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 ? |
3001 | (2^398398+774)/2 | 119930 | Stefano Morozzi | 09/2020 |
3002 | (10^119880+179)/9 | 119880 | Serge Batalov | 12/2014 |
3003 | (9*2^398145+1)/73 | 119853 | Lelio R Paula | 04/2014 |
3004 | (10^119836+77)/3 | 119836 | Robert Price | 01/2018 |
3005 | 2^398029+1872527 | 119819 | Enzo Doria | 05/2016 |
3006 | 2^397897-113541 | 119779 | Enzo Doria | 08/2014 |
3007 | (2^397696+1042)/2 | 119719 | Stefano Morozzi | 09/2020 |
3008 | (19*10^119716+101)/3 | 119717 | Robert Price | 09/2019 |
3009 | (21*2^397661-1)/27593 | 119705 | Lelio R Paula | 04/2014 |
3010 | 77*10^119688+3 | 119690 | Robert Price | 06/2020 |
3011 | (2^397504+1698)/2 | 119661 | Stefano Morozzi | 09/2020 |
3012 | (58^67853-1)/57 | 119653 | Paul Bourdelais | 09/2017 |
3013 | 2^397426+73849 | 119638 | Enzo Doria | 04/2016 |
3014 | 2^396933-112723 | 119489 | Enzo Doria | 06/2014 |
3015 | 11^114738-150 | 119488 | Enzo Doria | 10/2012 |
3016 | (4*10^119474+149)/9 | 119474 | Robert Price | 04/2018 |
3017 | 2^396790-21 | 119446 | Lelio R Paula | 04/2013 |
3018 | (2^396633-7)/5 | 119398 | Lelio R Paula | 04/2014 |
3019 | (2^396577+2522)/2 | 119382 | Stefano Morozzi | 09/2020 |
3020 | (2^396492+2230)/2 | 119356 | Stefano Morozzi | 01/2021 |
3022 | 105^59050+104 | 119352 | Norbert Schneider | 11/2016 |
3021 | 126^56824-125 | 119352 | Norbert Schneider | 12/2018 |
3023 | 139^55690-138 | 119345 | Norbert Schneider | 04/2019 |
3024 | 790^41179+41179^790 | 119322 | Norbert Schneider | 04/2019 |
3025 | 2^396374+73729 | 119321 | Enzo Doria | 04/2016 |
3026 | 88*10^119295+3 | 119297 | Robert Price | 05/2020 |
3027 | 2^396258-459491 | 119286 | Enzo Doria | 05/2015 |
3028 | (2^396248+3126)/2 | 119283 | Stefano Morozzi | 09/2020 |
3029 | 2^396128-396128-1 | 119247 | Karsten Bonath | 06/2018 |
3030 | (28*10^119206-67)/3 | 119207 | Robert Price | 10/2020 |
3031 | (2^395986+5)/76881 | 119199 | Lelio R Paula | 04/2014 |
3032 | (23*2^395942-1)/91 | 119190 | Lelio R Paula | 04/2014 |
3033 | (2^395902+202)/2 | 119179 | Stefano Morozzi | 09/2020 |
3034 | (2^395825+582)/2 | 119155 | Stefano Morozzi | 09/2020 |
3035 | (2^395733+1130)/2 | 119128 | Stefano Morozzi | 09/2020 |
3036 | (11*2^395548+1)/3 | 119073 | Lelio R Paula | 04/2014 |
3037 | (30^80599+1)/31 | 119054 | Robert Price | 07/2013 |
3038 | (43*10^119024-403)/9 | 119025 | Robert Price | 01/2019 |
3039 | 6*10^118964+43 | 118965 | Robert Price | 07/2020 |
3040 | (8*10^118960+49)/3 | 118961 | Robert Price | 05/2020 |
3041 | (194*10^118925-17)/3 | 118927 | Robert Price | 09/2020 |
