 | Frank Doornink | 01/2024 |
| 3343 | 2^585994-3311 | 176402 | Yves Minet | 12/2014 |
| 3344 | 8*10^176383-81 | 176384 | Robert Price | 02/2021 |
| 3345 | (23*10^176357+61)/3 | 176358 | Robert Price | 01/2022 |
| 3346 | 2^585832-112799 | 176354 | Enzo Doria | 07/2014 |
| 3347 | fibonacci(843383)+4 | 176257 | Jon Grantham | 08/2023 |
| 3348 | (29*10^176240+49)/3 | 176241 | Robert Price | 09/2021 |
| 3349 | (2^585387-11)/3 | 176219 | Stefano Morozzi | 08/2022 |
| 3350 | 2^585336-461039 | 176204 | Enzo Doria | 06/2015 |
| 3351 | (25*10^176164+143)/3 | 176165 | Robert Price | 08/2020 |
| 3352 | 2^585095-1848127 | 176132 | Enzo Doria | 08/2015 |
| 3353 | (5*10^176101-311)/9 | 176101 | Robert Price | 12/2021 |
| 3354 | 6*10^176092+77 | 176093 | Robert Price | 06/2020 |
| 3355 | 2^584950-115451 | 176088 | Enzo Doria | 07/2015 |
| 3356 | (646*10^176044+11)/9 | 176046 | Robert Price | 10/2021 |
| 3357 | (7^208283-5^208283)/2 | 176020 | Jon Grantham | 08/2023 |
| 3358 | 2^584671-3697 | 176004 | Yves Minet | 03/2017 |
| 3359 | (13*10^175965-37)/3 | 175966 | Robert Price | 01/2019 |
| 3360 | 31728^39089-39089^31728 | 175957 | Frank Doornink | 01/2024 |
| 3361 | 2*10^175860-87 | 175861 | Robert Price | 04/2018 |
| 3362 | 27009^39670-39670^27009 | 175798 | Frank Doornink | 01/2024 |
| 3363 | (86*10^175774-221)/9 | 175775 | Robert Price | 09/2019 |
| 3364 | 2^583614+216037 | 175686 | Enzo Doria | 11/2023 |
| 3365 | 9*10^175654+47 | 175655 | Robert Price | 01/2022 |
| 3366 | 25430^39869-39869^25430 | 175637 | Frank Doornink | 01/2024 |
| 3367 | 10^175608+2971 | 175609 | Mike Oakes | 05/2010 |
| 3368 | 10^175608-55829 | 175608 | Patrick De Geest | 09/2013 |
| 3369 | 2^583051+1879793 | 175516 | Enzo Doria | 09/2016 |
| 3370 | (76*10^175514+491)/9 | 175515 | Robert Price | 10/2021 |
| 3372 | 2^582860-23829 | 175459 | Enzo Doria | 08/2013 |
| 3371 | (200^76253-1)/199 | 175459 | Anonymous | 06/2024 |
| 3373 | (8*10^175428-53)/9 | 175428 | Serge Batalov | 11/2022 |
| 3374 | (28*10^175417-67)/3 | 175418 | Robert Price | 10/2020 |
| 3375 | 2^582670-23913 | 175402 | Enzo Doria | 08/2013 |
| 3376 | 2^582573-29563 | 175372 | Enzo Doria | 08/2015 |
| 3377 | 2^582435-1862767 | 175331 | Enzo Doria | 12/2015 |
| 3378 | 10^175295-40000001 | 175295 | Robert Price | 07/2022 |
| 3379 | 31093^39018-39018^31093 | 175295 | Frank Doornink | 01/2024 |
| 3380 | (11*10^175250-149)/3 | 175251 | Robert Price | 09/2018 |
| 3381 | (29*10^175236+31)/3 | 175237 | Robert Price | 04/2021 |
| 3382 | 2^582072-1848599 | 175222 | Enzo Doria | 08/2015 |
