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 ? |
| 3001 | Phi(293127,10) | 194041 | Makoto Kamada | 12/2023 |
| 3002 | 2*10^193981-43 | 193982 | Robert Price | 04/2018 |
| 3003 | 2^644382+75169 | 193979 | Enzo Doria | 04/2016 |
| 3004 | 2^644304-1861409 | 193955 | Enzo Doria | 11/2015 |
| 3005 | (29*10^193950-23)/3 | 193951 | Robert Price | 03/2022 |
| 3006 | 5*10^193907-9 | 193908 | Robert Price | 05/2015 |
| 3007 | 2^644043+1876703 | 193877 | Enzo Doria | 07/2016 |
| 3008 | 2*10^193841-63 | 193842 | Robert Price | 01/2022 |
| 3009 | 2^643735+1879733 | 193784 | Enzo Doria | 09/2016 |
| 3011 | (5^277183+1)/6 | 193742 | Paul Bourdelais | 07/2010 |
| 3010 | Phi(277183,-5) | 193742 | Henri Lifchitz | 12/2025 |
| 3012 | 9*10^193712+71 | 193713 | Robert Price | 08/2020 |
| 3013 | 2^643216-286565 | 193628 | Enzo Doria | 07/2015 |
| 3014 | 2^643204+215011 | 193624 | Enzo Doria | 11/2023 |
| 3015 | 10^193566+1126339 | 193567 | Patrick De Geest | 09/2022 |
| 3017 | 10^193566-10000000001 | 193566 | Robert Price | 04/2021 |
| 3016 | 10^193566-454041 | 193566 | Patrick De Geest | 08/2021 |
| 3019 | 2^642888-24965 | 193529 | Enzo Doria | 09/2013 |
| 3018 | 2^642889+25367 | 193529 | Enzo Doria | 11/2023 |
| 3020 | Phi(50157,p(50157)) | 193510 | Henri Lifchitz | 12/2025 |
| 3021 | 2^642448-114825 | 193397 | Enzo Doria | 05/2015 |
| 3022 | 2^642437-1856623 | 193393 | Enzo Doria | 10/2015 |
| 3023 | 2^642039-59797 | 193273 | Enzo Doria | 11/2015 |
| 3024 | 2^641987+39 | 193258 | Jon Grantham | 08/2023 |
| 3025 | 10^193230+513861 | 193231 | Patrick De Geest | 07/2018 |
| 3026 | 10^193230-8001 | 193230 | Robert Price | 03/2018 |
| 3027 | (62*10^193225-233)/9 | 193226 | Robert Price | 02/2022 |
| 3028 | Phi(86312,13#) | 193216 | Henri Lifchitz | 02/2026 |
| 3029 | 2^641692-286349 | 193169 | Enzo Doria | 07/2015 |
| 3030 | 2^641690-24737 | 193168 | Enzo Doria | 09/2013 |
| 3031 | 22268^44421+44421^22268 | 193129 | Norbert Schneider | 06/2023 |
| 3032 | 2^641514+1876549 | 193115 | Enzo Doria | 07/2016 |
| 3033 | 2^640888-2785619 | 192927 | Enzo Doria | 10/2015 |
| 3034 | 2^640874-2783741 | 192923 | Enzo Doria | 09/2015 |
| 3035 | 10^192922+192207 | 192923 | Patrick De Geest | 08/2021 |
| 3036 | 10^192922-7001 | 192922 | Robert Price | 04/2021 |
| 3037 | 2^640854+25 | 192917 | Stefano Morozzi | 05/2021 |
| 3038 | 34*10^192868-9 | 192870 | Robert Price | 06/2020 |
| 3039 | (46*10^192841+683)/9 | 192842 | Robert Price | 03/2019 |
| 3040 | 2^640181+467 | 192714 | Lelio R Paula | 08/2013 |
| 3041 | 2^640072+27541 | 192681 | Enzo Doria | 01/2024 |
| 3042 | (10^192700)/(192700^10) | 192648 | Henri Lifchitz | 05/2025 |
