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 ? |
2001 | 2^491851+1871453 | 148062 | Enzo Doria | 05/2016 |
2002 | (68*10^148060-257)/9 | 148061 | Robert Price | 07/2019 |
2003 | (151*10^148020-1)/3 | 148022 | Robert Price | 03/2020 |
2004 | 39153^6032+6032^39153 | 148017 | Hans Havermann | 10/2020 |
2005 | 36798^10507+10507^36798 | 147983 | Hans Havermann | 12/2020 |
2006 | (26*10^147973-23)/3 | 147974 | Robert Price | 12/2019 |
2007 | 2^491521-1865323 | 147963 | Enzo Doria | 02/2016 |
2008 | (2^491476+7)/71 | 147948 | Lelio R Paula | 10/2015 |
2009 | 2^491391-7807 | 147924 | Lelio R Paula | 01/2013 |
2010 | 39736^5279+5279^39736 | 147920 | Hans Havermann | 11/2020 |
2011 | 2^491370-113297 | 147918 | Enzo Doria | 07/2014 |
2012 | (32760^32760+32759^32759)/95546586737133567708319 | 147900 | Robert Price | 07/2016 |
2013 | 44368^2155+2155^44368 | 147899 | Hans Havermann | 12/2020 |
2014 | (17*10^147890+79)/3 | 147891 | Robert Price | 04/2019 |
2015 | 94*10^147871-3 | 147873 | Robert Price | 05/2020 |
2016 | 10^147843-50000000001 | 147843 | Robert Price | 12/2020 |
2017 | 35805^13346+13346^35805 | 147709 | Hans Havermann | 11/2020 |
2018 | 2^490660-17 | 147704 | Lelio R Paula | 12/2017 |
2019 | (229*10^147699-1)/3 | 147701 | Robert Price | 07/2020 |
2020 | 40656^4289+4289^40656 | 147678 | Hans Havermann | 11/2020 |
2021 | 2^490550+1179439 | 147671 | Enzo Doria | 06/2016 |
2022 | (23*10^147649-59)/9 | 147650 | Robert Price | 05/2018 |
2023 | 2^490303+2793653 | 147596 | Enzo Doria | 03/2016 |
2024 | 199^64181-2 | 147543 | Lelio R Paula | 11/2017 |
2025 | 32849^30958+30958^32849 | 147518 | Hans Havermann | 11/2020 |
2026 | 2^489871-286987 | 147466 | Enzo Doria | 07/2015 |
2027 | 2^489864-351899 | 147464 | Enzo Doria | 07/2015 |
2028 | 175^65724-2 | 147422 | Lelio R Paula | 11/2017 |
2029 | 2^489712+1874851 | 147419 | Enzo Doria | 06/2016 |
2030 | 34473^18830+18830^34473 | 147367 | Hans Havermann | 11/2020 |
2031 | 63^81896-2 | 147359 | Lelio R Paula | 10/2017 |
2032 | 34580^18237+18237^34580 | 147344 | Hans Havermann | 11/2020 |
2033 | (11*10^147313-107)/3 | 147314 | Robert Price | 09/2018 |
2034 | 2^489305-59815 | 147296 | Enzo Doria | 10/2015 |
2035 | (2^489280-7)/6381 | 147285 | Lelio R Paula | 09/2015 |
2036 | 2*10^147260+29 | 147261 | Robert Price | 02/2021 |
2037 | 33556^24457+24457^33556 | 147258 | Hans Havermann | 11/2020 |
2038 | 2^489125+80357 | 147242 | Enzo Doria | 06/2016 |
2039 | 32835^30452+30452^32835 | 147220 | Hans Havermann | 12/2020 |
2040 | (148*10^147139-1)/3 | 147141 | Robert Price | 03/2020 |