| 3383 | (404267#+465)/(3^4*5^2*31) | 175208 | Makoto Morimoto | 06/2019 |
| 3384 | Phi(262767,10)/259613797 | 175168 | Makoto Kamada | 12/2022 |
| 3385 | (2^581871-15)/7 | 175160 | Stefano Morozzi | 08/2022 |
| 3386 | (5*10^175117-311)/9 | 175117 | Robert Price | 12/2021 |
| 3387 | 31193^38966-38966^31193 | 175116 | Frank Doornink | 01/2024 |
| 3388 | 2^581695+225893 | 175108 | Enzo Doria | 08/2024 |
| 3389 | 10^175093-89 | 175093 | Riley Fisher | 10/2023 |
| 3390 | (220221^32768+1)/2 | 175075 | Serge Batalov | 03/2018 |
| 3391 | 2^581560+1874911 | 175068 | Enzo Doria | 06/2016 |
| 3392 | (25*10^174987-97)/9 | 174988 | Robert Price | 01/2022 |
| 3393 | (75^93307-1)/74 | 174955 | Paul Bourdelais | 09/2017 |
| 3394 | 26446^39555-39555^26446 | 174927 | Frank Doornink | 03/2024 |
| 3395 | 31522^38883-38883^31522 | 174920 | Frank Doornink | 01/2024 |
| 3396 | (27398^39601-1)/(27398^199-1) | 174855 | Ryan Propper | 11/2023 |
| 3397 | (861^59557-1)/860 | 174798 | Jinyuan Wang | 11/2023 |
| 3398 | 31492^38859-38859^31492 | 174796 | Frank Doornink | 01/2024 |
| 3399 | (47*10^174790+493)/9 | 174791 | Robert Price | 03/2019 |
| 3400 | 2^580528-114035 | 174757 | Enzo Doria | 11/2014 |
| 3401 | (19*10^174754+737)/9 | 174755 | Robert Price | 07/2018 |
| 3402 | V(1045270)/4181081/12543241/81333115129561 | 174730 | Serge Batalov | 07/2024 |
| 3403 | (35*10^174704-377)/9 | 174705 | Robert Price | 07/2018 |
| 3404 | 23181^40018-40018^23181 | 174684 | Norbert Schneider | 07/2025 |
| 3406 | (49*10^174681+293)/9 | 174682 | Robert Price | 03/2021 |
| 3405 | 6*10^174681+11 | 174682 | Robert Price | 07/2022 |
| 3407 | 3*10^174665-97 | 174666 | Robert Price | 07/2018 |
| 3408 | 2^580046-459161 | 174612 | Enzo Doria | 05/2015 |
| 3409 | 2^579992-39779 | 174595 | Enzo Doria | 09/2015 |
| 3410 | 25613^39582-39582^25613 | 174496 | Frank Doornink | 02/2024 |
| 3411 | (29*10^174481-149)/3 | 174482 | Robert Price | 08/2022 |
| 3412 | (4*10^174423+101)/3 | 174424 | Robert Price | 11/2021 |
| 3413 | (2^579371-17)/3 | 174408 | Stefano Morozzi | 08/2022 |
| 3414 | (2*10^174301-71)/3 | 174301 | Robert Price | 12/2017 |
| 3415 | 14^152048+3 | 174267 | Paul Bourdelais | 12/2020 |
| 3416 | (2^578696+1)/257 | 174203 | Lelio R Paula | 09/2015 |
| 3417 | 2^578656-1851809 | 174193 | Enzo Doria | 08/2015 |
| 3418 | 2^578583-113917 | 174171 | Enzo Doria | 10/2014 |
| 3419 | 2^578434-83 | 174126 | Yves Minet | 07/2017 |
| 3420 | 15^148048-2 | 174118 | Robert Price | 06/2015 |
| 3421 | 5*10^174101+81 | 174102 | Robert Price | 03/2019 |