| 3043 | 2^639536-1850249 | 192520 | Enzo Doria | 08/2015 |
| 3044 | 40714<>40712<>...<>1086421357911<>...<>40711<>40713 | 192464 | Hans Havermann | 12/2019 |
| 3045 | 5*10^192422-43 | 192423 | Robert Price | 01/2019 |
| 3046 | 2^639110-2792441 | 192392 | Enzo Doria | 02/2016 |
| 3047 | (241^80761-1)/240 | 192372 | Anonymous | 04/2024 |
| 3048 | 2^638955-29827 | 192345 | Enzo Doria | 08/2015 |
| 3049 | 15^163539+14 | 192337 | Alfred Reich | 10/2023 |
| 3050 | 2^638713-2784223 | 192272 | Enzo Doria | 10/2015 |
| 3051 | (10^192216+7)/149 | 192214 | Alfred Reich | 07/2021 |
| 3052 | 2^638490-113507 | 192205 | Enzo Doria | 08/2014 |
| 3053 | Phi(75881,561) | 192120 | Henri Lifchitz | 12/2025 |
| 3054 | 2^638189-39943 | 192115 | Enzo Doria | 09/2015 |
| 3055 | 2^638110+1877119 | 192091 | Enzo Doria | 07/2016 |
| 3056 | MillsLike(2,1,4,12,4,22,12,114,4,138,142,2956,6388,5248,17532,96930,83782,1464,897448,300832)+82440 | 192083 | Sergey Batalov | 05/2024 |
| 3057 | MillsLike(2,1,4,12,4,22,12,114,4,138,142,2956,6388,5248,17532,96930,83782,1464,897448,300832) | 192083 | Sergey Batalov | 05/2024 |
| 3058 | 2^638066-2785211 | 192078 | Enzo Doria | 10/2015 |
| 3059 | (35*10^191973-341)/9 | 191974 | Robert Price | 08/2021 |
| 3060 | (112*10^191900+17)/3 | 191902 | Robert Price | 03/2020 |
| 3061 | 2^637475-115309 | 191900 | Enzo Doria | 07/2015 |
| 3062 | (68*10^191754-113)/9 | 191755 | Robert Price | 03/2021 |
| 3063 | 2^636930-11 | 191736 | Stefano Morozzi | 07/2022 |
| 3064 | (47*10^191732+259)/9 | 191733 | Robert Price | 03/2022 |
| 3065 | 41459^41522-41522^41459 | 191733 | Norbert Schneider | 04/2025 |
| 3066 | 2^636661+1876577 | 191655 | Enzo Doria | 07/2016 |
| 3067 | (9^200807-2^200807)/7 | 191618 | Jon Grantham | 07/2023 |
| 3068 | (34^125101-1)/33 | 191589 | Paul Bourdelais | 11/2017 |
| 3069 | fibonacci(916540)-2 | 191546 | Jon Grantham | 08/2023 |
| 3070 | 2^636177-1855123 | 191509 | Enzo Doria | 09/2015 |
| 3071 | 2^636158-113627 | 191503 | Enzo Doria | 08/2014 |
| 3072 | (37*25^136966+63)/4 | 191472 | Curtis Bright | 05/2015 |
| 3073 | 2^636035+79823 | 191466 | Enzo Doria | 06/2016 |
| 3074 | 2^636011-1860847 | 191459 | Enzo Doria | 11/2015 |
| 3075 | 2^635926-1860611 | 191433 | Enzo Doria | 11/2015 |
| 3076 | 3*2^635755+5 | 191382 | Jon Grantham | 07/2023 |
| 3077 | 3*10^191152-7 | 191153 | Robert Price | 08/2015 |
| 3078 | 2^634883-1865677 | 191119 | Enzo Doria | 02/2016 |
| 3079 | 2^634801+1871957 | 191095 | Enzo Doria | 05/2016 |
| 3080 | (77*10^191080+157)/9 | 191081 | Robert Price | 05/2022 |
| 3081 | (35*10^191073+1)/9 | 191074 | Serge Batalov | 07/2010 |
| 3082 | 9*10^191018-41 | 191019 | Robert Price | 08/2020 |
