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 | (68*10^191754-113)/9 | 191755 | Robert Price | 03/2021 |
2002 | 2^636930-11 | 191736 | Stefano Morozzi | 07/2022 |
2003 | (47*10^191732+259)/9 | 191733 | Robert Price | 03/2022 |
2004 | 2^636661+1876577 | 191655 | Enzo Doria | 07/2016 |
2005 | (9^200807-2^200807)/7 | 191618 | Jon Grantham | 07/2023 |
2006 | (34^125101-1)/33 | 191589 | Paul Bourdelais | 11/2017 |
2007 | fibonacci(916540)-2 | 191546 | Jon Grantham | 08/2023 |
2008 | 2^636177-1855123 | 191509 | Enzo Doria | 09/2015 |
2009 | 2^636158-113627 | 191503 | Enzo Doria | 08/2014 |
2010 | (37*25^136966+63)/4 | 191472 | Curtis Bright | 05/2015 |
2011 | 2^636035+79823 | 191466 | Enzo Doria | 06/2016 |
2012 | 2^636011-1860847 | 191459 | Enzo Doria | 11/2015 |
2013 | 2^635926-1860611 | 191433 | Enzo Doria | 11/2015 |
2014 | 3*2^635755+5 | 191382 | Jon Grantham | 07/2023 |
2015 | 3*10^191152-7 | 191153 | Robert Price | 08/2015 |
2016 | 2^634883-1865677 | 191119 | Enzo Doria | 02/2016 |
2017 | 2^634801+1871957 | 191095 | Enzo Doria | 05/2016 |
2018 | (77*10^191080+157)/9 | 191081 | Robert Price | 05/2022 |
2019 | (35*10^191073+1)/9 | 191074 | Serge Batalov | 07/2010 |
2020 | 9*10^191018-41 | 191019 | Robert Price | 08/2020 |
2021 | 2^634470-59861 | 190995 | Enzo Doria | 11/2015 |
2022 | 2^634368-29519 | 190964 | Enzo Doria | 08/2015 |
2023 | (19*10^190935+11)/3 | 190936 | Robert Price | 07/2023 |
2024 | 2^634188-350609 | 190910 | Enzo Doria | 06/2015 |
2025 | (65*10^190899+493)/9 | 190900 | Robert Price | 08/2021 |
2026 | (62*10^190795-413)/9 | 190796 | Robert Price | 09/2021 |
2027 | (26*10^190789-131)/3 | 190790 | Robert Price | 10/2019 |
2028 | (13*10^190732-61)/3 | 190733 | Robert Price | 11/2018 |
2029 | 3*2^633524+5 | 190711 | Jon Grantham | 07/2023 |
2030 | (7*10^190683+143)/3 | 190684 | Robert Price | 03/2018 |
2031 | 2^633088-113879 | 190579 | Enzo Doria | 10/2014 |
2032 | 2^633064+1874911 | 190572 | Enzo Doria | 06/2016 |
2033 | 5*3^399402+2^399402 | 190564 | Jon Grantham | 08/2023 |
2034 | (43*10^190547+83)/9 | 190548 | Robert Price | 11/2018 |
2035 | 99*10^190543+7 | 190545 | Robert Price | 05/2021 |
2036 | (299*10^190533+1)/3 | 190535 | Robert Price | 04/2020 |
2037 | (1744*10^190519-1747)/3 | 190522 | Robert Price | 12/2020 |
2038 | 2^632815-15 | 190497 | Stefano Morozzi | 07/2022 |
2039 | 2^632795-351205 | 190491 | Enzo Doria | 06/2015 |
2040 | 41171^41272+41272^41171 | 190454 | Norbert Schneider | 05/2023 |
2041 | (16*10^190420-19)/3 | 190421 | Robert Price | 02/2019 |
2042 | (43*10^190351+101)/9 | 190352 | Robert Price | 11/2018 |
2043 | 2^632156+78391 | 190298 | Enzo Doria | 06/2016 |
2044 | 97*10^190295-3 | 190297 | Robert Price | 05/2020 |
2045 | 2^632082-1851971 | 190276 | Enzo Doria | 08/2015 |
2046 | 2^632020+1880311 | 190257 | Enzo Doria | 09/2016 |
2047 | (5*10^190242-77)/9 | 190242 | Robert Price | 04/2021 |
2048 | 7*10^190210+13 | 190211 | Robert Price | 03/2021 |
2049 | 2^631637-462373 | 190142 | Enzo Doria | 08/2015 |
2050 | 2*10^190126-21 | 190127 | Robert Price | 09/2020 |