| 3422 | 2^578346+2794249 | 174100 | Enzo Doria | 03/2016 |
| 3423 | 10^173947+1372429 | 173948 | Patrick De Geest | 08/2021 |
| 3425 | 10^173947-500000001 | 173947 | Robert Price | 12/2020 |
| 3424 | 10^173947-286391 | 173947 | Patrick De Geest | 08/2021 |
| 3426 | 5*10^173904-13 | 173905 | Robert Price | 01/2019 |
| 3427 | (8*10^173862-59)/3 | 173863 | Robert Price | 08/2021 |
| 3428 | (71*10^173854-287)/9 | 173855 | Robert Price | 09/2019 |
| 3429 | 20144^40391-40391^20144 | 173849 | Norbert Schneider | 10/2025 |
| 3430 | (43*10^173826+101)/9 | 173827 | Robert Price | 11/2018 |
| 3432 | 2*10^173809-87 | 173810 | Robert Price | 04/2018 |
| 3431 | (17^141263-1)/27122512 | 173810 | Paul Vanderveen | 05/2025 |
| 3433 | (2^577372-23)/7 | 173806 | Stefano Morozzi | 08/2022 |
| 3434 | (L!41532)/2 | 173772 | Serge Batalov | 07/2017 |
| 3435 | 2^577146-49877 | 173739 | Enzo Doria | 10/2015 |
| 3436 | 2^576889+49727 | 173661 | Enzo Doria | 09/2016 |
| 3437 | (19^173604)%(10^173604) | 173604 | Henri Lifchitz | 05/2025 |
| 3438 | 20048^40347-40347^20048 | 173576 | Norbert Schneider | 10/2025 |
| 3439 | (167*10^173571-17)/3 | 173573 | Robert Price | 03/2020 |
| 3440 | 37803^37916+37916^37803 | 173562 | Norbert Schneider | 06/2022 |
| 3441 | 5*10^173534+29 | 173535 | Robert Price | 12/2020 |
| 3442 | (2^576551-1)/4612409/64758208321/242584327930759 | 173528 | Dario Alpern | 06/2014 |
| 3443 | 4*10^173509+99 | 173510 | Robert Price | 12/2020 |
| 3444 | (13*10^173467-37)/3 | 173468 | Robert Price | 01/2019 |
| 3445 | 21518^40021-40021^21518 | 173404 | Norbert Schneider | 07/2025 |
| 3446 | 2^576016-113919 | 173399 | Enzo Doria | 11/2014 |
| 3447 | 2^575911-1849387 | 173367 | Enzo Doria | 08/2015 |
| 3448 | 21554^40005-40005^21554 | 173363 | Norbert Schneider | 04/2025 |
| 3449 | 2^575879+215843 | 173357 | Enzo Doria | 11/2023 |
| 3450 | 2^575871+28733 | 173355 | Enzo Doria | 01/2024 |
| 3451 | 515^63898+63898^515 | 173280 | Norbert Schneider | 12/2022 |
| 3452 | 2^575614+9 | 173278 | Stefano Morozzi | 04/2021 |
| 3453 | 2^575606-219533 | 173275 | Enzo Doria | 03/2024 |
| 3454 | (16*10^173251+107)/3 | 173252 | Robert Price | 03/2019 |
| 3455 | 26618^39149-39149^26618 | 173242 | Frank Doornink | 03/2024 |
| 3456 | 2^575196-114053 | 173152 | Enzo Doria | 11/2014 |
| 3457 | 7*10^173135-53 | 173136 | Robert Price | 06/2020 |
| 3458 | (301*10^173047+23)/9 | 173049 | Robert Price | 06/2021 |
| 3459 | 2^574591-115387 | 172970 | Enzo Doria | 07/2015 |
| 3460 | 25210^39297-39297^25210 | 172969 | Frank Doornink | 12/2023 |