3042 | 2^395062-109121 | 118926 | Enzo Doria | 04/2014 |
3043 | 2^395055-4627 | 118924 | Yves Minet | 05/2019 |
3044 | (2^395054+702)/2 | 118923 | Stefano Morozzi | 09/2020 |
3045 | (2^395011+2526)/2 | 118910 | Stefano Morozzi | 09/2020 |
3046 | (2^394918+78557)/3 | 118882 | Lelio R Paula | 01/2015 |
3047 | (2^394858+1018)/2 | 118864 | Stefano Morozzi | 09/2020 |
3048 | (16*10^118852+77)/3 | 118853 | Robert Price | 03/2019 |
3049 | (2^394739+3)/1883 | 118826 | Lelio R Paula | 04/2014 |
3050 | 10^118772-40000000001 | 118772 | Robert Price | 09/2020 |
3051 | (2^394513+2034)/2 | 118760 | Stefano Morozzi | 09/2020 |
3052 | 2^394266+71389 | 118686 | Enzo Doria | 04/2016 |
3053 | 2^394215-51261 | 118671 | Tony Prest / Never Odd Or Even | 04/2018 |
3054 | (2^394215-48793)/5/5 | 118670 | Tony Prest / Never Odd Or Even | 04/2018 |
3055 | (2^394215-17063)/3/3/5 | 118669 | Tony Prest / Never Odd Or Even | 04/2018 |
3056 | (2^394215-24393)/5/5/5/5/5 | 118668 | Tony Prest / Never Odd Or Even | 04/2018 |
3057 | (2^394215-35885)/3/71/173 | 118666 | Tony Prest / Never Odd Or Even | 04/2018 |
3058 | (2^394215-47925)/353/2269 | 118665 | Tony Prest / Never Odd Or Even | 04/2018 |
3059 | (2^394215-62819)/3/3/3/3/3/3/3/7/21737/1082153/1953673 | 118650 | Tony Prest / Never Odd Or Even | 04/2018 |
3060 | (2^394131+2214)/2 | 118645 | Stefano Morozzi | 09/2020 |
3061 | (17*10^118635+13)/3 | 118636 | Robert Price | 01/2019 |
3062 | 5^169727+2 | 118635 | Lelio R Paula | 08/2013 |
3063 | 2^394034-3 | 118617 | M. Frind & P. Underwood | 09/2002 |
3064 | (7*10^118607+179)/3 | 118608 | Robert Price | 09/2018 |
3065 | 3^248572-6242 | 118599 | Enzo Doria | 05/2014 |
3066 | (3*2^393964-1)/181 | 118594 | Lelio R Paula | 04/2014 |
3067 | 7^140303-6 | 118570 | Robert Price | 11/2014 |
3068 | (2^393864+2686)/2 | 118565 | Stefano Morozzi | 09/2020 |
3069 | (2^393862+3150)/2 | 118564 | Stefano Morozzi | 09/2020 |
3070 | (2^393849+3762)/2 | 118561 | Stefano Morozzi | 01/2021 |
3071 | (2^393661+2666)/2 | 118504 | Stefano Morozzi | 09/2020 |
3072 | (2^393650+1614)/2 | 118501 | Stefano Morozzi | 09/2020 |
3073 | 10^118485-50001 | 118485 | Robert Price | 08/2020 |
3074 | (2^393528+1030)/2 | 118464 | Stefano Morozzi | 09/2020 |
3075 | 2^393427+75683 | 118434 | Enzo Doria | 04/2016 |
3076 | 2^393376-114749 | 118418 | Enzo Doria | 04/2015 |
3077 | (2^393234+2010)/2 | 118375 | Stefano Morozzi | 09/2020 |
3078 | (2^393205+3030)/2 | 118367 | Stefano Morozzi | 09/2020 |
3079 | (2^393191+2174)/2 | 118362 | Stefano Morozzi | 09/2020 |
3080 | 2^393180+73 | 118359 | Lelio R Paula | 04/2013 |
3081 | p(23881)#/2+256 | 118328 | Robert Price | 05/2017 |