2041 | 34338^19255+19255^34338 | 147123 | Hans Havermann | 11/2020 |
2042 | 2^488648-112799 | 147098 | Enzo Doria | 07/2014 |
2043 | 2^488627-350317 | 147092 | Enzo Doria | 06/2015 |
2044 | 33432^25075+25075^33432 | 147076 | Hans Havermann | 12/2020 |
2045 | (2^488441-1)/(61543567*30051203516986199) | 147012 | Dario Alpern | 12/2014 |
2046 | 2^488280-2165 | 146987 | Yves Minet | 07/2014 |
2047 | 2^488278+45 | 146987 | Lelio R Paula | 12/2017 |
2048 | 2^487927-39 | 146881 | Lelio R Paula | 12/2017 |
2049 | (7*10^146860-13)/3 | 146861 | Robert Price | 07/2018 |
2050 | (26*10^146856-77)/3 | 146857 | Robert Price | 11/2019 |
2051 | 32948^28611+28611^32948 | 146834 | Hans Havermann | 11/2020 |
2052 | 42596^2799+2799^42596 | 146829 | Hans Havermann | 12/2020 |
2053 | 2^487726-1863341 | 146821 | Enzo Doria | 12/2015 |
2054 | 43838^2229+2229^43838 | 146775 | Hans Havermann | 11/2020 |
2055 | 35005^15594+15594^35005 | 146775 | Hans Havermann | 11/2020 |
2056 | ((10^73362-1)*10000/9+6701)*10^73362+(10^73362-1)/9 | 146728 | Hans Havermann | 09/2019 |
2057 | 32963^28234+28234^32963 | 146711 | Hans Havermann | 12/2020 |
2058 | (22*10^146661-19)/3 | 146662 | Robert Price | 04/2019 |
2059 | 33958^20825+20825^33958 | 146651 | Hans Havermann | 12/2020 |
2060 | 2^487121-39523 | 146639 | Enzo Doria | 09/2015 |
2061 | 61063^252+252^61063 | 146637 | Hans Havermann | 12/2020 |
2062 | (7*2^486967+1)/39 | 146591 | Lelio R Paula | 09/2015 |
2063 | 34099^19900+19900^34099 | 146587 | Hans Havermann | 11/2020 |
2064 | 2^486893-79 | 146570 | Yves Minet | 09/2017 |
2065 | 36818^9567+9567^36818 | 146565 | Hans Havermann | 12/2020 |
2066 | 2^486840+1879141 | 146554 | Enzo Doria | 08/2016 |
2067 | 2^486774-87 | 146534 | Lelio R Paula | 12/2017 |
2068 | 38655^6176+6176^38655 | 146530 | Hans Havermann | 11/2020 |
2069 | 8*10^146522-81 | 146523 | Robert Price | 02/2021 |
2070 | 32820^29059+29059^32820 | 146485 | Hans Havermann | 12/2020 |
2071 | 37261^8530+8530^37261 | 146472 | Hans Havermann | 12/2020 |
2072 | 66953^154+154^66953 | 146462 | Hans Havermann | 12/2020 |
2073 | 96^73877-95^73877 | 146445 | Robert Price | 10/2015 |
2074 | 8*10^146421+81 | 146422 | Robert Price | 08/2019 |
2075 | 33153^26090+26090^33153 | 146420 | Hans Havermann | 11/2020 |
2076 | (52^85259+1)/53 | 146304 | Paul Bourdelais | 09/2017 |
2077 | 179^64935+2 | 146290 | Lelio R Paula | 10/2017 |
2078 | 32595^30758+30758^32595 | 146285 | Hans Havermann | 11/2020 |
2079 | 34972^15157+15157^34972 | 146205 | Hans Havermann | 12/2020 |
2080 | (19*10^146181-49)/3 | 146182 | Robert Price | 05/2019 |
2081 | 2^485531-114621 | 146160 | Enzo Doria | 05/2015 |
