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 | (194*10^169461-17)/3 | 169463 | Robert Price | 09/2020 |
2002 | 2^562749+2793047 | 169405 | Enzo Doria | 02/2016 |
2003 | (154*10^169337-1)/9 | 169339 | Serge Batalov | 12/2014 |
2004 | (109*10^169337+11)/3 | 169339 | Robert Price | 06/2022 |
2005 | (56*10^169311+691)/9 | 169312 | Robert Price | 05/2019 |
2006 | 2^562369-29503 | 169290 | Enzo Doria | 08/2015 |
2007 | (17*10^169283+19)/9 | 169284 | Robert Price | 03/2021 |
2008 | 10^169074-5000000001 | 169074 | Robert Price | 09/2022 |
2009 | 8*10^169071+59 | 169072 | Robert Price | 11/2021 |
2010 | (10^169044+173)/3 | 169044 | Robert Price | 05/2021 |
2011 | (22*10^169019+383)/9 | 169020 | Robert Price | 05/2018 |
2012 | (64*10^168935+143)/9 | 168936 | Robert Price | 05/2021 |
2013 | (22*10^168906+47)/3 | 168907 | Robert Price | 06/2021 |
2014 | 5*10^168807+87 | 168808 | Robert Price | 02/2019 |
2015 | (77*10^168803-149)/9 | 168804 | Robert Price | 03/2022 |
2016 | 4*10^168788-47 | 168789 | Robert Price | 10/2021 |
2017 | (154*10^168781+17)/9 | 168783 | Robert Price | 06/2022 |
2018 | (418*10^168751-31)/9 | 168753 | Robert Price | 05/2022 |
2019 | (29*10^168711+691)/9 | 168712 | Robert Price | 08/2021 |
2020 | 2^560405-39943 | 168699 | Enzo Doria | 09/2015 |
2021 | (397*10^168634+17)/9 | 168636 | Robert Price | 03/2022 |
2022 | 89*10^168614-7 | 168616 | Robert Price | 04/2020 |
2023 | (23*10^168608-83)/3 | 168609 | Robert Price | 08/2019 |
2024 | (94*10^168541+11)/3 | 168543 | Robert Price | 04/2020 |
2025 | 2^559686-350471 | 168483 | Enzo Doria | 06/2015 |
2026 | 2^559672-1855889 | 168479 | Enzo Doria | 09/2015 |
2027 | (5*10^168379-473)/9 | 168379 | Robert Price | 01/2018 |
2028 | 2^559000-59939 | 168276 | Enzo Doria | 11/2015 |
2029 | (16*10^168181+41)/3 | 168182 | Robert Price | 03/2019 |
2030 | (19*10^168173+77)/3 | 168174 | Robert Price | 05/2019 |
2031 | (74*10^168170+529)/9 | 168171 | Robert Price | 02/2022 |
2032 | 2^558640-114149 | 168168 | Enzo Doria | 12/2014 |
2033 | 2^558498-1849277 | 168125 | Enzo Doria | 08/2015 |
2034 | 2^558476-1850249 | 168119 | Enzo Doria | 08/2015 |
2035 | (7*10^168067+71)/3 | 168068 | Robert Price | 05/2018 |
2036 | 2^558150-1852211 | 168020 | Enzo Doria | 08/2015 |
2037 | 2^558125-113661 | 168013 | Enzo Doria | 09/2014 |
2038 | 393^64747-392^64747 | 167980 | Jean-Louis Charton | 08/2012 |
2039 | (13*10^167911+179)/3 | 167912 | Robert Price | 11/2018 |
2040 | (26*10^167861-167)/3 | 167862 | Robert Price | 10/2019 |
2041 | 39^105456+2 | 167788 | Robert Price | 04/2015 |
2042 | (28^115877-1)/27 | 167691 | Paul Bourdelais | 12/2017 |
2043 | 6*10^167662-13 | 167663 | Robert Price | 07/2022 |
2044 | 10^167639+121911 | 167640 | Patrick De Geest | 07/2018 |
2045 | 10^167639-8001 | 167639 | Robert Price | 03/2018 |
2046 | 2^556504-462059 | 167525 | Enzo Doria | 07/2015 |
2047 | (88*10^167522-637)/9 | 167523 | Robert Price | 11/2020 |
2048 | 5*10^167510+9 | 167511 | Robert Price | 08/2015 |
2049 | 7*10^167437-93 | 167438 | Robert Price | 08/2020 |