3082 | (10^118324+359)/9 | 118324 | Serge Batalov | 12/2014 |
3083 | (2^393048+570)/2 | 118319 | Stefano Morozzi | 09/2020 |
3084 | p(23879)#/2+64 | 118317 | Robert Price | 03/2017 |
3085 | (238*10^118265-1)/3 | 118267 | Robert Price | 04/2020 |
3087 | (53*10^118216+1)/9 | 118217 | Serge Batalov | 11/2010 |
3086 | (2^392707+1710)/2 | 118217 | Stefano Morozzi | 09/2020 |
3088 | (7*2^392697+1)/15 | 118214 | Renaud Lifchitz | 03/2011 |
3089 | (157^53819-1)/156 | 118179 | Alexander Zhirkov | 06/2016 |
3090 | (2^392567+2750)/2 | 118175 | Stefano Morozzi | 01/2021 |
3091 | (19*2^392551+1)/3 | 118171 | Lelio R Paula | 04/2014 |
3092 | 2^392516-29219 | 118160 | Enzo Doria | 07/2015 |
3093 | 66*10^118155+7 | 118157 | Robert Price | 07/2020 |
3094 | Phi(177231,10) | 118152 | Maksym Voznyy | 09/2015 |
3095 | 2^392413-112123 | 118129 | Enzo Doria | 06/2014 |
3096 | (2^392342+5)/27 | 118106 | Lelio R Paula | 04/2014 |
3097 | (2^392333+1782)/2 | 118104 | Stefano Morozzi | 01/2021 |
3098 | 2^392300-1841429 | 118095 | Enzo Doria | 06/2015 |
3099 | 2^392220+75241 | 118070 | Enzo Doria | 04/2016 |
3100 | 10^118009+664447 | 118010 | Patrick De Geest | 07/2018 |
3101 | 10^118009-20001 | 118009 | Robert Price | 01/2018 |
3102 | (14*10^117991+61)/3 | 117992 | Robert Price | 12/2018 |
3103 | 2^391933+5897 | 117984 | Yves Minet | 02/2011 |
3104 | 2^391912-1849259 | 117978 | Enzo Doria | 08/2015 |
3105 | (2^391812+5)/14181 | 117944 | Lelio R Paula | 04/2014 |
3106 | (7*10^117930-79)/9 | 117930 | Robert Price | 01/2021 |
3107 | (46*10^117902+683)/9 | 117903 | Robert Price | 03/2019 |
3108 | (7*2^391521+1)/285 | 117858 | Lelio R Paula | 04/2014 |
3109 | 2^391324+90901 | 117801 | Yves Minet | 02/2011 |
3110 | 2^391301-1859923 | 117794 | Enzo Doria | 03/2016 |
3111 | 2^391261-4131 | 117782 | Yves Minet | 02/2018 |
3112 | (2^391211+554)/2 | 117766 | Stefano Morozzi | 09/2020 |
3113 | 38434^1157+1157^38434 | 117737 | Hans Havermann | 01/2021 |
3114 | (2^391115-7)/11791 | 117734 | Lelio R Paula | 04/2014 |
3115 | (85*10^117719+473)/9 | 117720 | Robert Price | 01/2020 |
3116 | (2^391049+582)/2 | 117718 | Stefano Morozzi | 09/2020 |
3117 | 2^390960+1880881 | 117691 | Enzo Doria | 11/2016 |
3118 | 2^390897-1860553 | 117672 | Enzo Doria | 11/2015 |
3119 | 2^390889-115003 | 117670 | Enzo Doria | 06/2015 |
3120 | 2^390779-30337 | 117637 | Paul Bourdelais | 02/2012 |
3121 | 11^112916-780 | 117590 | Enzo Doria | 12/2012 |
3122 | (2^390473+1002)/2 | 117544 | Stefano Morozzi | 09/2020 |
3123 | (16^97613-5^97613)/11 | 117537 | Robert Price | 07/2013 |
