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 ? |
2001 | 32820^29059+29059^32820 | 146485 | Hans Havermann | 12/2020 |
2002 | 37261^8530+8530^37261 | 146472 | Hans Havermann | 12/2020 |
2003 | 66953^154+154^66953 | 146462 | Hans Havermann | 12/2020 |
2004 | 96^73877-95^73877 | 146445 | Robert Price | 10/2015 |
2005 | 8*10^146421+81 | 146422 | Robert Price | 08/2019 |
2006 | 33153^26090+26090^33153 | 146420 | Hans Havermann | 11/2020 |
2007 | (52^85259+1)/53 | 146304 | Paul Bourdelais | 09/2017 |
2008 | 179^64935+2 | 146290 | Lelio R Paula | 10/2017 |
2009 | 32595^30758+30758^32595 | 146285 | Hans Havermann | 11/2020 |
2010 | 34972^15157+15157^34972 | 146205 | Hans Havermann | 12/2020 |
2011 | (19*10^146181-49)/3 | 146182 | Robert Price | 05/2019 |
2012 | 2^485531-114621 | 146160 | Enzo Doria | 05/2015 |
2013 | (2^485461-9)/4249 | 146135 | Lelio R Paula | 09/2015 |
2014 | 2^485389+81887 | 146117 | Enzo Doria | 08/2016 |
2015 | 32554^30773+30773^32554 | 146108 | Hans Havermann | 11/2020 |
2016 | (10^146063+71)/9 | 146063 | Serge Batalov | 07/2015 |
2017 | (2^485119+7)/2745 | 146032 | Lelio R Paula | 09/2015 |
2018 | (22*10^146029+257)/9 | 146030 | Robert Price | 06/2018 |
2019 | 7*10^145982-53 | 145983 | Robert Price | 06/2020 |
2020 | 2^484743+1872923 | 145923 | Enzo Doria | 05/2016 |
2021 | 37875^7112+7112^37875 | 145895 | Hans Havermann | 01/2021 |
2022 | 32862^27473+27473^32862 | 145872 | Hans Havermann | 01/2021 |
2023 | 32836^27587+27587^32836 | 145815 | Hans Havermann | 01/2021 |
2024 | 2^484341+1879157 | 145802 | Enzo Doria | 08/2016 |
2025 | 2*10^145784-91 | 145785 | Robert Price | 04/2020 |
2026 | 2^484271-51 | 145781 | Lelio R Paula | 11/2017 |
2027 | 70^78989-69^78989 | 145743 | Robert Price | 01/2014 |
2028 | 51^85334+2 | 145714 | Robert Price | 07/2015 |
2029 | 2^483886-1857101 | 145665 | Enzo Doria | 10/2015 |
2030 | (19*10^145585+131)/3 | 145586 | Robert Price | 06/2020 |
2031 | 2^483563-1863457 | 145567 | Enzo Doria | 12/2015 |
2032 | 2^483516-1861409 | 145553 | Enzo Doria | 11/2015 |
2033 | 2^483384-115019 | 145514 | Enzo Doria | 06/2015 |
2034 | (259*10^145501-1)/3 | 145503 | Robert Price | 12/2020 |
2035 | (19*10^145499-61)/3 | 145500 | Robert Price | 05/2019 |
2036 | 2^483302-3221 | 145489 | Lelio R Paula | 12/2012 |
2037 | 35523^12434+12434^35523 | 145453 | Hans Havermann | 01/2021 |
2038 | 2^483046+1874029 | 145412 | Enzo Doria | 06/2016 |
2039 | 2^482991-2789317 | 145395 | Enzo Doria | 12/2015 |
2040 | 2^482978-2789711 | 145391 | Enzo Doria | 12/2015 |
2041 | 2^482966-2789711 | 145388 | Enzo Doria | 12/2015 |
2042 | 2^482817-1858843 | 145343 | Enzo Doria | 10/2015 |
2043 | 2^482751-113635 | 145323 | Enzo Doria | 08/2014 |
2044 | 2^482705-459763 | 145309 | Enzo Doria | 05/2015 |
2045 | (4^241306+11)/3 | 145281 | Robert Price | 03/2017 |
2046 | (238*10^145276-1)/3 | 145278 | Robert Price | 04/2020 |
2047 | (19*2^482534+1)/7 | 145258 | Lelio R Paula | 09/2015 |
2048 | 53^84239-52^84239 | 145252 | Robert Price | 01/2013 |
2049 | 47952^1067+1067^47952 | 145207 | Hans Havermann | 01/2021 |