2050 | 11^160663+6 | 167314 | Paul Bourdelais | 02/2021 |
2051 | (7*10^167308-493)/9 | 167308 | Robert Price | 04/2021 |
2052 | 2^555343+1874333 | 167175 | Enzo Doria | 06/2016 |
2053 | 2^555210-461261 | 167135 | Enzo Doria | 07/2015 |
2054 | 9*10^166996+59 | 166997 | Robert Price | 01/2020 |
2055 | (68*10^166976-329)/9 | 166977 | Robert Price | 03/2021 |
2056 | (64*10^166901+683)/9 | 166902 | Robert Price | 06/2022 |
2057 | 2^554372+1880527 | 166883 | Enzo Doria | 11/2016 |
2058 | 2^554299+72173 | 166861 | Enzo Doria | 04/2016 |
2059 | (185*10^166837+7)/3 | 166839 | Robert Price | 03/2020 |
2060 | (14*10^166813+139)/9 | 166814 | Robert Price | 11/2021 |
2061 | 2^554037-2797063 | 166782 | Enzo Doria | 03/2016 |
2062 | (49*10^166721-139)/9 | 166722 | Robert Price | 03/2019 |
2063 | 2*10^166646+69 | 166647 | Robert Price | 06/2020 |
2064 | 2^553484-29819 | 166616 | Enzo Doria | 08/2015 |
2065 | (188*10^166611+7)/3 | 166613 | Robert Price | 03/2020 |
2066 | 2^553383-2784217 | 166585 | Enzo Doria | 10/2015 |
2067 | 2^553069+76427 | 166491 | Enzo Doria | 05/2016 |
2068 | 2*10^166458+51 | 166459 | Robert Price | 05/2018 |
2069 | 589^60089-588^60089 | 166454 | Jean-Louis Charton | 09/2012 |
2070 | 11^159756-2424 | 166369 | Mike Oakes | 06/2010 |
2071 | 2^552649-361275 | 166364 | Enzo Doria | 05/2014 |
2072 | 2^552628-113669 | 166358 | Enzo Doria | 09/2014 |
2073 | (19*10^166325+137)/3 | 166326 | Robert Price | 03/2022 |
2074 | 10^166324+550603 | 166325 | Patrick De Geest | 07/2018 |
2075 | 10^166324-801 | 166324 | Robert Price | 02/2018 |
2076 | (85*10^166297-1)/3 | 166299 | Robert Price | 02/2020 |
2077 | 8*10^166227+87 | 166228 | Robert Price | 10/2019 |
2078 | 2^552137+1879397 | 166210 | Enzo Doria | 09/2016 |
2079 | (103*10^166126+11)/3 | 166128 | Robert Price | 01/2021 |
2080 | (22*10^166077+179)/3 | 166078 | Robert Price | 02/2022 |
2081 | (23^121949-1)/22 | 166061 | Paul Bourdelais | 10/2017 |
2082 | 2^551542+19249 | 166031 | Payam Samidoost | 08/2002 |
2083 | 2^551535-23821 | 166029 | Enzo Doria | 08/2013 |
2084 | 2^551507-23557 | 166021 | Enzo Doria | 08/2013 |
2085 | 10^165947-7000000001 | 165947 | Robert Price | 12/2021 |
2086 | 4*10^165938-47 | 165939 | Robert Price | 10/2021 |
2087 | 7*10^165916-3 | 165917 | Robert Price | 08/2015 |
2088 | (29*10^165796+67)/3 | 165797 | Robert Price | 12/2019 |
2089 | 81^86837-80^86837 | 165728 | Robert Price | 01/2015 |
2090 | (71*10^165716-17)/9 | 165717 | Serge Batalov | 01/2023 |
2091 | (71*10^165663-539)/9 | 165664 | Robert Price | 07/2019 |
2092 | 2^550270-69551 | 165648 | Enzo Doria | 12/2015 |
2093 | (202*10^165569-13)/9 | 165571 | Robert Price | 01/2020 |
2094 | 2^550000-23625 | 165567 | Enzo Doria | 08/2013 |
2095 | 2^549675-1852057 | 165469 | Enzo Doria | 08/2015 |
2096 | (529*10^165394-7)/9 | 165396 | Robert Price | 08/2021 |
2097 | -tau(103^14938) | 165374 | Nik Lygeros / Olivier Rozier | 10/2013 |
2098 | (28*10^165358-1)/9 | 165359 | Serge Batalov | 10/2014 |
2099 | 2^549278+1879669 | 165350 | Enzo Doria | 09/2016 |