3124 | (2^390438+1774)/2 | 117534 | Stefano Morozzi | 01/2021 |
3125 | (2^390418+1722)/2 | 117528 | Stefano Morozzi | 09/2020 |
3126 | (2^390403+654)/2 | 117523 | Stefano Morozzi | 09/2020 |
3127 | (2^390343+2246)/2 | 117505 | Stefano Morozzi | 01/2021 |
3128 | (17*2^390272+1)/1983 | 117482 | Lelio R Paula | 04/2014 |
3129 | 31^78759-30 | 117459 | Theodore Burton | 02/2007 |
3130 | (2^390176+1222)/2 | 117455 | Stefano Morozzi | 09/2020 |
3131 | (2^390133+1242)/2 | 117442 | Stefano Morozzi | 09/2020 |
3132 | (2^390013+2634)/2 | 117406 | Stefano Morozzi | 09/2020 |
3133 | 3*10^117402+61 | 117403 | Robert Price | 09/2020 |
3134 | 2^389998+37 | 117402 | Lelio R Paula | 03/2013 |
3135 | (2^389977+314)/2 | 117395 | Stefano Morozzi | 09/2020 |
3136 | (2^389921+342)/2 | 117378 | Stefano Morozzi | 01/2021 |
3137 | (2^389768+1582)/2 | 117332 | Stefano Morozzi | 09/2020 |
3138 | 29^80192-1*80192^29 | 117273 | Dylan Delgado | 12/2019 |
3139 | (13*10^117249+161)/3 | 117250 | Robert Price | 11/2018 |
3140 | 2^389478+43 | 117245 | Lelio R Paula | 03/2013 |
3141 | (298*10^117229-7)/3 | 117231 | Robert Price | 06/2020 |
3142 | (2^389416+342)/2 | 117226 | Stefano Morozzi | 01/2021 |
3143 | (29*2^389351-1)/9 | 117207 | Lelio R Paula | 04/2014 |
3144 | 5*10^117205+23 | 117206 | Robert Price | 03/2019 |
3145 | (2^389341-7)/3865 | 117200 | Lelio R Paula | 03/2014 |
3146 | (2^389276+2650)/2 | 117184 | Stefano Morozzi | 09/2020 |
3147 | 10^117142+209223 | 117143 | Patrick De Geest | 04/2018 |
3149 | 10^117142-1000000001 | 117142 | Robert Price | 03/2018 |
3148 | 10^117142-966057 | 117142 | Patrick De Geest | 04/2018 |
3150 | 10^117135+360577 | 117136 | Patrick De Geest | 07/2018 |
3151 | 10^117135-401 | 117135 | Robert Price | 01/2018 |
3152 | 4*10^117123+79 | 117124 | Robert Price | 08/2018 |
3153 | 2^389033-12797413 | 117111 | Enzo Doria | 04/2016 |
3154 | (2^388987+2634)/2 | 117097 | Stefano Morozzi | 09/2020 |
3155 | 2^388960-109505 | 117089 | Enzo Doria | 04/2014 |
3156 | 7*10^117082-57 | 117083 | Robert Price | 06/2020 |
3157 | (2^388928-5)/569 | 117077 | Lelio R Paula | 03/2014 |
3158 | 2^388888-1529 | 117067 | Yves Minet | 01/2013 |
3159 | 10^117059+28879 | 117060 | Patrick De Geest | 05/2016 |
3160 | (10^117059-71)/1 | 117059 | Serge Batalov | 12/2014 |
3161 | (2^388854+3630)/2 | 117057 | Stefano Morozzi | 09/2020 |
3162 | (2^388842+2098)/2 | 117053 | Stefano Morozzi | 09/2020 |
3163 | (2^388789+3150)/2 | 117037 | Stefano Morozzi | 09/2020 |
3164 | (693^41189-1)/692 | 117005 | Michael Stocker | 05/2020 |
3165 | (26*10^116999-503)/9 | 117000 | Robert Price | 07/2018 |