2050 | (2^482333+5)/9287 | 145193 | Lelio R Paula | 09/2015 |
2051 | 2^482087+72623 | 145123 | Enzo Doria | 04/2016 |
2052 | 2^482016-286565 | 145102 | Enzo Doria | 07/2015 |
2053 | (19*10^145093+101)/3 | 145094 | Robert Price | 09/2019 |
2054 | 2^481840+15 | 145049 | Robert Price | 09/2015 |
2055 | 2^481801-481801 | 145037 | Henri Lifchitz | 03/2005 |
2056 | 2^481775+1869083 | 145029 | Enzo Doria | 05/2016 |
2057 | 29^99167+2 | 145022 | Ray Chandler | 08/2011 |
2058 | 2^481663+1875653 | 144996 | Enzo Doria | 07/2016 |
2059 | 36486^9379+9379^36486 | 144929 | Hans Havermann | 12/2020 |
2060 | 35111^13378+13378^35111 | 144882 | Hans Havermann | 01/2021 |
2061 | 2^481149-459073 | 144841 | Enzo Doria | 05/2015 |
2062 | 34619^15246+15246^34619 | 144817 | Hans Havermann | 01/2021 |
2063 | 2^481045-113215 | 144809 | Enzo Doria | 07/2014 |
2064 | (64*10^144797+287)/9 | 144798 | Robert Price | 04/2019 |
2065 | 2^480859-113695 | 144753 | Enzo Doria | 09/2014 |
2066 | 2^480766-2787191 | 144725 | Enzo Doria | 11/2015 |
2067 | (21*27^101106-151)/26 | 144720 | Curtis Bright | 05/2015 |
2068 | 2^480678+1874239 | 144699 | Enzo Doria | 06/2016 |
2069 | (8*10^144683+1)/9 | 144683 | Serge Batalov | 05/2010 |
2070 | L(692147) | 144651 | Henri Lifchitz | 10/2008 |
2071 | (76*10^144648+77)/9 | 144649 | Robert Price | 10/2019 |
2072 | 32672^26721+26721^32672 | 144635 | Hans Havermann | 01/2021 |
2073 | 2^480417+35 | 144620 | Robert Price | 11/2015 |
2074 | 2^480296-59879 | 144584 | Enzo Doria | 10/2015 |
2075 | 33157^22878+22878^33157 | 144546 | Hans Havermann | 12/2020 |
2076 | 45494^1497+1497^45494 | 144454 | Hans Havermann | 12/2020 |
2077 | 2^479844+3 | 144448 | Donovan Johnson | 06/2006 |
2078 | 33442^20803+20803^33442 | 144407 | Hans Havermann | 01/2021 |
2079 | (27*2^479642+1)/733 | 144386 | Lelio R Paula | 08/2015 |
2080 | 44917^1638+1638^44917 | 144378 | Hans Havermann | 12/2020 |
2081 | (21*2^479605-1)/11 | 144376 | Lelio R Paula | 09/2015 |
2082 | 2^479348-2792219 | 144299 | Enzo Doria | 02/2016 |
2083 | 3^302389-6242 | 144277 | Enzo Doria | 05/2014 |
2084 | 33778^18641+18641^33778 | 144248 | Hans Havermann | 01/2021 |
2085 | 2*10^144220-3 | 144221 | Robert Price | 07/2015 |
2086 | 2^478904-109109 | 144165 | Enzo Doria | 04/2014 |
2087 | 43698^1987+1987^43698 | 144125 | Hans Havermann | 12/2020 |
2088 | 36128^9723+9723^36128 | 144072 | Hans Havermann | 01/2021 |
2089 | 32817^24518+24518^32817 | 144050 | Hans Havermann | 12/2020 |
2090 | 2^478337-115333 | 143994 | Enzo Doria | 07/2015 |
2091 | (7*2^478330-1)/3 | 143993 | Lelio R Paula | 08/2015 |
2092 | 88^74047-87^74047 | 143984 | Robert Price | 05/2015 |
2093 | 33242^21441+21441^33242 | 143980 | Hans Havermann | 01/2021 |
2094 | (19*10^143963+467)/9 | 143964 | Robert Price | 02/2018 |
2095 | (71^77761+1)/72 | 143954 | Paul Bourdelais | 09/2017 |
2096 | 33490^19859+19859^33490 | 143939 | Hans Havermann | 01/2021 |
2097 | 2^478088+27 | 143919 | Robert Price | 11/2015 |
2098 | 7*10^143857-93 | 143858 | Robert Price | 08/2020 |
2099 | 3*10^143844+61 | 143845 | Robert Price | 09/2020 |
