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 ? |
9001 | 2^299949-1005 | 90294 | Lelio R Paula | 11/2013 |
9002 | (2^299947+2054)/2 | 90293 | Stefano Morozzi | 01/2020 |
9003 | 21740^14229+14229^21740 | 90291 | Hans Havermann | 08/2020 |
9004 | 10^90279+62457 | 90280 | Patrick De Geest | 07/2019 |
9005 | 10^90279-1551 | 90279 | Patrick De Geest | 09/2018 |
9006 | (2^299892+1986)/2 | 90277 | Stefano Morozzi | 01/2020 |
9007 | (2^299893+3866)/2 | 90277 | Stefano Morozzi | 01/2020 |
9008 | 2^299868-109613 | 90270 | Enzo Doria | 04/2014 |
9009 | 10^90259+359299 | 90260 | Patrick De Geest | 07/2019 |
9010 | 10^90259-9443 | 90259 | Patrick De Geest | 09/2018 |
9011 | (2^299831+3786)/2 | 90258 | Stefano Morozzi | 01/2020 |
9012 | 10^90252+102873 | 90253 | Patrick De Geest | 07/2019 |
9013 | 10^90252-8103 | 90252 | Patrick De Geest | 09/2018 |
9014 | 5*2^299799-3 | 90250 | Lelio R Paula | 08/2014 |
9015 | (2^299803+3326)/2 | 90250 | Stefano Morozzi | 01/2020 |
9016 | 13^81008-2 | 90239 | Lelio R Paula | 09/2013 |
9017 | 80697!4-256 | 90234 | Robert Price | 11/2019 |
9019 | 64^49957-63^49957 | 90232 | Robert Price | 07/2013 |
9018 | (2^299745+2994)/2 | 90232 | Stefano Morozzi | 01/2020 |
9020 | 10^90227+56281 | 90228 | Patrick De Geest | 06/2016 |
9024 | 2^299725+315 | 90227 | Lelio R Paula | 11/2013 |
9021 | 10^90227-1000000001 | 90227 | Robert Price | 05/2016 |
9022 | 10^90227-68763 | 90227 | Patrick De Geest | 06/2016 |
9023 | 9*10^90226+29 | 90227 | Robert Price | 03/2022 |
9025 | 5^129082+48 | 90225 | Lelio R Paula | 03/2016 |
9026 | 2^299699+195 | 90219 | Lelio R Paula | 10/2012 |
9027 | (55*10^90211-1)/9 | 90212 | Ray Chandler | 02/2012 |
9028 | 22131^11918+11918^22131 | 90211 | Hans Havermann | 08/2020 |
9029 | 10^90209+8971 | 90210 | Patrick De Geest | 09/2018 |
9030 | 10^90209-724797 | 90209 | Patrick De Geest | 07/2019 |
9031 | 10^90208+189531 | 90209 | Patrick De Geest | 07/2019 |
9032 | 10^90208-2247 | 90208 | Patrick De Geest | 09/2018 |
9033 | (629^32233-1)/628 | 90207 | Michael Stocker | 05/2020 |
9034 | 116^43693-115 | 90203 | Norbert Schneider | 08/2018 |
9035 | 10^90202+3553 | 90203 | Patrick De Geest | 09/2018 |
9036 | 3^189053+280 | 90202 | Lelio R Paula | 04/2016 |
9037 | 10^90202-67793 | 90202 | Patrick De Geest | 07/2019 |
9038 | 2^299639-187 | 90201 | Lelio R Paula | 10/2012 |
9039 | 10^90199+254031 | 90200 | Patrick De Geest | 07/2019 |
9040 | 10^90199-2891 | 90199 | Patrick De Geest | 09/2018 |
9042 | 2^299631+183 | 90198 | Lelio R Paula | 10/2012 |