3166 | (2^388609+1554)/2 | 116983 | Stefano Morozzi | 09/2020 |
3167 | 3^245154-6256 | 116969 | Enzo Doria | 05/2014 |
3168 | 5*10^116966-43 | 116967 | Robert Price | 01/2019 |
3169 | 2^388536-113495 | 116961 | Enzo Doria | 08/2014 |
3170 | 2^388469-43 | 116941 | Lelio R Paula | 03/2013 |
3171 | (7*2^388372-1)/111 | 116911 | Lelio R Paula | 04/2014 |
3172 | (7*2^388335-1)/25 | 116900 | Lelio R Paula | 04/2014 |
3173 | 2^388180+93 | 116854 | Donovan Johnson | 07/2008 |
3174 | (2^388167+2510)/2 | 116850 | Stefano Morozzi | 09/2020 |
3175 | (85*10^116842+293)/9 | 116843 | Robert Price | 12/2019 |
3176 | (2^388081+7)/1179 | 116821 | Lelio R Paula | 03/2014 |
3177 | 2^387946-113261 | 116784 | Enzo Doria | 07/2014 |
3178 | 2^387745+75557 | 116723 | Enzo Doria | 04/2016 |
3179 | (206*10^116687-17)/9 | 116689 | Robert Price | 01/2020 |
3180 | (7*10^116684+179)/3 | 116685 | Robert Price | 09/2018 |
3181 | 2^387554+71455 | 116666 | Enzo Doria | 04/2016 |
3182 | (2^387447+78557)/365 | 116631 | Lelio R Paula | 12/2014 |
3183 | (145*10^116610-1)/9 | 116612 | Serge Batalov | 12/2014 |
3184 | (25*10^116601-331)/9 | 116602 | Robert Price | 08/2018 |
3185 | (17*2^387284-1)/59 | 116584 | Lelio R Paula | 04/2014 |
3186 | (2^387284+3726)/2 | 116584 | Stefano Morozzi | 09/2020 |
3187 | 10^116557+987654321 | 116558 | Robert Price | 12/2019 |
3188 | 10^116557+9141 | 116558 | Patrick De Geest | 02/2020 |
3189 | 10^116557-609431 | 116557 | Patrick De Geest | 02/2020 |
3190 | (2^387076+2046)/2 | 116522 | Stefano Morozzi | 09/2020 |
3191 | (2^387073+2570)/2 | 116521 | Stefano Morozzi | 09/2020 |
3192 | (31*2^386968+1)/7 | 116490 | Lelio R Paula | 03/2014 |
3193 | 2^386947+1869143 | 116483 | Enzo Doria | 05/2016 |
3194 | (30^78857-1)/29 | 116480 | Robert Price | 12/2011 |
3195 | (2^386903-3)/5 | 116469 | Lelio R Paula | 03/2014 |
3196 | 2^386389-1864783 | 116315 | Enzo Doria | 03/2016 |
3197 | (31*2^386231-1)/1711 | 116266 | Lelio R Paula | 03/2014 |
3198 | 297^47012+47012^297 | 116250 | Norbert Schneider | 07/2018 |
3199 | (2^386087-5)/40959 | 116220 | Lelio R Paula | 03/2014 |
3200 | (2^386054+714)/2 | 116214 | Stefano Morozzi | 09/2020 |
3201 | 2^385969-823 | 116189 | Yves Minet | 05/2012 |
3202 | 57^66162+2 | 116173 | Lelio R Paula | 08/2013 |
3203 | 2^385854+75049 | 116154 | Enzo Doria | 04/2016 |
3204 | (2^385832+2590)/2 | 116147 | Stefano Morozzi | 09/2020 |
3205 | 2^385775-109587 | 116130 | Enzo Doria | 04/2014 |
3206 | (14*10^116112-101)/3 | 116113 | Robert Price | 02/2019 |
3207 | Phi(236354,10)/24817171 | 116109 | Makoto Kamada | 08/2019 |
3208 | 2^385622+71839 | 116084 | Enzo Doria | 04/2016 |