2051 | (28*10^190105+191)/3 | 190106 | Robert Price | 02/2020 |
2052 | 2*10^190093+33 | 190094 | Robert Price | 11/2021 |
2053 | (68*10^190053-113)/9 | 190054 | Robert Price | 03/2021 |
2054 | 2^631183+213413 | 190006 | Enzo Doria | 09/2023 |
2055 | 2^631122-39251 | 189987 | Enzo Doria | 08/2015 |
2056 | (26*10^189928-89)/9 | 189929 | Robert Price | 06/2018 |
2057 | 2^630839-97 | 189902 | Yves Minet | 07/2017 |
2058 | 99*10^189876-7 | 189878 | Robert Price | 07/2020 |
2059 | 2^630359+1788953 | 189757 | Enzo Doria | 05/2016 |
2060 | 2^630228+1871971 | 189718 | Enzo Doria | 05/2016 |
2061 | 2^630108+781 | 189682 | Lelio R Paula | 08/2013 |
2062 | 2^629870-1867811 | 189610 | Enzo Doria | 04/2016 |
2063 | (359^74189-1)/358 | 189558 | Alexander Zhirkov | 11/2017 |
2064 | 2^629326+2792929 | 189447 | Enzo Doria | 02/2016 |
2065 | 4*10^189401+13 | 189402 | Robert Price | 02/2021 |
2066 | (10^189385-61)/3 | 189385 | Serge Batalov | 12/2014 |
2067 | 2^629036+1874011 | 189359 | Enzo Doria | 06/2016 |
2068 | 2^628930+80779 | 189327 | Enzo Doria | 07/2016 |
2069 | (134*10^189309-11)/3 | 189311 | Robert Price | 01/2022 |
2070 | 9*10^189306-11 | 189307 | Mischa Rodermond | 04/2023 |
2071 | 2^628582+1179439 | 189223 | Enzo Doria | 06/2016 |
2072 | (44*10^188999-503)/9 | 189000 | Robert Price | 12/2018 |
2073 | (16*10^188823+167)/3 | 188824 | Robert Price | 06/2019 |
2074 | (10^188796+7)/4967/71933 | 188788 | Alfred Reich | 08/2021 |
2075 | 2^627037-2785843 | 188757 | Enzo Doria | 10/2015 |
2076 | (10^188566+7)/311640753181661 | 188552 | Alfred Reich | 04/2022 |
2077 | (16*10^188457+161)/3 | 188458 | Robert Price | 02/2019 |
2078 | L(901657) | 188436 | Renaud Lifchitz | 11/2008 |
2079 | (19*10^188375+137)/3 | 188376 | Robert Price | 03/2022 |
2080 | 2^625651+74213 | 188340 | Enzo Doria | 04/2016 |
2081 | 2^625584-460919 | 188320 | Enzo Doria | 06/2015 |
2082 | (8*10^188308-71)/9 | 188308 | Serge Batalov | 11/2022 |
2083 | 5^269399-3^269399-1 | 188302 | Jon Grantham | 08/2023 |
2084 | (14*10^188218+139)/9 | 188219 | Robert Price | 11/2021 |
2085 | (10^188164+359)/9 | 188164 | Serge Batalov | 12/2014 |
2086 | 2^624481-113611 | 187988 | Enzo Doria | 08/2014 |
2087 | (2^624389+125582)/2 | 187960 | Stefano Morozzi | 09/2019 |
2088 | (2^624389+166982)/2 | 187960 | Stefano Morozzi | 09/2019 |
2089 | (2^624389+179886)/2 | 187960 | Stefano Morozzi | 09/2019 |
2090 | (2^624387+64466)/2 | 187959 | Stefano Morozzi | 09/2019 |
2091 | (2^624378+112990)/2 | 187957 | Stefano Morozzi | 09/2019 |
2092 | (2^624380+161986)/2 | 187957 | Stefano Morozzi | 09/2019 |
2094 | (2^624376+55762)/2 | 187956 | Stefano Morozzi | 09/2019 |
2093 | (2^624377+169506)/2 | 187956 | Stefano Morozzi | 09/2019 |
2095 | (2^624355+80178)/2 | 187950 | Stefano Morozzi | 09/2019 |
2096 | (2^624351+61350)/2 | 187949 | Stefano Morozzi | 09/2019 |
2097 | (2^624341+63462)/2 | 187946 | Stefano Morozzi | 09/2019 |
2098 | (2^624344+135990)/2 | 187946 | Stefano Morozzi | 09/2019 |
2099 | (2^624344+183898)/2 | 187946 | Stefano Morozzi | 09/2019 |
2101 | (2^624340+126982)/2 | 187945 | Stefano Morozzi | 09/2019 |