2100 | 41358^3005+3005^41358 | 143837 | Hans Havermann | 01/2021 |
2101 | 2^477475-22071 | 143735 | Enzo Doria | 07/2013 |
2102 | 2^477417-22123 | 143717 | Enzo Doria | 07/2013 |
2103 | 2^477359-3577 | 143700 | Yves Minet | 03/2017 |
2104 | 39435^4402+4402^39435 | 143688 | Hans Havermann | 01/2021 |
2105 | (14*10^143662-71)/3 | 143663 | Robert Price | 12/2018 |
2106 | 46371^1252+1252^46371 | 143640 | Hans Havermann | 12/2020 |
2107 | 34975^12748+12748^34975 | 143588 | Hans Havermann | 01/2021 |
2108 | 2^476936-396765 | 143573 | Enzo Doria | 07/2013 |
2109 | 2^476888-2791229 | 143558 | Enzo Doria | 01/2016 |
2110 | 2^476868-1859939 | 143552 | Enzo Doria | 11/2015 |
2111 | 40807^3294+3294^40807 | 143548 | Hans Havermann | 12/2020 |
2112 | 755^49878+49878^755 | 143547 | Norbert Schneider | 04/2019 |
2113 | 2^476558-113627 | 143459 | Enzo Doria | 08/2014 |
2114 | 2^476478-45 | 143435 | Lelio R Paula | 11/2017 |
2115 | (2^475854+9)/73 | 143245 | Lelio R Paula | 08/2015 |
2116 | 2^475813-113239 | 143234 | Enzo Doria | 07/2014 |
2117 | 2^475624-114059 | 143178 | Enzo Doria | 11/2014 |
2118 | (11*2^475553-7)/15 | 143156 | Renaud Lifchitz | 03/2011 |
2119 | 2^475480-2786129 | 143134 | Enzo Doria | 10/2015 |
2120 | (19*10^143129-43)/3 | 143130 | Robert Price | 05/2019 |
2121 | (14*10^143097+73)/3 | 143098 | Robert Price | 12/2018 |
2122 | 2^475299-111937 | 143080 | Enzo Doria | 05/2014 |
2123 | 2^475160-2797859 | 143038 | Enzo Doria | 04/2016 |
2124 | (2^475069+3)/395 | 143008 | Lelio R Paula | 08/2015 |
2125 | 2^475027-461437 | 142998 | Enzo Doria | 07/2015 |
2126 | (2^474958+78557)/3 | 142977 | Lelio R Paula | 07/2015 |
2127 | (8*10^142932+43)/3 | 142933 | Robert Price | 06/2018 |
2128 | (54^82471-1)/53 | 142871 | Paul Bourdelais | 09/2017 |
2129 | (11*2^474432-1)/95 | 142818 | Lelio R Paula | 09/2015 |
2130 | 2^474417-59923 | 142814 | Enzo Doria | 10/2015 |
2131 | 4*10^142810-21 | 142811 | Robert Price | 09/2018 |
2132 | (3*2^474166+1)/7 | 142738 | Lelio R Paula | 08/2015 |
2133 | 2^474053+2897 | 142705 | Lelio R Paula | 08/2012 |
2134 | (10^142699+359)/9 | 142699 | Serge Batalov | 12/2014 |
2135 | 898^48311-897^48311 | 142676 | Jean-Louis Charton | 12/2012 |
2136 | 2^473740-1856639 | 142610 | Enzo Doria | 09/2015 |
2137 | (25*10^142573-403)/9 | 142574 | Robert Price | 12/2020 |
2138 | 2^473617-114223 | 142573 | Enzo Doria | 12/2014 |
2139 | 2^473257+1873217 | 142465 | Enzo Doria | 06/2016 |
2140 | (19*2^473242+1)/11 | 142461 | Lelio R Paula | 09/2015 |
2141 | (61*10^142446-1)/3 | 142448 | Robert Price | 01/2020 |
2142 | (14*10^142440+61)/3 | 142441 | Robert Price | 12/2018 |
2143 | (10^142425+359)/9 | 142425 | Serge Batalov | 12/2014 |
2144 | 2^473101+773027 | 142418 | Enzo Doria | 05/2016 |
2145 | (13*2^472838-1)/3 | 142340 | Lelio R Paula | 09/2015 |
2146 | (17*2^472789-1)/187347 | 142320 | Lelio R Paula | 09/2015 |
2147 | 2^472634-109511 | 142278 | Enzo Doria | 04/2014 |
2148 | (48^84589+1)/49 | 142213 | Robert Price | 01/2014 |
2149 | 7*10^142189+87 | 142190 | Robert Price | 07/2019 |