9041 | 21567^15212+15212^21567 | 90198 | Hans Havermann | 08/2020 |
9043 | 10^90196+187539 | 90197 | Patrick De Geest | 07/2019 |
9045 | 10^90196-9683 | 90196 | Patrick De Geest | 09/2018 |
9044 | 21708^14287+14287^21708 | 90196 | Hans Havermann | 08/2020 |
9046 | 10^90193+104973 | 90194 | Patrick De Geest | 07/2019 |
9047 | 10^90193-9809 | 90193 | Patrick De Geest | 09/2018 |
9048 | (21*2^299611-1)/29 | 90192 | Lelio R Paula | 02/2014 |
9049 | (2277^26863-1)/2276 | 90186 | Serge Batalov | 08/2020 |
9050 | 2^299567+113 | 90179 | Lelio R Paula | 10/2012 |
9051 | 2^299550+189 | 90174 | Lelio R Paula | 10/2012 |
9052 | 10^90170+376063 | 90171 | Patrick De Geest | 07/2019 |
9053 | 10^90170-6749 | 90170 | Patrick De Geest | 09/2018 |
9054 | (2^299535+3)/29 | 90168 | Lelio R Paula | 01/2013 |
9055 | (2^299521+13)/15 | 90164 | Stefano Morozzi | 03/2020 |
9056 | 22316^10971+10971^22316 | 90163 | Hans Havermann | 08/2020 |
9057 | 10^90160+271947 | 90161 | Patrick De Geest | 07/2019 |
9058 | 10^90160-5613 | 90160 | Patrick De Geest | 09/2018 |
9059 | (97*10^90155-1)/3 | 90157 | Serge Batalov | 12/2014 |
9060 | (29*10^90153+583)/9 | 90154 | Robert Price | 06/2021 |
9062 | 3^188938+8680 | 90147 | Lelio R Paula | 04/2016 |
9061 | 10^90146+160209 | 90147 | Patrick De Geest | 07/2019 |
9063 | 10^90146-2027 | 90146 | Patrick De Geest | 09/2018 |
9064 | 6^115840-55 | 90142 | Lelio R Paula | 05/2016 |
9065 | (5*10^90142-473)/9 | 90142 | Robert Price | 12/2017 |
9066 | 8*7^106659+5 | 90139 | Lelio R Paula | 10/2014 |
9067 | 2^299406-16653 | 90131 | Enzo Doria | 05/2013 |
9068 | (317057^16384+1)/2 | 90131 | Richard Fischer | 10/2019 |
9069 | (2^299409+834)/2 | 90131 | Stefano Morozzi | 01/2020 |
9070 | 10^90128+288673 | 90129 | Patrick De Geest | 07/2019 |
9072 | 208049#+(216091#)/(208049#) | 90128 | Makoto Morimoto | 04/2016 |
9071 | 10^90128-8387 | 90128 | Patrick De Geest | 09/2018 |
9074 | (F(5)^128941-F(4)^128941)/F(3) | 90126 | Wayne Halsdorf | 03/2012 |
9073 | (5^128941-3^128941)/2 | 90126 | Kellen Shenton | 05/2022 |
9075 | 6^115819-53 | 90125 | Lelio R Paula | 05/2016 |
9076 | 10^90123+6313 | 90124 | Patrick De Geest | 09/2018 |
9077 | 6^115816-16853 | 90123 | Enzo Doria | 02/2013 |
9078 | 10^90123-22617 | 90123 | Patrick De Geest | 07/2019 |
9079 | 6^115813-23 | 90121 | Lelio R Paula | 05/2016 |
9080 | 2^299351-933075 | 90114 | Paul Bourdelais | 04/2021 |
9081 | (22*10^90110+161)/3 | 90111 | Robert Price | 02/2017 |
9083 | (2^299330-509203)/69 | 90106 | Lelio R Paula | 01/2013 |
9082 | 10^90105+915759 | 90106 | Patrick De Geest | 07/2019 |