2100 | (7*10^165350-439)/9 | 165350 | Robert Price | 02/2018 |
2101 | 2^549252-351269 | 165342 | Enzo Doria | 06/2015 |
2102 | 67*10^165327-9 | 165329 | Robert Price | 03/2021 |
2104 | (14*10^165282+73)/3 | 165283 | Robert Price | 12/2018 |
2103 | (8*10^165283-629)/9 | 165283 | Robert Price | 01/2022 |
2105 | 2^549049+2797607 | 165281 | Enzo Doria | 04/2016 |
2106 | (218*10^165258+7)/9 | 165260 | Robert Price | 06/2022 |
2107 | (7*10^165239+107)/3 | 165240 | Robert Price | 02/2018 |
2108 | (44^100511-1)/43 | 165184 | Paul Bourdelais | 11/2017 |
2109 | (7*10^165134+107)/3 | 165135 | Robert Price | 02/2018 |
2110 | (301*10^165082+23)/9 | 165084 | Robert Price | 06/2021 |
2111 | (23*10^165060-83)/3 | 165061 | Robert Price | 08/2019 |
2112 | 7*10^165023-27 | 165024 | Robert Price | 11/2022 |
2113 | (76*10^164984+77)/9 | 164985 | Robert Price | 10/2019 |
2114 | (4*10^164929-211)/9 | 164929 | Robert Price | 03/2022 |
2115 | 10^164878-500001 | 164878 | Robert Price | 03/2022 |
2116 | 9*10^164791-89 | 164792 | Robert Price | 04/2020 |
2117 | 2^547178-1856501 | 164717 | Enzo Doria | 10/2015 |
2118 | 11^158160-60 | 164707 | Enzo Doria | 07/2012 |
2119 | (26*10^164643-167)/3 | 164644 | Robert Price | 10/2019 |
2120 | (10^164643+83)/3 | 164643 | Robert Price | 08/2021 |
2121 | 2^546398-59201 | 164483 | Enzo Doria | 10/2015 |
2122 | 2^546332-114893 | 164463 | Enzo Doria | 05/2015 |
2123 | (65*10^164427+277)/9 | 164428 | Robert Price | 08/2021 |
2124 | 2^545958+1876189 | 164350 | Enzo Doria | 07/2016 |
2125 | 2^545878-49841 | 164326 | Enzo Doria | 10/2015 |
2126 | (10^164284+77)/3 | 164284 | Robert Price | 01/2018 |
2127 | 2^545668-4019 | 164263 | Yves Minet | 04/2017 |
2128 | 10^164216+780639 | 164217 | Patrick De Geest | 08/2021 |
2130 | 10^164216-50000000001 | 164216 | Robert Price | 12/2020 |
2129 | 10^164216-891051 | 164216 | Patrick De Geest | 08/2021 |
2131 | (13*53^238539*617^9)/10^247183 | 164151 | Hans Havermann | 07/2017 |
2132 | (16*10^164122+197)/3 | 164123 | Robert Price | 04/2019 |
2133 | 2^545185+76757 | 164118 | Enzo Doria | 05/2016 |
2134 | 2^545076-114329 | 164085 | Enzo Doria | 01/2015 |
2135 | (2*10^164043-143)/3 | 164043 | Robert Price | 03/2018 |
2136 | (29^112153+1)/30 | 164011 | Lelio R Paula | 03/2017 |
2137 | 2^544757+1874777 | 163989 | Enzo Doria | 06/2016 |
2138 | 2^544444-457199 | 163894 | Enzo Doria | 01/2015 |
2139 | 2^544264-343529 | 163840 | Enzo Doria | 07/2015 |
2140 | 7*10^163780+9 | 163781 | Robert Price | 09/2015 |
2141 | 2^543989+2793047 | 163758 | Enzo Doria | 02/2016 |
2142 | (28*10^163709+113)/3 | 163710 | Robert Price | 03/2020 |
2143 | (23*10^163523+301)/9 | 163524 | Robert Price | 06/2021 |
2144 | (68*10^163521-329)/9 | 163522 | Robert Price | 03/2021 |
2145 | 35820^35899+35899^35820 | 163489 | Norbert Schneider | 11/2021 |
2146 | 2^542870-69551 | 163421 | Enzo Doria | 12/2015 |
2147 | 2^542395+72533 | 163278 | Enzo Doria | 04/2016 |
2148 | 2^542072-3683 | 163180 | Yves Minet | 03/2017 |
2149 | (682*10^163106-7)/9 | 163108 | Robert Price | 08/2021 |