2150 | 2^472258+7 | 142164 | Lelio R Paula | 08/2015 |
2151 | (23*10^142105+91)/3 | 142106 | Robert Price | 08/2019 |
2152 | tau(317^10330) | 142098 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
2153 | (11*2^471913-1)/21 | 142060 | Lelio R Paula | 09/2015 |
2154 | (16*10^142058+197)/3 | 142059 | Robert Price | 04/2019 |
2155 | 9*10^142037+91 | 142038 | Robert Price | 10/2020 |
2156 | 2^471792+1880881 | 142024 | Enzo Doria | 11/2016 |
2157 | (31*2^471580-1)/135 | 141960 | Lelio R Paula | 08/2015 |
2158 | (4*10^141926-79)/3 | 141927 | Robert Price | 02/2018 |
2159 | (904*19^110984-1)/3 | 141924 | Curtis Bright | 05/2015 |
2160 | (14*10^141853+37)/3 | 141854 | Robert Price | 12/2018 |
2161 | (41*10^141836-59)/9 | 141837 | Robert Price | 10/2018 |
2162 | 4*5^202913-9 | 141831 | Lelio R Paula | 04/2015 |
2163 | 2*7^167812-3 | 141818 | Lelio R Paula | 12/2014 |
2164 | 2^471043+35 | 141799 | Robert Price | 11/2015 |
2165 | (11*2^470956-1)/7945 | 141770 | Lelio R Paula | 08/2015 |
2166 | (11*10^141718-131)/3 | 141719 | Robert Price | 09/2018 |
2167 | 2^470692+76771 | 141693 | Enzo Doria | 05/2016 |
2168 | (21*2^470391+1)/533 | 141601 | Lelio R Paula | 09/2015 |
2169 | 2^470305-459373 | 141576 | Enzo Doria | 05/2015 |
2170 | 2^470183+72623 | 141540 | Enzo Doria | 04/2016 |
2171 | 2^470173-16465 | 141537 | Enzo Doria | 04/2013 |
2172 | 44*10^141476+9 | 141478 | Robert Price | 10/2020 |
2173 | 2^469974-1841 | 141477 | Enzo Doria | 05/2014 |
2174 | (23*2^469939+1)/25 | 141466 | Lelio R Paula | 08/2015 |
2175 | 2*10^141448+63 | 141449 | Robert Price | 05/2018 |
2176 | 2^469824+43 | 141432 | Lelio R Paula | 09/2017 |
2177 | 2*10^141407-27 | 141408 | Robert Price | 05/2018 |
2178 | (214*10^141385-1)/3 | 141387 | Robert Price | 01/2021 |
2179 | 2^469630-59711 | 141373 | Enzo Doria | 10/2015 |
2180 | (25*10^141366-331)/9 | 141367 | Robert Price | 08/2018 |
2181 | (2^469564-3)/611 | 141351 | Lelio R Paula | 08/2015 |
2182 | 197^61601+2 | 141342 | Lelio R Paula | 08/2017 |
2183 | 2^469372-112949 | 141296 | Enzo Doria | 07/2014 |
2185 | 143^(2^16)+106^(2^16) | 141253 | Valeryi Kuryshev | 02/2011 |
2184 | 143^65536+106^65536 | 141253 | Serge Batalov | 03/2018 |
2186 | 2^469198+782569 | 141243 | Enzo Doria | 10/2016 |
2187 | 2^469076-2786939 | 141206 | Enzo Doria | 11/2015 |
2188 | 2^468804+1874941 | 141125 | Enzo Doria | 06/2016 |
2189 | 2^468787+81353 | 141119 | Enzo Doria | 08/2016 |
2190 | 2^468595-350407 | 141062 | Enzo Doria | 06/2015 |
2191 | 2^468564+87855 | 141052 | Lelio R Paula | 01/2016 |
2192 | 2^468551-21987 | 141048 | Enzo Doria | 07/2013 |
2193 | 2^468523-7 | 141040 | Lelio R Paula | 08/2015 |
2194 | (25*10^141019-13)/3 | 141020 | Robert Price | 09/2019 |
2195 | 2^468405+87855 | 141004 | Lelio R Paula | 01/2016 |
2196 | 2^467978-113051 | 140876 | Enzo Doria | 07/2014 |
2197 | 2^467780-344429 | 140816 | Enzo Doria | 07/2015 |
2198 | 9*10^140802-89 | 140803 | Robert Price | 04/2020 |
2199 | 2^467605-1855603 | 140764 | Enzo Doria | 09/2015 |
2200 | 5*10^140744-7 | 140745 | Robert Price | 09/2015 |