2100 | (2^624338+196110)/2 | 187945 | Stefano Morozzi | 09/2019 |
2102 | (2^624337+105162)/2 | 187944 | Stefano Morozzi | 09/2019 |
2103 | (2^624333+44594)/2 | 187943 | Stefano Morozzi | 09/2019 |
2104 | (2^624321+88742)/2 | 187940 | Stefano Morozzi | 09/2019 |
2105 | (2^624324+135178)/2 | 187940 | Stefano Morozzi | 09/2019 |
2106 | (2^624317+14186)/2 | 187938 | Stefano Morozzi | 09/2019 |
2107 | (2^624313+24450)/2 | 187937 | Stefano Morozzi | 09/2019 |
2108 | (2^624309+184590)/2 | 187936 | Stefano Morozzi | 09/2019 |
2110 | (2^624307+152658)/2 | 187935 | Stefano Morozzi | 09/2019 |
2109 | (2^624307+185978)/2 | 187935 | Stefano Morozzi | 09/2019 |
2111 | (2^624303+40814)/2 | 187934 | Stefano Morozzi | 09/2019 |
2112 | (2^624304+99238)/2 | 187934 | Stefano Morozzi | 09/2019 |
2113 | 2^624077-112723 | 187866 | Enzo Doria | 07/2014 |
2114 | 2^624008+1874941 | 187846 | Enzo Doria | 06/2016 |
2115 | 2^623870+78679 | 187804 | Enzo Doria | 05/2016 |
2116 | 2^623714-114035 | 187757 | Enzo Doria | 11/2014 |
2117 | 2^623651-1834251 | 187738 | Enzo Doria | 07/2015 |
2118 | (7*10^187645+53)/3 | 187646 | Robert Price | 05/2018 |
2119 | 21650^43269+43269^21650 | 187591 | Norbert Schneider | 02/2023 |
2120 | 2^623146+1879609 | 187586 | Enzo Doria | 09/2016 |
2121 | (401*10^187512+13)/9 | 187514 | Robert Price | 09/2021 |
2122 | (5*10^187471+91)/3 | 187472 | Robert Price | 04/2018 |
2123 | 2^622500+81931 | 187392 | Enzo Doria | 08/2016 |
2124 | (13*10^187382-157)/9 | 187383 | Robert Price | 05/2021 |
2125 | 2^622300-1861739 | 187331 | Enzo Doria | 12/2015 |
2126 | (22*10^187317-91)/3 | 187318 | Robert Price | 02/2022 |
2127 | (38*10^187281+349)/9 | 187282 | Robert Price | 10/2018 |
2128 | 2^622052+76771 | 187257 | Enzo Doria | 05/2016 |
2129 | 2^622005-24091 | 187243 | Enzo Doria | 08/2013 |
2130 | 7*10^187205-57 | 187206 | Robert Price | 06/2020 |
2131 | 2^621879-351787 | 187205 | Enzo Doria | 07/2015 |
2132 | (5*10^187168-53)/3 | 187169 | Robert Price | 09/2020 |
2133 | (23*10^187164-143)/3 | 187165 | Robert Price | 12/2019 |
2134 | 2^621700+79411 | 187151 | Enzo Doria | 06/2016 |
2135 | 2^621597+39 | 187120 | Jon Grantham | 08/2023 |
2136 | (7*10^187101+131)/3 | 187102 | Robert Price | 06/2020 |
2137 | (2*10^187062-71)/3 | 187062 | Robert Price | 12/2017 |
2138 | 2^620929-113203 | 186919 | Enzo Doria | 07/2014 |
2139 | 10^186842+168223 | 186843 | Patrick De Geest | 07/2018 |
2140 | 10^186842-8001 | 186842 | Robert Price | 03/2018 |
2141 | 2^620347-1777 | 186744 | Lelio R Paula | 07/2013 |
2142 | (11*10^186659+133)/9 | 186660 | Robert Price | 04/2022 |
2143 | 11^179143-4170 | 186559 | Enzo Doria | 12/2012 |
2144 | (5^266863+1)/6 | 186529 | Paul Bourdelais | 07/2010 |
2145 | 2*10^186517+33 | 186518 | Robert Price | 11/2021 |
2146 | 2^619297-2792353 | 186427 | Enzo Doria | 02/2016 |
2147 | (49*10^186409+383)/9 | 186410 | Robert Price | 04/2019 |
2148 | 2^618940-460139 | 186320 | Enzo Doria | 06/2015 |
2149 | 2^618046+74179 | 186051 | Enzo Doria | 04/2016 |
2150 | 3*10^186019+11 | 186020 | Robert Price | 07/2018 |