| 3461 | (202*10^172940-13)/9 | 172942 | Robert Price | 01/2020 |
| 3462 | 45^104608+104608^45 | 172940 | Norbert Schneider | 06/2021 |
| 3463 | 2^574328+81121 | 172890 | Enzo Doria | 07/2016 |
| 3464 | (115*10^172862-1)/3 | 172864 | Serge Batalov | 12/2014 |
| 3465 | 3*10^172812+13 | 172813 | Robert Price | 07/2018 |
| 3466 | 4*5^247217-9 | 172798 | Lelio R Paula | 10/2015 |
| 3467 | 6*10^172772-31 | 172773 | Robert Price | 07/2020 |
| 3468 | 69061#+(467617#)/(69061#) | 172766 | Makoto Morimoto | 10/2017 |
| 3469 | (34*10^172693-547)/9 | 172694 | Robert Price | 03/2021 |
| 3470 | 5*10^172692+57 | 172693 | Robert Price | 12/2018 |
| 3471 | 2^573668-2782319 | 172692 | Enzo Doria | 09/2015 |
| 3472 | 2^573444-1855109 | 172624 | Enzo Doria | 09/2015 |
| 3473 | 25500^39173-39173^25500 | 172618 | Frank Doornink | 02/2024 |
| 3474 | 27180^38917-38917^27180 | 172568 | Frank Doornink | 01/2024 |
| 3475 | (14*10^172532-29)/3 | 172533 | Robert Price | 09/2021 |
| 3476 | 3^361608+2 | 172531 | Robert Price | 08/2014 |
| 3477 | (54^99563+1)/55 | 172481 | Paul Bourdelais | 11/2017 |
| 3478 | 2^572592-114893 | 172368 | Enzo Doria | 05/2015 |
| 3479 | 4*10^172354-39 | 172355 | Robert Price | 09/2020 |
| 3480 | 37535^37674+37674^37535 | 172338 | Norbert Schneider | 05/2022 |
| 3481 | (11*10^172322-119)/9 | 172323 | Robert Price | 11/2021 |
| 3482 | 5^246497-4^246497 | 172295 | Jean-Louis Charton | 11/2008 |
| 3483 | (38^109037-1)/37 | 172254 | Paul Bourdelais | 10/2017 |
| 3484 | (4*10^172241+89)/3 | 172242 | Robert Price | 08/2021 |
| 3485 | 3*10^172219+73 | 172220 | Robert Price | 08/2018 |
| 3486 | 2^571810+28549 | 172132 | Enzo Doria | 01/2024 |
| 3487 | A352736(15) | 172032 | Kellen Shenton | 05/2022 |
| 3488 | 32298^38149-38149^32298 | 172021 | Frank Doornink | 02/2024 |
| 3489 | 2^571300+25861 | 171979 | Enzo Doria | 11/2023 |
| 3490 | (36^110503+1)/37 | 171975 | Lelio R Paula | 04/2017 |
| 3491 | 8*10^171934+71 | 171935 | Robert Price | 05/2022 |
| 3492 | (11*10^171933+601)/9 | 171934 | Robert Price | 02/2022 |
| 3493 | (22*10^171929-73)/3 | 171930 | Robert Price | 04/2019 |
| 3494 | (49*10^171892-319)/9 | 171893 | Robert Price | 10/2021 |
| 3495 | 4*10^171891-63 | 171892 | Robert Price | 10/2018 |
| 3496 | 67*10^171882-9 | 171884 | Robert Price | 03/2021 |
| 3497 | 2^570980-2782289 | 171883 | Enzo Doria | 09/2015 |
| 3498 | (25*10^171849-37)/3 | 171850 | Robert Price | 10/2019 |
| 3499 | 10^171832+387909 | 171833 | Patrick De Geest | 08/2021 |
| 3500 | 10^171832-10001 | 171832 | Robert Price | 02/2021 |