2082 | (2^485461-9)/4249 | 146135 | Lelio R Paula | 09/2015 |
2083 | 2^485389+81887 | 146117 | Enzo Doria | 08/2016 |
2084 | 32554^30773+30773^32554 | 146108 | Hans Havermann | 11/2020 |
2085 | (10^146063+71)/9 | 146063 | Serge Batalov | 07/2015 |
2086 | (2^485119+7)/2745 | 146032 | Lelio R Paula | 09/2015 |
2087 | (22*10^146029+257)/9 | 146030 | Robert Price | 06/2018 |
2088 | 7*10^145982-53 | 145983 | Robert Price | 06/2020 |
2089 | 2^484743+1872923 | 145923 | Enzo Doria | 05/2016 |
2090 | 33140^25287+25287^33140 | 145913 | Hans Havermann | 01/2021 |
2091 | 37875^7112+7112^37875 | 145895 | Hans Havermann | 01/2021 |
2092 | 32862^27473+27473^32862 | 145872 | Hans Havermann | 01/2021 |
2093 | 32836^27587+27587^32836 | 145815 | Hans Havermann | 01/2021 |
2094 | 2^484341+1879157 | 145802 | Enzo Doria | 08/2016 |
2095 | 2*10^145784-91 | 145785 | Robert Price | 04/2020 |
2096 | 2^484271-51 | 145781 | Lelio R Paula | 11/2017 |
2097 | 70^78989-69^78989 | 145743 | Robert Price | 01/2014 |
2098 | 51^85334+2 | 145714 | Robert Price | 07/2015 |
2099 | 2^483886-1857101 | 145665 | Enzo Doria | 10/2015 |
2100 | 33350^23307+23307^33350 | 145656 | Hans Havermann | 01/2021 |
2101 | (19*10^145585+131)/3 | 145586 | Robert Price | 06/2020 |
2102 | 2^483563-1863457 | 145567 | Enzo Doria | 12/2015 |
2103 | 2^483516-1861409 | 145553 | Enzo Doria | 11/2015 |
2104 | 2^483384-115019 | 145514 | Enzo Doria | 06/2015 |
2105 | (259*10^145501-1)/3 | 145503 | Robert Price | 12/2020 |
2106 | (19*10^145499-61)/3 | 145500 | Robert Price | 05/2019 |
2107 | 2^483302-3221 | 145489 | Lelio R Paula | 12/2012 |
2108 | 35523^12434+12434^35523 | 145453 | Hans Havermann | 01/2021 |
2109 | 2^483046+1874029 | 145412 | Enzo Doria | 06/2016 |
2110 | 2^482991-2789317 | 145395 | Enzo Doria | 12/2015 |
2111 | 2^482978-2789711 | 145391 | Enzo Doria | 12/2015 |
2112 | 2^482966-2789711 | 145388 | Enzo Doria | 12/2015 |
2113 | 2^482817-1858843 | 145343 | Enzo Doria | 10/2015 |
2114 | 2^482751-113635 | 145323 | Enzo Doria | 08/2014 |
2115 | 2^482705-459763 | 145309 | Enzo Doria | 05/2015 |
2116 | (4^241306+11)/3 | 145281 | Robert Price | 03/2017 |
2117 | (238*10^145276-1)/3 | 145278 | Robert Price | 04/2020 |
2118 | (19*2^482534+1)/7 | 145258 | Lelio R Paula | 09/2015 |
2119 | 53^84239-52^84239 | 145252 | Robert Price | 01/2013 |
2120 | 47952^1067+1067^47952 | 145207 | Hans Havermann | 01/2021 |
2121 | (2^482333+5)/9287 | 145193 | Lelio R Paula | 09/2015 |
2122 | 2^482087+72623 | 145123 | Enzo Doria | 04/2016 |
2123 | 2^482016-286565 | 145102 | Enzo Doria | 07/2015 |