9084 | 10^90105-767 | 90105 | Patrick De Geest | 09/2018 |
9085 | 10^90104+48189 | 90105 | Patrick De Geest | 07/2019 |
9086 | (6050^23827-1)/6049 | 90105 | Serge Batalov | 08/2020 |
9088 | 10^90104-7703 | 90104 | Patrick De Geest | 09/2018 |
9089 | 10^90103+9159 | 90104 | Patrick De Geest | 09/2018 |
9087 | (797*10^90102-23)/9 | 90104 | Robert Price | 01/2022 |
9090 | 10^90103-208457 | 90103 | Patrick De Geest | 07/2019 |
9091 | 10^90102+73011 | 90103 | Patrick De Geest | 07/2019 |
9092 | 10^90102-602433 | 90102 | Patrick De Geest | 07/2019 |
9093 | 10^90101+1225537 | 90102 | Patrick De Geest | 07/2019 |
9095 | 10^90101-3207 | 90101 | Patrick De Geest | 09/2018 |
9094 | 21730^14007+14007^21730 | 90101 | Hans Havermann | 08/2020 |
9096 | (29*10^90096-17)/3 | 90097 | Erik Branger | 09/2014 |
9097 | (49*10^90094+221)/9 | 90095 | Robert Price | 02/2022 |
9098 | 10^90089+143181 | 90090 | Patrick De Geest | 07/2019 |
9099 | 10^90089-3573 | 90089 | Patrick De Geest | 09/2018 |
9100 | 2^299247+1870529 | 90083 | Enzo Doria | 05/2016 |
9101 | 10^90079+1609 | 90080 | Patrick De Geest | 09/2018 |
9102 | 10^90079-230043 | 90079 | Patrick De Geest | 07/2019 |
9103 | 2^299229-2797783 | 90077 | Enzo Doria | 04/2016 |
9104 | 2^299220-16317 | 90075 | Enzo Doria | 04/2013 |
9105 | 11^86491-1350 | 90072 | Enzo Doria | 12/2012 |
9106 | 10^90071+52333 | 90072 | Patrick De Geest | 07/2019 |
9108 | 10^90071-1233 | 90071 | Patrick De Geest | 09/2018 |
9107 | (314401^16384+1)/2 | 90071 | Richard Fischer | 10/2019 |
9109 | 10^90069+158527 | 90070 | Patrick De Geest | 07/2019 |
9110 | 10^90069-7841 | 90069 | Patrick De Geest | 09/2018 |
9111 | (31*10^90064-1)/3 | 90066 | Robert Price | 03/2017 |
9112 | 10^90060+149431 | 90061 | Patrick De Geest | 07/2019 |
9113 | 10^90060-839 | 90060 | Patrick De Geest | 09/2018 |
9114 | 10^90052+2361 | 90053 | Patrick De Geest | 09/2018 |
9115 | 10^90052-90929 | 90052 | Patrick De Geest | 07/2019 |
9116 | 10^90047+619 | 90048 | Patrick De Geest | 09/2018 |
9117 | 10^90047-151071 | 90047 | Patrick De Geest | 07/2019 |
9118 | L(430949)/861899/1440387561539 | 90045 | Serge Batalov | 07/2017 |
9119 | 22996^8233+8233^22996 | 90043 | Hans Havermann | 08/2020 |
9120 | 2^299112+73951 | 90042 | Enzo Doria | 04/2016 |
9121 | 8^99702+7^(99702-1) | 90040 | Robert Price | 02/2016 |
9122 | 10^90038+157803 | 90039 | Patrick De Geest | 07/2019 |
9123 | 10^90038-9687 | 90038 | Patrick De Geest | 09/2018 |
9125 | 5*10^90033+3 | 90034 | Dmitry Domanov | 11/2010 |
9124 | (17574^21211-1)/17573 | 90034 | Serge Batalov | 08/2020 |