| 3083 | 2^634470-59861 | 190995 | Enzo Doria | 11/2015 |
| 3084 | 2^634368-29519 | 190964 | Enzo Doria | 08/2015 |
| 3085 | (19*10^190935+11)/3 | 190936 | Robert Price | 07/2023 |
| 3086 | 2^634188-350609 | 190910 | Enzo Doria | 06/2015 |
| 3087 | (65*10^190899+493)/9 | 190900 | Robert Price | 08/2021 |
| 3088 | 405^73192+404 | 190845 | Alfred Reich | 04/2026 |
| 3089 | (62*10^190795-413)/9 | 190796 | Robert Price | 09/2021 |
| 3090 | (26*10^190789-131)/3 | 190790 | Robert Price | 10/2019 |
| 3091 | (13*10^190732-61)/3 | 190733 | Robert Price | 11/2018 |
| 3092 | 3*2^633524+5 | 190711 | Jon Grantham | 07/2023 |
| 3093 | (7*10^190683+143)/3 | 190684 | Robert Price | 03/2018 |
| 3094 | 2^633088-113879 | 190579 | Enzo Doria | 10/2014 |
| 3095 | 2^633064+1874911 | 190572 | Enzo Doria | 06/2016 |
| 3096 | 5*3^399402+2^399402 | 190564 | Jon Grantham | 08/2023 |
| 3097 | (43*10^190547+83)/9 | 190548 | Robert Price | 11/2018 |
| 3098 | 99*10^190543+7 | 190545 | Robert Price | 05/2021 |
| 3099 | (299*10^190533+1)/3 | 190535 | Robert Price | 04/2020 |
| 3100 | (1744*10^190519-1747)/3 | 190522 | Robert Price | 12/2020 |
| 3101 | 2^632815-15 | 190497 | Stefano Morozzi | 07/2022 |
| 3102 | 2^632795-351205 | 190491 | Enzo Doria | 06/2015 |
| 3103 | 41171^41272+41272^41171 | 190454 | Norbert Schneider | 05/2023 |
| 3104 | (16*10^190420-19)/3 | 190421 | Robert Price | 02/2019 |
| 3105 | (43*10^190351+101)/9 | 190352 | Robert Price | 11/2018 |
| 3106 | 2^632156+78391 | 190298 | Enzo Doria | 06/2016 |
| 3107 | 97*10^190295-3 | 190297 | Robert Price | 05/2020 |
| 3108 | 2^632082-1851971 | 190276 | Enzo Doria | 08/2015 |
| 3109 | 2^632020+1880311 | 190257 | Enzo Doria | 09/2016 |
| 3110 | (5*10^190242-77)/9 | 190242 | Robert Price | 04/2021 |
| 3111 | 7*10^190210+13 | 190211 | Robert Price | 03/2021 |
| 3112 | 2^631637-462373 | 190142 | Enzo Doria | 08/2015 |
| 3113 | 2*10^190126-21 | 190127 | Robert Price | 09/2020 |
| 3114 | (28*10^190105+191)/3 | 190106 | Robert Price | 02/2020 |
| 3115 | 2*10^190093+33 | 190094 | Robert Price | 11/2021 |
| 3116 | (68*10^190053-113)/9 | 190054 | Robert Price | 03/2021 |
| 3117 | 2^631183+213413 | 190006 | Enzo Doria | 09/2023 |
| 3118 | 2^631122-39251 | 189987 | Enzo Doria | 08/2015 |
| 3119 | (26*10^189928-89)/9 | 189929 | Robert Price | 06/2018 |
| 3120 | 2^630839-97 | 189902 | Yves Minet | 07/2017 |
| 3121 | 99*10^189876-7 | 189878 | Robert Price | 07/2020 |
| 3122 | 41147^41147+30^30 | 189867 | Henri Lifchitz | 09/2024 |
| 3123 | 2^630359+1788953 | 189757 | Enzo Doria | 05/2016 |
| 3124 | 2^630228+1871971 | 189718 | Enzo Doria | 05/2016 |
| 3125 | 2^630108+781 | 189682 | Lelio R Paula | 08/2013 |