2124 | (19*10^145093+101)/3 | 145094 | Robert Price | 09/2019 |
2125 | 2^481840+15 | 145049 | Robert Price | 09/2015 |
2126 | 2^481801-481801 | 145037 | Henri Lifchitz | 03/2005 |
2127 | 2^481775+1869083 | 145029 | Enzo Doria | 05/2016 |
2128 | 29^99167+2 | 145022 | Ray Chandler | 08/2011 |
2129 | 2^481663+1875653 | 144996 | Enzo Doria | 07/2016 |
2130 | 36486^9379+9379^36486 | 144929 | Hans Havermann | 12/2020 |
2131 | 35111^13378+13378^35111 | 144882 | Hans Havermann | 01/2021 |
2132 | 2^481149-459073 | 144841 | Enzo Doria | 05/2015 |
2133 | 34619^15246+15246^34619 | 144817 | Hans Havermann | 01/2021 |
2134 | 2^481045-113215 | 144809 | Enzo Doria | 07/2014 |
2135 | (64*10^144797+287)/9 | 144798 | Robert Price | 04/2019 |
2136 | 2^480859-113695 | 144753 | Enzo Doria | 09/2014 |
2137 | 2^480766-2787191 | 144725 | Enzo Doria | 11/2015 |
2138 | (21*27^101106-151)/26 | 144720 | Curtis Bright | 05/2015 |
2139 | 2^480678+1874239 | 144699 | Enzo Doria | 06/2016 |
2140 | (8*10^144683+1)/9 | 144683 | Serge Batalov | 05/2010 |
2141 | L(692147) | 144651 | Henri Lifchitz | 10/2008 |
2142 | (76*10^144648+77)/9 | 144649 | Robert Price | 10/2019 |
2143 | 32672^26721+26721^32672 | 144635 | Hans Havermann | 01/2021 |
2144 | 2^480417+35 | 144620 | Robert Price | 11/2015 |
2145 | 2^480296-59879 | 144584 | Enzo Doria | 10/2015 |
2146 | 33157^22878+22878^33157 | 144546 | Hans Havermann | 12/2020 |
2147 | 45494^1497+1497^45494 | 144454 | Hans Havermann | 12/2020 |
2148 | 2^479844+3 | 144448 | Donovan Johnson | 06/2006 |
2149 | 33442^20803+20803^33442 | 144407 | Hans Havermann | 01/2021 |
2150 | (27*2^479642+1)/733 | 144386 | Lelio R Paula | 08/2015 |
2151 | 44917^1638+1638^44917 | 144378 | Hans Havermann | 12/2020 |
2152 | (21*2^479605-1)/11 | 144376 | Lelio R Paula | 09/2015 |
2153 | 2^479348-2792219 | 144299 | Enzo Doria | 02/2016 |
2154 | 3^302389-6242 | 144277 | Enzo Doria | 05/2014 |
2155 | 33778^18641+18641^33778 | 144248 | Hans Havermann | 01/2021 |
2156 | 2*10^144220-3 | 144221 | Robert Price | 07/2015 |
2157 | 2^478904-109109 | 144165 | Enzo Doria | 04/2014 |
2158 | 43698^1987+1987^43698 | 144125 | Hans Havermann | 12/2020 |
2159 | 36128^9723+9723^36128 | 144072 | Hans Havermann | 01/2021 |
2160 | 32817^24518+24518^32817 | 144050 | Hans Havermann | 12/2020 |
2161 | 34295^15828+15828^34295 | 144020 | Hans Havermann | 01/2021 |
2162 | 2^478337-115333 | 143994 | Enzo Doria | 07/2015 |
2163 | (7*2^478330-1)/3 | 143993 | Lelio R Paula | 08/2015 |
2164 | 88^74047-87^74047 | 143984 | Robert Price | 05/2015 |
2165 | 33242^21441+21441^33242 | 143980 | Hans Havermann | 01/2021 |