9126 | 10^90030+225813 | 90031 | Patrick De Geest | 07/2019 |
9127 | 10^90030-1461 | 90030 | Patrick De Geest | 09/2018 |
9128 | 243^37738+37738^243 | 90029 | Norbert Schneider | 01/2015 |
9129 | 10^90025+70569 | 90026 | Patrick De Geest | 07/2019 |
9130 | 10^90025-8703 | 90025 | Patrick De Geest | 09/2018 |
9131 | 2^299048-113297 | 90023 | Enzo Doria | 07/2014 |
9132 | 10^90015+849 | 90016 | Patrick De Geest | 09/2018 |
9133 | 10^90015-228387 | 90015 | Patrick De Geest | 07/2019 |
9134 | 10^90010+6811 | 90011 | Patrick De Geest | 09/2018 |
9135 | 10^90010-183597 | 90010 | Patrick De Geest | 11/2018 |
9136 | 10^90009+58321 | 90010 | Patrick De Geest | 11/2018 |
9137 | 10^90008+847 | 90009 | Patrick De Geest | 09/2018 |
9138 | 10^90009-356013 | 90009 | Patrick De Geest | 11/2018 |
9139 | 25696^3183+3183^25696 | 90009 | Hans Havermann | 08/2020 |
9141 | 3^188648+92 | 90008 | Lelio R Paula | 12/2015 |
9140 | 10^90008-706049 | 90008 | Patrick De Geest | 11/2018 |
9142 | 5*2^298991+7 | 90006 | Lelio R Paula | 08/2014 |
9143 | (25*2^298997+1)/3267 | 90005 | Lelio R Paula | 01/2013 |
9144 | 10^90000+8869 | 90001 | Patrick De Geest | 12/2009 |
9146 | 10^89999+82939 | 90000 | Dirk Augustin | 04/2007 |
9145 | 10^90000-140343 | 90000 | Patrick De Geest | 12/2009 |
9147 | 10^89999-194891 | 89999 | Patrick De Geest | 11/2009 |
9148 | (2^298918+1798)/2 | 89983 | Stefano Morozzi | 01/2020 |
9149 | 19^70365-78 | 89980 | Lelio R Paula | 12/2015 |
9150 | Phi(186062,10)/134609632483003347004451 | 89977 | Makoto Kamada | 04/2021 |
9151 | NaturalLogarithmOf2(89973) | 89973 | Eric W. Weisstein | 10/2015 |
9152 | 2^298852-64875 | 89964 | Lelio R Paula | 11/2013 |
9153 | (2^298825+1010)/2 | 89955 | Stefano Morozzi | 01/2020 |
9154 | 18^71659+37 | 89952 | Lelio R Paula | 12/2015 |
9155 | (2908*29^61501-3)/7 | 89942 | Curtis Bright | 05/2015 |
9156 | (2^298769+1202)/2 | 89939 | Stefano Morozzi | 01/2020 |
9157 | (5941^23827-1)/5940 | 89916 | Serge Batalov | 08/2020 |
9158 | 17^73067+20 | 89906 | Lelio R Paula | 11/2015 |
9159 | 2^298637-99 | 89899 | Lelio R Paula | 10/2012 |
9160 | 25513^3336+3336^25513 | 89889 | Hans Havermann | 08/2020 |
9161 | 29748^1051+1051^29748 | 89887 | Hans Havermann | 08/2020 |
9162 | 9*10^89882-91 | 89883 | Robert Price | 06/2021 |
9163 | 9^94188+3*8^94188-2*4^94188-3*2^94188 | 89879 | Mike Oakes | 01/2011 |
9164 | 2*10^89876+83 | 89877 | Robert Price | 12/2021 |
9165 | (2^298558+3654)/2 | 89875 | Stefano Morozzi | 01/2020 |
9166 | 9*10^89858+19 | 89859 | Robert Price | 05/2016 |
9167 | 2^298501+1870067 | 89858 | Enzo Doria | 05/2016 |