3210 | 2^385564+781 | 116067 | Lelio R Paula | 01/2012 |
3209 | (2^385565+1562)/2 | 116067 | Stefano Morozzi | 09/2020 |
3211 | (2^385557+2214)/2 | 116064 | Stefano Morozzi | 09/2020 |
3212 | (266*10^116024-17)/3 | 116026 | Robert Price | 07/2020 |
3213 | 2^385410+71893 | 116020 | Enzo Doria | 04/2016 |
3214 | (25*2^385091+1)/27 | 115924 | Lelio R Paula | 04/2014 |
3215 | (11*10^115910-131)/3 | 115911 | Robert Price | 09/2018 |
3216 | 2^384953-111943 | 115883 | Paul Bourdelais | 02/2012 |
3217 | 43^70942-2 | 115882 | Lelio R Paula | 04/2014 |
3218 | (94*10^115879-7)/3 | 115881 | Robert Price | 03/2020 |
3219 | 7*10^115863+3 | 115864 | Robert Price | 08/2015 |
3220 | 2^384834+93 | 115847 | Donovan Johnson | 07/2008 |
3221 | 2^384809+71297 | 115840 | Enzo Doria | 04/2016 |
3222 | (7*2^384759-1)/5 | 115825 | Lelio R Paula | 04/2014 |
3223 | (71*10^115810-17)/9 | 115811 | Ray Chandler | 08/2011 |
3224 | (97*10^115794-1)/3 | 115796 | Serge Batalov | 12/2014 |
3225 | 2*10^115784-27 | 115785 | Robert Price | 05/2018 |
3226 | 10^115773+126739 | 115774 | Patrick De Geest | 04/2018 |
3228 | 10^115773-1000000001 | 115773 | Robert Price | 03/2018 |
3227 | 10^115773-396221 | 115773 | Patrick De Geest | 04/2018 |
3229 | (2^384582+778)/2 | 115771 | Stefano Morozzi | 09/2020 |
3230 | 2^384530+1874305 | 115756 | Enzo Doria | 06/2016 |
3231 | (3*2^384524-1)/7631 | 115750 | Lelio R Paula | 03/2014 |
3232 | (44*10^115697-413)/9 | 115698 | Robert Price | 03/2019 |
3233 | (2^384317+1026)/2 | 115691 | Stefano Morozzi | 09/2020 |
3234 | 2^384312+81091 | 115690 | Yves Minet | 02/2010 |
3235 | (2^384265+2150)/2 | 115675 | Stefano Morozzi | 09/2020 |
3236 | (2^384246+2182)/2 | 115670 | Stefano Morozzi | 09/2020 |
3237 | (4*10^115668-79)/3 | 115669 | Robert Price | 02/2018 |
3238 | (2^384193+242)/2 | 115654 | Stefano Morozzi | 09/2020 |
3239 | (266947#+1267)/(7^2*181) | 115647 | Makoto Morimoto | 06/2019 |
3240 | (266947#+273808)/(2*109*157) | 115646 | Makoto Morimoto | 10/2017 |
3241 | 18^92125-92125^18 | 115642 | Dylan Delgado | 08/2019 |
3242 | 3^242337-4042 | 115625 | Enzo Doria | 05/2014 |
3243 | (2^383931+674)/2 | 115575 | Stefano Morozzi | 09/2020 |
3244 | 7*10^115561+79 | 115562 | Robert Price | 07/2019 |
3245 | (2^383872+3046)/2 | 115557 | Stefano Morozzi | 09/2020 |
3246 | (11*2^383859-13)/15 | 115553 | Renaud Lifchitz | 03/2011 |
3247 | 2^383780-3149 | 115530 | Yves Minet | 10/2014 |
3248 | (41*10^115523-383)/9 | 115524 | Robert Price | 11/2018 |
3249 | (2^383618+618)/2 | 115481 | Stefano Morozzi | 09/2020 |
3250 | 2^383560-455 | 115464 | Yves Minet | 03/2012 |