2166 | (19*10^143963+467)/9 | 143964 | Robert Price | 02/2018 |
2167 | (71^77761+1)/72 | 143954 | Paul Bourdelais | 09/2017 |
2168 | 33490^19859+19859^33490 | 143939 | Hans Havermann | 01/2021 |
2169 | 2^478088+27 | 143919 | Robert Price | 11/2015 |
2170 | 7*10^143857-93 | 143858 | Robert Price | 08/2020 |
2171 | 3*10^143844+61 | 143845 | Robert Price | 09/2020 |
2172 | 41358^3005+3005^41358 | 143837 | Hans Havermann | 01/2021 |
2173 | 2^477475-22071 | 143735 | Enzo Doria | 07/2013 |
2174 | 2^477417-22123 | 143717 | Enzo Doria | 07/2013 |
2175 | 2^477359-3577 | 143700 | Yves Minet | 03/2017 |
2176 | 39435^4402+4402^39435 | 143688 | Hans Havermann | 01/2021 |
2177 | (14*10^143662-71)/3 | 143663 | Robert Price | 12/2018 |
2178 | 6^184614+5 | 143658 | Paul Bourdelais | 01/2021 |
2179 | 46371^1252+1252^46371 | 143640 | Hans Havermann | 12/2020 |
2180 | 34975^12748+12748^34975 | 143588 | Hans Havermann | 01/2021 |
2181 | 2^476936-396765 | 143573 | Enzo Doria | 07/2013 |
2182 | 2^476888-2791229 | 143558 | Enzo Doria | 01/2016 |
2183 | 2^476868-1859939 | 143552 | Enzo Doria | 11/2015 |
2184 | 40807^3294+3294^40807 | 143548 | Hans Havermann | 12/2020 |
2185 | 755^49878+49878^755 | 143547 | Norbert Schneider | 04/2019 |
2186 | (19^112247-4^112247)/15 | 143536 | Robert Price | 03/2021 |
2187 | 2^476558-113627 | 143459 | Enzo Doria | 08/2014 |
2188 | 2^476478-45 | 143435 | Lelio R Paula | 11/2017 |
2189 | (2^475854+9)/73 | 143245 | Lelio R Paula | 08/2015 |
2190 | 2^475813-113239 | 143234 | Enzo Doria | 07/2014 |
2191 | 2^475624-114059 | 143178 | Enzo Doria | 11/2014 |
2192 | (11*2^475553-7)/15 | 143156 | Renaud Lifchitz | 03/2011 |
2193 | 2^475480-2786129 | 143134 | Enzo Doria | 10/2015 |
2194 | (19*10^143129-43)/3 | 143130 | Robert Price | 05/2019 |
2195 | (14*10^143097+73)/3 | 143098 | Robert Price | 12/2018 |
2196 | 2^475299-111937 | 143080 | Enzo Doria | 05/2014 |
2197 | 2^475160-2797859 | 143038 | Enzo Doria | 04/2016 |
2198 | (2^475069+3)/395 | 143008 | Lelio R Paula | 08/2015 |
2199 | 2^475027-461437 | 142998 | Enzo Doria | 07/2015 |
2200 | (2^474958+78557)/3 | 142977 | Lelio R Paula | 07/2015 |
2201 | (8*10^142932+43)/3 | 142933 | Robert Price | 06/2018 |
2202 | (54^82471-1)/53 | 142871 | Paul Bourdelais | 09/2017 |
2203 | (11*2^474432-1)/95 | 142818 | Lelio R Paula | 09/2015 |
2204 | 2^474417-59923 | 142814 | Enzo Doria | 10/2015 |
2205 | 4*10^142810-21 | 142811 | Robert Price | 09/2018 |
2206 | (3*2^474166+1)/7 | 142738 | Lelio R Paula | 08/2015 |
2207 | 2^474053+2897 | 142705 | Lelio R Paula | 08/2012 |