9168 | (286*10^89852-1)/3 | 89854 | Robert Price | 11/2020 |
9169 | 11^86280-98 | 89852 | Lelio R Paula | 11/2015 |
9170 | 2^298473-110059 | 89850 | Enzo Doria | 04/2014 |
9171 | 19^70263-32 | 89850 | Lelio R Paula | 12/2015 |
9172 | 21208^17221+17221^21208 | 89839 | Hans Havermann | 08/2020 |
9173 | (299*10^89836-17)/3 | 89838 | Robert Price | 01/2017 |
9174 | 3^188286+14 | 89836 | Robert Price | 09/2013 |
9175 | (2^298408+450)/2 | 89830 | Stefano Morozzi | 01/2020 |
9176 | gcd(V(1091095),5*F(218219)*(F(218219)+1)+1) | 89822 | Serge Batalov | 04/2015 |
9177 | 2^298367+58905 | 89818 | Lelio R Paula | 11/2013 |
9178 | 21966^12269+12269^21966 | 89815 | Hans Havermann | 08/2020 |
9179 | (2^298333+3126)/2 | 89807 | Stefano Morozzi | 01/2020 |
9180 | 2^298319-109657 | 89803 | Enzo Doria | 04/2014 |
9181 | 7*10^89801+13 | 89802 | Robert Price | 03/2021 |
9182 | 8*10^89800+81 | 89801 | Robert Price | 05/2017 |
9183 | 21354^16037+16037^21354 | 89797 | Hans Havermann | 08/2020 |
9184 | (2^298249+714)/2 | 89782 | Stefano Morozzi | 01/2020 |
9185 | 21227^16954+16954^21227 | 89775 | Hans Havermann | 08/2020 |
9186 | (733^31333-1)/732 | 89770 | Serge Batalov | 08/2020 |
9189 | 18^71505-91 | 89759 | Lelio R Paula | 12/2015 |
9187 | (2^298171+1586)/2 | 89759 | Stefano Morozzi | 01/2020 |
9188 | (4788^24391-1)/4787 | 89759 | Serge Batalov | 08/2020 |
9190 | (2^298161+3530)/2 | 89756 | Stefano Morozzi | 01/2020 |
9191 | (7^106187+1)/8 | 89738 | Paul Bourdelais | 04/2008 |
9192 | 2^298079-109531 | 89731 | Enzo Doria | 04/2014 |
9193 | (9255^22621-1)/9254 | 89720 | Serge Batalov | 08/2020 |
9194 | 117^43380+116 | 89718 | Norbert Schneider | 08/2018 |
9195 | (22699*2^298016+1)/385 | 89714 | Lelio R Paula | 06/2014 |
9196 | 156*(10^(4*22427)-1)/101-1 | 89709 | Robert Price | 05/2014 |
9197 | 2^298003-66945 | 89708 | Lelio R Paula | 11/2013 |
9198 | (23*10^89705-83)/3 | 89706 | Robert Price | 09/2017 |
9199 | 17^72900-54 | 89700 | Lelio R Paula | 11/2015 |
9200 | (2^297968+2482)/2 | 89698 | Stefano Morozzi | 01/2020 |
9201 | 3*2^297956+5 | 89695 | Lelio R Paula | 08/2014 |
9202 | (2^297952+3558)/2 | 89693 | Stefano Morozzi | 01/2020 |
9203 | (3992^24907-1)/3991 | 89692 | Serge Batalov | 08/2020 |
9204 | 20^68938-11 | 89691 | Lelio R Paula | 01/2016 |
9205 | 89*10^89687-7 | 89689 | Robert Price | 03/2017 |
9206 | (6882^23371-1)/6881 | 89688 | Serge Batalov | 08/2020 |
9207 | (297847^16384+1)/2 | 89686 | Richard Fischer | 08/2019 |
9208 | 2^297923+77783 | 89684 | Enzo Doria | 05/2016 |