2151 | (16*10^185978-19)/3 | 185979 | Robert Price | 02/2019 |
2152 | (22*10^185950-7)/3 | 185951 | Robert Price | 12/2022 |
2153 | 2*10^185930-3 | 185931 | Robert Price | 07/2015 |
2154 | (82^97151+1)/83 | 185928 | Paul Bourdelais | 10/2017 |
2155 | 4*10^185905+91 | 185906 | Robert Price | 08/2018 |
2156 | (34*10^185902-601)/9 | 185903 | Robert Price | 07/2022 |
2157 | (19*10^185873+131)/3 | 185874 | Robert Price | 06/2020 |
2158 | (37*10^185811+377)/9 | 185812 | Robert Price | 11/2018 |
2159 | 2^617250+9385 | 185811 | Lelio R Paula | 07/2013 |
2160 | 2^617130+9 | 185775 | Stefano Morozzi | 05/2021 |
2161 | 39^116749+2 | 185756 | Robert Price | 04/2015 |
2162 | 2^616924-39719 | 185713 | Enzo Doria | 09/2015 |
2163 | 2^616712+1867621 | 185649 | Enzo Doria | 04/2016 |
2164 | (3^388897+2^388897)/5 | 185551 | Jean-Louis Charton | 09/2011 |
2165 | 8*10^185529-67 | 185530 | Robert Price | 08/2019 |
2166 | (5*10^185520-11)/3 | 185521 | Tyler Busby | 02/2023 |
2167 | 5*2^616124+3 | 185473 | Jon Grantham | 07/2023 |
2168 | 2^615916+26791 | 185410 | Enzo Doria | 12/2023 |
2169 | 4*10^185355+9 | 185356 | Robert Price | 05/2015 |
2170 | (16*10^185347+89)/3 | 185348 | Robert Price | 03/2021 |
2171 | 2^615622+76639 | 185321 | Enzo Doria | 05/2016 |
2172 | 2^615591+1879223 | 185312 | Enzo Doria | 08/2016 |
2173 | 2^615587+217013 | 185311 | Enzo Doria | 12/2023 |
2174 | 2^615503+1876643 | 185285 | Enzo Doria | 07/2016 |
2175 | 5^265007-4^265007 | 185232 | Jean-Louis Charton | 02/2013 |
2177 | (4*10^185221-91)/3 | 185222 | Robert Price | 03/2018 |
2176 | (7*10^185221+131)/3 | 185222 | Robert Price | 06/2020 |
2178 | (62*10^185216-413)/9 | 185217 | Robert Price | 09/2021 |
2179 | (232*10^185180+17)/3 | 185182 | Robert Price | 04/2021 |
2180 | 56*10^185139-3 | 185141 | Robert Price | 03/2020 |
2181 | (28*10^185105+47)/3 | 185106 | Robert Price | 02/2021 |
2182 | 2^614783-2790097 | 185069 | Enzo Doria | 12/2015 |
2183 | 2^614745-29323 | 185057 | Enzo Doria | 07/2015 |
2184 | (((((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840 | 185052 | Serge Batalov | 07/2013 |
2185 | (49*10^184993-67)/9 | 184994 | Robert Price | 03/2019 |
2186 | (5*10^184954-191)/3 | 184955 | Robert Price | 05/2021 |
2187 | (14*10^184950+211)/9 | 184951 | Robert Price | 04/2018 |
2188 | 2^614360+215221 | 184941 | Enzo Doria | 11/2023 |
2189 | 2^614315+81353 | 184928 | Enzo Doria | 08/2016 |
2190 | 2^614294-1849961 | 184921 | Enzo Doria | 08/2015 |
2191 | 2^614266+1877809 | 184913 | Enzo Doria | 08/2016 |
2192 | (185*10^184878-11)/3 | 184880 | Robert Price | 10/2020 |
2193 | 91*10^184867-9 | 184869 | Robert Price | 06/2021 |
2194 | (14789^44521-1)/(14789^211-1) | 184770 | Ryan Propper | 11/2023 |
2195 | (85*10^184768+77)/9 | 184769 | Robert Price | 07/2022 |
2196 | 2^613715-1860727 | 184747 | Enzo Doria | 11/2015 |
2197 | (4*10^184730+89)/3 | 184731 | Robert Price | 08/2021 |
2198 | 2*10^184723+41 | 184724 | Robert Price | 05/2018 |
2199 | (49*10^184703+293)/9 | 184704 | Robert Price | 03/2021 |
2200 | 2^613487-349987 | 184678 | Enzo Doria | 07/2015 |