2208 | (10^142699+359)/9 | 142699 | Serge Batalov | 12/2014 |
2209 | 898^48311-897^48311 | 142676 | Jean-Louis Charton | 12/2012 |
2210 | 2^473740-1856639 | 142610 | Enzo Doria | 09/2015 |
2211 | (25*10^142573-403)/9 | 142574 | Robert Price | 12/2020 |
2212 | 2^473617-114223 | 142573 | Enzo Doria | 12/2014 |
2213 | 2^473257+1873217 | 142465 | Enzo Doria | 06/2016 |
2214 | (19*2^473242+1)/11 | 142461 | Lelio R Paula | 09/2015 |
2215 | (61*10^142446-1)/3 | 142448 | Robert Price | 01/2020 |
2216 | (14*10^142440+61)/3 | 142441 | Robert Price | 12/2018 |
2217 | (10^142425+359)/9 | 142425 | Serge Batalov | 12/2014 |
2218 | 2^473101+773027 | 142418 | Enzo Doria | 05/2016 |
2219 | (13*2^472838-1)/3 | 142340 | Lelio R Paula | 09/2015 |
2220 | (17*2^472789-1)/187347 | 142320 | Lelio R Paula | 09/2015 |
2221 | 2^472634-109511 | 142278 | Enzo Doria | 04/2014 |
2222 | (48^84589+1)/49 | 142213 | Robert Price | 01/2014 |
2223 | 7*10^142189+87 | 142190 | Robert Price | 07/2019 |
2224 | 2^472258+7 | 142164 | Lelio R Paula | 08/2015 |
2225 | (23*10^142105+91)/3 | 142106 | Robert Price | 08/2019 |
2226 | tau(317^10330) | 142098 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
2227 | (11*2^471913-1)/21 | 142060 | Lelio R Paula | 09/2015 |
2228 | (16*10^142058+197)/3 | 142059 | Robert Price | 04/2019 |
2229 | 9*10^142037+91 | 142038 | Robert Price | 10/2020 |
2230 | 2^471792+1880881 | 142024 | Enzo Doria | 11/2016 |
2231 | (31*2^471580-1)/135 | 141960 | Lelio R Paula | 08/2015 |
2232 | (4*10^141926-79)/3 | 141927 | Robert Price | 02/2018 |
2233 | (904*19^110984-1)/3 | 141924 | Curtis Bright | 05/2015 |
2234 | (14*10^141853+37)/3 | 141854 | Robert Price | 12/2018 |
2235 | (41*10^141836-59)/9 | 141837 | Robert Price | 10/2018 |
2236 | 4*5^202913-9 | 141831 | Lelio R Paula | 04/2015 |
2237 | 2*7^167812-3 | 141818 | Lelio R Paula | 12/2014 |
2238 | 2^471043+35 | 141799 | Robert Price | 11/2015 |
2239 | (11*2^470956-1)/7945 | 141770 | Lelio R Paula | 08/2015 |
2240 | (11*10^141718-131)/3 | 141719 | Robert Price | 09/2018 |
2241 | 2^470692+76771 | 141693 | Enzo Doria | 05/2016 |
2242 | (21*2^470391+1)/533 | 141601 | Lelio R Paula | 09/2015 |
2243 | 2^470305-459373 | 141576 | Enzo Doria | 05/2015 |
2244 | 2^470183+72623 | 141540 | Enzo Doria | 04/2016 |
2245 | 2^470173-16465 | 141537 | Enzo Doria | 04/2013 |
2246 | 44*10^141476+9 | 141478 | Robert Price | 10/2020 |
2247 | 2^469974-1841 | 141477 | Enzo Doria | 05/2014 |
2248 | (23*2^469939+1)/25 | 141466 | Lelio R Paula | 08/2015 |
2249 | 2*10^141448+63 | 141449 | Robert Price | 05/2018 |
2250 | 2^469824+43 | 141432 | Lelio R Paula | 09/2017 |