9209 | (2^297893+1694)/2 | 89675 | Stefano Morozzi | 01/2020 |
9210 | 11^86107-24576000 | 89672 | Enzo Doria | 02/2013 |
9211 | 2^297878-111911 | 89671 | Enzo Doria | 05/2014 |
9212 | 20823^20236+20236^20823 | 89667 | Hans Havermann | 08/2020 |
9213 | 25708^3075+3075^25708 | 89666 | Hans Havermann | 08/2020 |
9214 | 21002^18585+18585^21002 | 89661 | Hans Havermann | 08/2020 |
9215 | (2^297830+3670)/2 | 89656 | Stefano Morozzi | 01/2020 |
9216 | 7^106084+40 | 89652 | Lelio R Paula | 10/2015 |
9217 | (2^297815+618)/2 | 89651 | Stefano Morozzi | 01/2020 |
9218 | 21223^16752+16752^21223 | 89648 | Hans Havermann | 08/2020 |
9220 | (86*10^89640-311)/9 | 89641 | Robert Price | 10/2017 |
9219 | (2^297781+2094)/2 | 89641 | Stefano Morozzi | 01/2020 |
9221 | (2^297741+3326)/2 | 89629 | Stefano Morozzi | 01/2020 |
9222 | (2^297724+3606)/2 | 89624 | Stefano Morozzi | 01/2020 |
9223 | 3*10^89608-59 | 89609 | Robert Price | 10/2017 |
9224 | (2^297650+3250)/2 | 89602 | Stefano Morozzi | 01/2020 |
9225 | (34*10^89599+191)/9 | 89600 | Robert Price | 08/2021 |
9226 | 3^187780-11038 | 89594 | Enzo Doria | 05/2014 |
9227 | (13*2^297560-1)/49151787 | 89568 | Lelio R Paula | 02/2014 |
9228 | 5^128141-36 | 89567 | Lelio R Paula | 10/2015 |
9229 | 4*7^105951+9 | 89540 | Lelio R Paula | 10/2014 |
9230 | (7*10^89539-439)/9 | 89539 | Robert Price | 09/2017 |
9231 | (9772^22441-1)/9771 | 89536 | Serge Batalov | 08/2020 |
9232 | 22823^8374+8374^22823 | 89534 | Hans Havermann | 08/2020 |
9233 | (2^297401+2954)/2 | 89527 | Stefano Morozzi | 01/2020 |
9234 | (2^297403+1670)/2 | 89527 | Stefano Morozzi | 01/2020 |
9235 | 2^297397+177257 | 89526 | Enzo Doria | 05/2016 |
9236 | (77*10^89518+247)/9 | 89519 | Robert Price | 03/2021 |
9237 | 2^297346-16241 | 89511 | Enzo Doria | 04/2013 |
9238 | (5*10^89510+79)/3 | 89511 | Robert Price | 11/2017 |
9240 | 4^148663-3^148663 | 89505 | Jean-Louis Charton | 11/2005 |
9239 | 2^297328-869688105 | 89505 | Lelio R Paula | 07/2016 |
9241 | 2^297325-459043 | 89504 | Enzo Doria | 05/2015 |
9242 | 2^297310-112391 | 89500 | Enzo Doria | 06/2014 |
9243 | 2^297296-112157 | 89496 | Enzo Doria | 06/2014 |
9244 | 57^50966-2 | 89490 | Lelio R Paula | 08/2013 |
9245 | (266*10^89482+1)/3 | 89484 | Robert Price | 02/2016 |
9246 | (14*10^89480-101)/3 | 89481 | Robert Price | 04/2017 |
9247 | 2618^26175+26175^2618 | 89466 | Norbert Schneider | 01/2020 |
9248 | (2^297150+3922)/2 | 89451 | Stefano Morozzi | 01/2020 |
9249 | (2^297141+1842)/2 | 89449 | Stefano Morozzi | 01/2020 |
9250 | (2^297140+1458)/2 | 89448 | Stefano Morozzi | 01/2020 |