2150 | 2^541779-1867357 | 163092 | Enzo Doria | 04/2016 |
2151 | (29*10^163056+313)/9 | 163057 | Robert Price | 04/2021 |
2152 | 8*10^162924+9 | 162925 | Robert Price | 12/2015 |
2153 | 2^540905+81557 | 162829 | Enzo Doria | 08/2016 |
2154 | 2^540628-2939 | 162746 | Yves Minet | 09/2014 |
2155 | 919^54919-918^54919 | 162743 | Jean-Louis Charton | 01/2013 |
2156 | (203*10^162689+1)/3 | 162691 | Robert Price | 03/2020 |
2157 | 3*10^162620-59 | 162621 | Robert Price | 09/2018 |
2158 | 3^340694-112960 | 162553 | Enzo Doria | 07/2014 |
2159 | 2^539807-114081 | 162499 | Enzo Doria | 12/2014 |
2160 | 414^62093+62093^414 | 162498 | Norbert Schneider | 03/2022 |
2161 | 2^539654-114893 | 162453 | Enzo Doria | 05/2015 |
2162 | 9*10^162449-71 | 162450 | Robert Price | 11/2020 |
2163 | 2^539617-1851583 | 162441 | Enzo Doria | 08/2015 |
2164 | (509*10^162422+31)/9 | 162424 | Robert Price | 08/2022 |
2165 | (91*10^162410+11)/3 | 162412 | Robert Price | 02/2020 |
2166 | 2^539346+81049 | 162360 | Enzo Doria | 07/2016 |
2167 | 2^539220-115499 | 162322 | Enzo Doria | 07/2015 |
2168 | 2^539217-113713 | 162321 | Enzo Doria | 09/2014 |
2169 | 9^170099-8^170099 | 162316 | Jean-Louis Charton | 05/2005 |
2170 | 4*10^162263-39 | 162264 | Robert Price | 09/2020 |
2171 | (115*10^162153-1)/3 | 162155 | Serge Batalov | 12/2014 |
2172 | (10^162138+11)/3 | 162138 | Serge Batalov | 07/2015 |
2173 | (61*10^162100-403)/9 | 162101 | Robert Price | 10/2021 |
2174 | 2^538473-112723 | 162097 | Enzo Doria | 06/2014 |
2175 | 2^538157+79337 | 162002 | Enzo Doria | 06/2016 |
2176 | 2^538136-114749 | 161996 | Enzo Doria | 05/2015 |
2177 | 2^538104+1531 | 161986 | Lelio R Paula | 01/2013 |
2178 | (9*29^110727-1199)/14 | 161927 | Curtis Bright | 05/2015 |
2179 | 2^537815-22737 | 161899 | Enzo Doria | 07/2013 |
2180 | 2^537804-39839 | 161896 | Enzo Doria | 09/2015 |
2181 | (754*10^161816-7)/9 | 161818 | Robert Price | 05/2020 |
2182 | 2^537458-59201 | 161791 | Enzo Doria | 10/2015 |
2183 | 2*10^161766+51 | 161767 | Robert Price | 05/2018 |
2184 | (7*10^161760+131)/3 | 161761 | Robert Price | 06/2020 |
2185 | 2^537209-49849 | 161717 | Enzo Doria | 10/2015 |
2186 | (58*10^161691+329)/9 | 161692 | Robert Price | 05/2019 |
2187 | 23*10^161668-3 | 161670 | Robert Price | 10/2021 |
2188 | 2^536937+1879397 | 161635 | Enzo Doria | 09/2016 |
2189 | 34*10^161593-9 | 161595 | Robert Price | 06/2020 |
2190 | (31*10^161582-1)/3 | 161584 | Robert Price | 02/2020 |
2191 | 2^536569-22881 | 161524 | Enzo Doria | 07/2013 |
2192 | (46*10^161505+593)/9 | 161506 | Robert Price | 04/2022 |
2193 | 10^161253+573493 | 161254 | Patrick De Geest | 07/2018 |
2194 | 10^161253-8001 | 161253 | Robert Price | 03/2018 |
2195 | 97*10^161247-3 | 161249 | Robert Price | 05/2020 |
2196 | 7*10^161192-9 | 161193 | Robert Price | 10/2015 |
2197 | 2^535415-19 | 161176 | Stefano Morozzi | 07/2022 |
2198 | 2^535269-113803 | 161133 | Enzo Doria | 09/2014 |
2199 | 2^535205-23295 | 161113 | Enzo Doria | 08/2013 |
2200 | (49*10^161105+437)/9 | 161106 | Robert Price | 09/2021 |