| 3126 | 2^629870-1867811 | 189610 | Enzo Doria | 04/2016 |
| 3127 | (359^74189-1)/358 | 189558 | Alexander Zhirkov | 11/2017 |
| 3128 | 2^629326+2792929 | 189447 | Enzo Doria | 02/2016 |
| 3129 | 4*10^189401+13 | 189402 | Robert Price | 02/2021 |
| 3130 | (10^189385-61)/3 | 189385 | Serge Batalov | 12/2014 |
| 3131 | 2^629036+1874011 | 189359 | Enzo Doria | 06/2016 |
| 3132 | 2^628930+80779 | 189327 | Enzo Doria | 07/2016 |
| 3133 | (134*10^189309-11)/3 | 189311 | Robert Price | 01/2022 |
| 3134 | 9*10^189306-11 | 189307 | Mischa Rodermond | 04/2023 |
| 3135 | 3^396740-396740^3 | 189294 | Henri Lifchitz | 07/2024 |
| 3136 | 2^628582+1179439 | 189223 | Enzo Doria | 06/2016 |
| 3137 | (44*10^188999-503)/9 | 189000 | Robert Price | 12/2018 |
| 3138 | (16*10^188823+167)/3 | 188824 | Robert Price | 06/2019 |
| 3139 | (10^188796+7)/4967/71933 | 188788 | Alfred Reich | 08/2021 |
| 3140 | 2^627037-2785843 | 188757 | Enzo Doria | 10/2015 |
| 3141 | (10^188566+7)/311640753181661 | 188552 | Alfred Reich | 04/2022 |
| 3142 | (16*10^188457+161)/3 | 188458 | Robert Price | 02/2019 |
| 3143 | L(901657) | 188436 | Renaud Lifchitz | 11/2008 |
| 3144 | (19*10^188375+137)/3 | 188376 | Robert Price | 03/2022 |
| 3145 | 2^625651+74213 | 188340 | Enzo Doria | 04/2016 |
| 3146 | 2^625584-460919 | 188320 | Enzo Doria | 06/2015 |
| 3147 | (8*10^188308-71)/9 | 188308 | Serge Batalov | 11/2022 |
| 3148 | 5^269399-3^269399-1 | 188302 | Jon Grantham | 08/2023 |
| 3149 | (14*10^188218+139)/9 | 188219 | Robert Price | 11/2021 |
| 3150 | 40760^40821-40821^40760 | 188195 | Norbert Schneider | 07/2024 |
| 3151 | (10^188164+359)/9 | 188164 | Serge Batalov | 12/2014 |
| 3152 | 2^624481-113611 | 187988 | Enzo Doria | 08/2014 |
| 3153 | (2^624389+125582)/2 | 187960 | Stefano Morozzi | 09/2019 |
| 3154 | (2^624389+166982)/2 | 187960 | Stefano Morozzi | 09/2019 |
| 3155 | (2^624389+179886)/2 | 187960 | Stefano Morozzi | 09/2019 |
| 3156 | (2^624387+64466)/2 | 187959 | Stefano Morozzi | 09/2019 |
| 3157 | (2^624378+112990)/2 | 187957 | Stefano Morozzi | 09/2019 |
| 3158 | (2^624380+161986)/2 | 187957 | Stefano Morozzi | 09/2019 |
| 3160 | (2^624376+55762)/2 | 187956 | Stefano Morozzi | 09/2019 |
| 3159 | (2^624377+169506)/2 | 187956 | Stefano Morozzi | 09/2019 |
| 3161 | (2^624355+80178)/2 | 187950 | Stefano Morozzi | 09/2019 |
| 3162 | (2^624351+61350)/2 | 187949 | Stefano Morozzi | 09/2019 |
| 3163 | (2^624341+63462)/2 | 187946 | Stefano Morozzi | 09/2019 |
| 3164 | (2^624344+135990)/2 | 187946 | Stefano Morozzi | 09/2019 |
| 3165 | (2^624344+183898)/2 | 187946 | Stefano Morozzi | 09/2019 |
| 3167 | (2^624340+126982)/2 | 187945 | Stefano Morozzi | 09/2019 |