2201 | (95^93377+1)/96 | 184672 | Paul Bourdelais | 11/2017 |
2202 | 14^161089-161089^14 | 184629 | Serge Batalov | 05/2016 |
2203 | 2^613033-114621 | 184542 | Enzo Doria | 05/2015 |
2204 | 3*2^612759+5 | 184460 | Jon Grantham | 07/2023 |
2205 | 2^612587+213413 | 184408 | Enzo Doria | 09/2023 |
2206 | (9^193247+1)/10 | 184404 | Paul Bourdelais | 02/2010 |
2207 | 2^612424+212341 | 184358 | Enzo Doria | 08/2023 |
2208 | 2^612341-1604809 | 184334 | Enzo Doria | 05/2015 |
2209 | (11*10^184308-161)/3 | 184309 | Robert Price | 11/2021 |
2210 | (2^611999-1)/18464214225958267477777390354183 | 184199 | Dario Alpern | 06/2017 |
2211 | (14*10^184150+37)/3 | 184151 | Robert Price | 12/2018 |
2212 | 2^611731-4837 | 184150 | Yves Minet | 08/2018 |
2213 | 3*10^184124-7 | 184125 | Robert Price | 08/2015 |
2214 | 2^611644+29371 | 184124 | Enzo Doria | 02/2024 |
2215 | 9*10^184122-11 | 184123 | Mischa Rodermond | 04/2023 |
2216 | 2^611628-24537 | 184119 | Enzo Doria | 08/2013 |
2217 | 3*10^184110+13 | 184111 | Robert Price | 07/2018 |
2218 | (14280^44521-1)/(14280^211-1) | 184097 | Ryan Propper | 11/2023 |
2219 | (194*10^184093+1)/3 | 184095 | Robert Price | 01/2022 |
2220 | 2^611460+71881 | 184068 | Enzo Doria | 04/2016 |
2221 | 4*10^184004-63 | 184005 | Robert Price | 10/2018 |
2222 | 2^610982-1867301 | 183924 | Enzo Doria | 03/2016 |
2223 | (10^183917+1)/11/9952852373 | 183906 | Alfred Reich | 02/2024 |
2224 | 3*10^183872-49 | 183873 | Robert Price | 07/2018 |
2225 | 2^610597+1178897 | 183809 | Enzo Doria | 05/2016 |
2226 | (8*10^183773-269)/9 | 183773 | Robert Price | 09/2021 |
2227 | (65*10^183768-29)/9 | 183769 | Robert Price | 01/2021 |
2228 | (17*10^183764+31)/3 | 183765 | Robert Price | 01/2019 |
2229 | (28*10^183760-1)/9 | 183761 | Serge Batalov | 10/2014 |
2230 | 2^610401+1277 | 183750 | Lelio R Paula | 07/2013 |
2231 | 2^610356-1849019 | 183736 | Enzo Doria | 08/2015 |
2232 | (63^102103+1)/64 | 183717 | Paul Bourdelais | 10/2017 |
2233 | 23*10^183710-7 | 183712 | Robert Price | 02/2020 |
2234 | (2^610289+1)/(3*528882550291) | 183704 | Gord Palameta | 08/2018 |
2235 | (4*10^183659+149)/9 | 183659 | Robert Price | 04/2018 |
2236 | (4*10^183657+41)/9 | 183657 | Serge Batalov | 11/2022 |
2237 | 7*10^183596+39 | 183597 | Robert Price | 01/2023 |
2238 | 2^609846-1834251 | 183582 | Enzo Doria | 07/2015 |
2239 | 10^183546+1547323 | 183547 | Patrick De Geest | 02/2021 |
2241 | 10^183546-800000001 | 183546 | Robert Price | 09/2020 |
2240 | 10^183546-584297 | 183546 | Patrick De Geest | 02/2021 |
2242 | 7*10^183511+39 | 183512 | Robert Price | 01/2023 |
2243 | (23*10^183477-71)/3 | 183478 | Robert Price | 07/2020 |
2244 | 11^176176-240 | 183469 | Enzo Doria | 11/2012 |
2245 | (41^113749+1)/42 | 183451 | Paul Bourdelais | 11/2017 |
2246 | 6*10^183429+53 | 183430 | Robert Price | 05/2020 |
2247 | (28*10^183403+191)/3 | 183404 | Robert Price | 02/2020 |
2248 | (101^91463+1)/102 | 183320 | Paul Bourdelais | 11/2017 |
2249 | 2^608965-113989 | 183317 | Enzo Doria | 10/2014 |
2250 | A005165(43592) | 183312 | Serge Batalov | 07/2017 |