2201 | (5^201359-1)/4 | 140744 | Paul Bourdelais | 12/2008 |
2202 | 2^467401-1850023 | 140702 | Enzo Doria | 08/2015 |
2203 | (266*10^140681-17)/3 | 140683 | Robert Price | 07/2020 |
2204 | (2^467270+78557)/23247 | 140658 | Lelio R Paula | 07/2015 |
2205 | (29*2^467205+1)/2780517107 | 140635 | Lelio R Paula | 08/2015 |
2206 | 2^467166-1854941 | 140631 | Enzo Doria | 09/2015 |
2207 | (203*10^140615-11)/3 | 140617 | Robert Price | 12/2020 |
2208 | (85*10^140609+473)/9 | 140610 | Robert Price | 01/2020 |
2209 | 2^466866-1840571 | 140541 | Enzo Doria | 06/2015 |
2210 | 2^466829-459523 | 140530 | Enzo Doria | 05/2015 |
2211 | 2^466692-113213 | 140489 | Enzo Doria | 07/2014 |
2212 | (44*10^140410-503)/9 | 140411 | Robert Price | 12/2018 |
2213 | 153^64265-2 | 140400 | Lelio R Paula | 11/2017 |
2214 | (112*10^140357-1)/3 | 140359 | Serge Batalov | 12/2014 |
2215 | (29*10^140337-17)/3 | 140338 | Robert Price | 09/2019 |
2216 | 2^466181-112735 | 140335 | Enzo Doria | 07/2014 |
2217 | (5*10^140281-119)/3 | 140282 | Robert Price | 03/2018 |
2218 | (27*2^465872+1)/691 | 140241 | Lelio R Paula | 08/2015 |
2219 | 2^465803-114817 | 140221 | Enzo Doria | 05/2015 |
2220 | floor(Pi*10^140344)%10^140165 | 140165 | Serge Batalov | 08/2012 |
2221 | 2^465503-112735 | 140131 | Enzo Doria | 07/2014 |
2222 | (16*10^140091+167)/3 | 140092 | Robert Price | 06/2019 |
2223 | 6^179964+179964^6-1 | 140040 | Robert Price | 04/2014 |
2224 | (89*10^140008-179)/9 | 140009 | Robert Price | 12/2019 |
2225 | (29*2^465077-1)/9 | 140003 | Lelio R Paula | 08/2015 |
2226 | 2^465072-344789 | 140001 | Enzo Doria | 07/2015 |
2227 | 5*10^139945+81 | 139946 | Robert Price | 03/2019 |
2228 | 2^464864-17 | 139939 | Lelio R Paula | 09/2017 |
2229 | 8*10^139920+51 | 139921 | Robert Price | 08/2019 |
2230 | (85*10^139913-1)/3 | 139915 | Robert Price | 02/2020 |
2231 | (17^113621+1)/18 | 139804 | Paul Bourdelais | 09/2011 |
2232 | 2*10^139800-21 | 139801 | Robert Price | 09/2020 |
2233 | (13*10^139782+191)/3 | 139783 | Robert Price | 11/2018 |
2234 | (19*2^464312-1)/3 | 139773 | Lelio R Paula | 08/2015 |
2235 | (74*10^139723+133)/9 | 139724 | Robert Price | 06/2019 |
2236 | 3*10^139722+13 | 139723 | Robert Price | 07/2018 |
2237 | 77*10^139692+3 | 139694 | Robert Price | 06/2020 |
2238 | (31*2^463970+1)/5 | 139670 | Lelio R Paula | 08/2015 |
2239 | (18203^(2^15)+(18201)^(2^15))/2 | 139597 | Robert Price | 05/2018 |
2240 | 2^463633-2788423 | 139568 | Enzo Doria | 11/2015 |
2241 | 2^463271-109501 | 139459 | Enzo Doria | 04/2014 |
2242 | (5*10^139457+79)/3 | 139458 | Robert Price | 03/2018 |
2243 | 2^463237-99 | 139449 | Lelio R Paula | 09/2017 |
2244 | (49*10^139442+383)/9 | 139443 | Robert Price | 04/2019 |
2245 | (7*10^139417-547)/9 | 139417 | Robert Price | 01/2018 |
2246 | (10^139398+191)/3 | 139398 | Robert Price | 02/2018 |
2247 | 2^463019+79463 | 139383 | Enzo Doria | 06/2016 |
2248 | 3^292129-112756 | 139381 | Enzo Doria | 07/2014 |
2249 | (15*2^462951+1)/527 | 139361 | Lelio R Paula | 08/2015 |
2250 | 2^462855-1865707 | 139334 | Enzo Doria | 02/2016 |