2201 | (89^82609+1)/90 | 161036 | Paul Bourdelais | 10/2017 |
2202 | 2^534429-19 | 160880 | Stefano Morozzi | 07/2022 |
2203 | 2^534359-23229 | 160859 | Enzo Doria | 08/2013 |
2204 | 10^160836-1001 | 160836 | Robert Price | 10/2022 |
2205 | 457^60454+60454^457 | 160803 | Norbert Schneider | 07/2021 |
2206 | 8*10^160782+53 | 160783 | Robert Price | 08/2020 |
2207 | 67^88037-66^88037 | 160763 | Robert Price | 10/2013 |
2208 | (88*10^160761-637)/9 | 160762 | Robert Price | 11/2020 |
2209 | 2^533948-1852559 | 160735 | Enzo Doria | 08/2015 |
2210 | 2^533665-23335 | 160650 | Enzo Doria | 08/2013 |
2211 | (19*10^160619+77)/3 | 160620 | Robert Price | 05/2019 |
2212 | 2^533350-5 | 160555 | Stefano Morozzi | 07/2022 |
2213 | (38*10^160508+151)/9 | 160509 | Robert Price | 03/2022 |
2214 | 3^336353-2^336353 | 160482 | Mike Oakes | 10/2007 |
2215 | 2*10^160339-63 | 160340 | Robert Price | 01/2022 |
2216 | (91*10^160249+11)/3 | 160251 | Robert Price | 02/2020 |
2217 | (772*10^160239-7)/9 | 160241 | Robert Price | 06/2022 |
2218 | (2*10^160226+187)/9 | 160226 | Robert Price | 06/2022 |
2219 | 2^532253+1868417 | 160225 | Enzo Doria | 04/2016 |
2220 | 2^532108-291989 | 160181 | Enzo Doria | 07/2015 |
2221 | (28*10^160171-79)/3 | 160172 | Robert Price | 02/2021 |
2222 | (26*10^160166-131)/3 | 160167 | Robert Price | 10/2019 |
2223 | 55*10^160145+9 | 160147 | Robert Price | 03/2022 |
2224 | (17*10^160143+67)/3 | 160144 | Robert Price | 01/2019 |
2225 | (19*10^159921-49)/3 | 159922 | Robert Price | 05/2019 |
2226 | 2^531110-1853771 | 159881 | Enzo Doria | 09/2015 |
2227 | (275^65536+53^65536)/2 | 159864 | Kellen Shenton | 11/2020 |
2228 | 2^530992-1845299 | 159845 | Enzo Doria | 07/2015 |
2229 | 2^530810+25 | 159790 | Stefano Morozzi | 04/2021 |
2230 | (136*10^159736-1)/9 | 159738 | Serge Batalov | 12/2014 |
2231 | 2^530417-113833 | 159672 | Enzo Doria | 09/2014 |
2232 | 2^530406-69701 | 159669 | Enzo Doria | 02/2016 |
2233 | 5*10^159656+41 | 159657 | Robert Price | 03/2019 |
2234 | 8*10^159614+29 | 159615 | Robert Price | 11/2021 |
2235 | 7^188858-188858^7 | 159604 | Norbert Schneider | 05/2018 |
2236 | 7*10^159553-27 | 159554 | Robert Price | 11/2022 |
2237 | (24^115597-1)/23 | 159547 | Paul Bourdelais | 09/2017 |
2238 | (17*10^159438-377)/9 | 159439 | Robert Price | 03/2018 |
2239 | (203*10^159381+1)/3 | 159383 | Robert Price | 03/2020 |
2240 | 89*10^159258-9 | 159260 | Robert Price | 05/2020 |
2241 | 2^528944-461315 | 159229 | Enzo Doria | 07/2015 |
2242 | (13*10^159147+161)/3 | 159148 | Robert Price | 11/2018 |
2243 | 2^528620-113879 | 159131 | Enzo Doria | 10/2014 |
2244 | (47*10^159083+259)/9 | 159084 | Robert Price | 03/2022 |
2245 | 2^528000-23307 | 158944 | Enzo Doria | 08/2013 |
2246 | (29*10^158909+61)/9 | 158910 | Robert Price | 08/2018 |
2247 | 2^527817+1876817 | 158889 | Enzo Doria | 07/2016 |
2248 | 2^527774-49451 | 158876 | Enzo Doria | 10/2015 |
2249 | 2^527745-291553 | 158868 | Enzo Doria | 07/2015 |
2250 | (91*10^158862+11)/3 | 158864 | Robert Price | 02/2020 |