| 3166 | (2^624338+196110)/2 | 187945 | Stefano Morozzi | 09/2019 |
| 3168 | (2^624337+105162)/2 | 187944 | Stefano Morozzi | 09/2019 |
| 3169 | (2^624333+44594)/2 | 187943 | Stefano Morozzi | 09/2019 |
| 3170 | (2^624321+88742)/2 | 187940 | Stefano Morozzi | 09/2019 |
| 3171 | (2^624324+135178)/2 | 187940 | Stefano Morozzi | 09/2019 |
| 3172 | (2^624317+14186)/2 | 187938 | Stefano Morozzi | 09/2019 |
| 3173 | (2^624313+24450)/2 | 187937 | Stefano Morozzi | 09/2019 |
| 3174 | (2^624309+184590)/2 | 187936 | Stefano Morozzi | 09/2019 |
| 3176 | (2^624307+152658)/2 | 187935 | Stefano Morozzi | 09/2019 |
| 3175 | (2^624307+185978)/2 | 187935 | Stefano Morozzi | 09/2019 |
| 3177 | (2^624303+40814)/2 | 187934 | Stefano Morozzi | 09/2019 |
| 3178 | (2^624304+99238)/2 | 187934 | Stefano Morozzi | 09/2019 |
| 3179 | 2^624077-112723 | 187866 | Enzo Doria | 07/2014 |
| 3180 | 2^624008+1874941 | 187846 | Enzo Doria | 06/2016 |
| 3181 | 2^623870+78679 | 187804 | Enzo Doria | 05/2016 |
| 3182 | 2^623714-114035 | 187757 | Enzo Doria | 11/2014 |
| 3183 | 2^623651-1834251 | 187738 | Enzo Doria | 07/2015 |
| 3184 | (7*10^187645+53)/3 | 187646 | Robert Price | 05/2018 |
| 3185 | 21650^43269+43269^21650 | 187591 | Norbert Schneider | 02/2023 |
| 3186 | 2^623146+1879609 | 187586 | Enzo Doria | 09/2016 |
| 3187 | (401*10^187512+13)/9 | 187514 | Robert Price | 09/2021 |
| 3188 | (5*10^187471+91)/3 | 187472 | Robert Price | 04/2018 |
| 3189 | 2^622500+81931 | 187392 | Enzo Doria | 08/2016 |
| 3190 | (13*10^187382-157)/9 | 187383 | Robert Price | 05/2021 |
| 3191 | 2^622300-1861739 | 187331 | Enzo Doria | 12/2015 |
| 3192 | (22*10^187317-91)/3 | 187318 | Robert Price | 02/2022 |
| 3193 | (38*10^187281+349)/9 | 187282 | Robert Price | 10/2018 |
| 3194 | 2^622052+76771 | 187257 | Enzo Doria | 05/2016 |
| 3195 | 2^622005-24091 | 187243 | Enzo Doria | 08/2013 |
| 3196 | 7*10^187205-57 | 187206 | Robert Price | 06/2020 |
| 3197 | 2^621879-351787 | 187205 | Enzo Doria | 07/2015 |
| 3198 | (5*10^187168-53)/3 | 187169 | Robert Price | 09/2020 |
| 3199 | (23*10^187164-143)/3 | 187165 | Robert Price | 12/2019 |
| 3200 | 2^621700+79411 | 187151 | Enzo Doria | 06/2016 |
| 3201 | 2^621597+39 | 187120 | Jon Grantham | 08/2023 |
| 3202 | Phi(271207,-5) | 187112 | Henri Lifchitz | 12/2025 |
| 3203 | (7*10^187101+131)/3 | 187102 | Robert Price | 06/2020 |
| 3204 | (2*10^187062-71)/3 | 187062 | Robert Price | 12/2017 |
| 3205 | V(1118570) | 187013 | Serge Batalov | 07/2024 |
| 3206 | 2^620929-113203 | 186919 | Enzo Doria | 07/2014 |
| 3207 | 10^186842+168223 | 186843 | Patrick De Geest | 07/2018 |
| 3208 | 10^186842-8001 | 186842 | Robert Price | 03/2018 |
| 3209 | V(1117640)/13411681/26823361/575709775681/989889277441 | 186814 | Serge Batalov | 07/2024 |
| 3210 | 2^620347-1777 | 186744 | Lelio R Paula | 07/2013 |
| 3211 | (11*10^186659+133)/9 | 186660 | Robert Price | 04/2022 |
| 3212 | 11^179143-4170 | 186559 | Enzo Doria | 12/2012 |
| 3213 | (13^186535)%(10^186535) | 186535 | Henri Lifchitz | 05/2025 |
| 3215 | (5^266863+1)/6 | 186529 | Paul Bourdelais | 07/2010 |
| 3214 | Phi(266863,-5) | 186529 | Henri Lifchitz | 12/2025 |
| 3216 | 2*10^186517+33 | 186518 | Robert Price | 11/2021 |
| 3217 | 2^619297-2792353 | 186427 | Enzo Doria | 02/2016 |
| 3218 | (49*10^186409+383)/9 | 186410 | Robert Price | 04/2019 |
| 3219 | 2^618940-460139 | 186320 | Enzo Doria | 06/2015 |
| 3220 | Phi(343570,2^5) | 186109 | Henri Lifchitz | 12/2025 |
| 3221 | 2^618046+74179 | 186051 | Enzo Doria | 04/2016 |
| 3222 | 3*10^186019+11 | 186020 | Robert Price | 07/2018 |
| 3223 | (16*10^185978-19)/3 | 185979 | Robert Price | 02/2019 |
| 3224 | (22*10^185950-7)/3 | 185951 | Robert Price | 12/2022 |
| 3225 | 2*10^185930-3 | 185931 | Robert Price | 07/2015 |
| 3226 | (82^97151+1)/83 | 185928 | Paul Bourdelais | 10/2017 |
| 3227 | 4*10^185905+91 | 185906 | Robert Price | 08/2018 |
| 3228 | (34*10^185902-601)/9 | 185903 | Robert Price | 07/2022 |
| 3229 | (19*10^185873+131)/3 | 185874 | Robert Price | 06/2020 |
| 3230 | (37*10^185811+377)/9 | 185812 | Robert Price | 11/2018 |
| 3231 | 2^617250+9385 | 185811 | Lelio R Paula | 07/2013 |
| 3232 | 2^617130+9 | 185775 | Stefano Morozzi | 05/2021 |
| 3233 | 39^116749+2 | 185756 | Robert Price | 04/2015 |
| 3234 | 2^616924-39719 | 185713 | Enzo Doria | 09/2015 |
| 3235 | 2^616712+1867621 | 185649 | Enzo Doria | 04/2016 |
| 3236 | (3^388897+2^388897)/5 | 185551 | Jean-Louis Charton | 09/2011 |
| 3237 | 8*10^185529-67 | 185530 | Robert Price | 08/2019 |
| 3238 | (5*10^185520-11)/3 | 185521 | Tyler Busby | 02/2023 |
| 3239 | 5*2^616124+3 | 185473 | Jon Grantham | 07/2023 |
| 3240 | 2^615916+26791 | 185410 | Enzo Doria | 12/2023 |
| 3241 | 4*10^185355+9 | 185356 | Robert Price | 05/2015 |
| 3242 | (16*10^185347+89)/3 | 185348 | Robert Price | 03/2021 |
| 3243 | 40247^40247-56^56 | 185327 | Henri Lifchitz | 08/2024 |
| 3244 | 2^615622+76639 | 185321 | Enzo Doria | 05/2016 |
| 3245 | 2^615591+1879223 | 185312 | Enzo Doria | 08/2016 |
| 3246 | 2^615587+217013 | 185311 | Enzo Doria | 12/2023 |
| 3247 | 2^615503+1876643 | 185285 | Enzo Doria | 07/2016 |
| 3248 | 5^265007-4^265007 | 185232 | Jean-Louis Charton | 02/2013 |
| 3250 | (4*10^185221-91)/3 | 185222 | Robert Price | 03/2018 |
| 3249 | (7*10^185221+131)/3 | 185222 | Robert Price | 06/2020 |
