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 | 23277^12058+12058^23277 | 95000 | Hans Havermann | 09/2020 |
9002 | 1*10^94983+21 | 94984 | Robert Price | 03/2011 |
9003 | 10^94983-717 | 94983 | Patrick De Geest | 08/2013 |
9004 | 17^77187-82 | 94975 | Lelio R Paula | 07/2016 |
9005 | 3*10^94964+37 | 94965 | Robert Price | 10/2017 |
9006 | (2^315481-1017)/20627285 | 94962 | Lelio R Paula | 06/2013 |
9007 | 2^315423-110329 | 94952 | Enzo Doria | 05/2014 |
9008 | 11^91169-7230 | 94943 | Enzo Doria | 02/2013 |
9009 | 7*10^94938+61 | 94939 | Robert Price | 02/2017 |
9010 | 2^315351+1025 | 94931 | Lelio R Paula | 03/2016 |
9011 | 2^315350-112253 | 94930 | Enzo Doria | 06/2014 |
9012 | (2^315343+1766)/2 | 94928 | Stefano Morozzi | 09/2020 |
9013 | (2^315343+3066)/2 | 94928 | Stefano Morozzi | 09/2020 |
9016 | ((1001327^(15817)+1)/1001328) | 94906 | Faysal YAtim | 10/2010 |
9015 | 2^315268+7 | 94906 | Lelio R Paula | 10/2012 |
9014 | 11^91133+62 | 94906 | Lelio R Paula | 05/2016 |
9017 | (2^315264+1558)/2 | 94904 | Stefano Morozzi | 09/2020 |
9018 | 30^64244+41 | 94897 | Lelio R Paula | 08/2012 |
9019 | (151*10^94885-7)/9 | 94887 | Robert Price | 07/2017 |
9020 | 4^157583+22147 | 94875 | Mike Oakes | 01/2011 |
9021 | 2^315098+123 | 94854 | Lelio R Paula | 11/2012 |
9022 | 3^198791-34 | 94848 | Robert Price | 11/2013 |
9023 | 3*2^315067+7 | 94846 | Lelio R Paula | 08/2014 |
9024 | (9*2^315057+1)/73 | 94841 | Lelio R Paula | 09/2013 |
9025 | 22576^15875+15875^22576 | 94836 | Hans Havermann | 09/2020 |
9026 | (16*10^94822+53)/3 | 94823 | Robert Price | 12/2016 |
9027 | 8*6^121849-7 | 94818 | Lelio R Paula | 10/2014 |
9028 | (85*10^94802+473)/9 | 94803 | Robert Price | 03/2017 |
9029 | 21767^21852-21852^21767 | 94790 | Norbert Schneider | 06/2020 |
9030 | 2^314870-108761 | 94786 | Enzo Doria | 03/2014 |
9031 | 5*2^314841-3 | 94778 | Lelio R Paula | 08/2014 |
9032 | (2^314842+550)/2 | 94777 | Stefano Morozzi | 09/2020 |
9033 | 11738^23289-23289^11738 | 94777 | Norbert Schneider | 05/2022 |
9034 | (2^314787+2630)/2 | 94761 | Stefano Morozzi | 09/2020 |
9035 | 6^121772+113 | 94758 | Lelio R Paula | 05/2016 |
9036 | (2^314740+3642)/2 | 94746 | Stefano Morozzi | 09/2020 |
9037 | 6^121756-16253 | 94745 | Enzo Doria | 02/2013 |
9038 | 2^314718-112793 | 94740 | Enzo Doria | 06/2014 |
9039 | 2^314697-1847293 | 94734 | Enzo Doria | 07/2015 |
9040 | 22836^14065+14065^22836 | 94727 | Hans Havermann | 09/2020 |
9042 | 2^314548-249 | 94689 | Yves Minet | 02/2012 |
9041 | 5^135469+42 | 94689 | Lelio R Paula | 04/2016 |
9043 | gcd(V(1137755),5*F(227551)*(F(227551)-1)+1) | 94676 | Serge Batalov | 04/2015 |
9044 | 25773^4712+4712^25773 | 94670 | Hans Havermann | 08/2020 |
9045 | (2^314485+3614)/2 | 94670 | Stefano Morozzi | 09/2020 |
9046 | (2*10^94604+547)/9 | 94604 | Robert Price | 10/2017 |
9047 | (2^314164+2562)/2 | 94573 | Stefano Morozzi | 09/2020 |
9048 | 3*2^314152+5 | 94570 | Lelio R Paula | 08/2014 |
9049 | 2*6^121530-5 | 94570 | Lelio R Paula | 10/2014 |
9050 | 7*10^94563+87 | 94564 | Robert Price | 11/2016 |
9051 | 2^314114-109677 | 94558 | Enzo Doria | 04/2014 |
9052 | (218389#+240867)/(3*11*811) | 94557 | Makoto Morimoto | 10/2017 |
9053 | 2^314059+76583 | 94542 | Enzo Doria | 05/2016 |
9054 | 2^314041-4035 | 94536 | Lelio R Paula | 03/2016 |
9055 | (2^314028+3102)/2 | 94532 | Stefano Morozzi | 09/2020 |
9056 | (2^314012+618)/2 | 94527 | Stefano Morozzi | 09/2020 |
9057 | (7*2^314003-11)/15 | 94524 | Renaud Lifchitz | 03/2011 |
9058 | 11^90760-4710 | 94517 | Enzo Doria | 01/2013 |
9059 | 21675^21796+21796^21675 | 94507 | Norbert Schneider | 04/2020 |
9060 | 2^313942+100280245065 | 94506 | Paul Bourdelais | 10/2021 |
9061 | (7*2^313926-1)/148821 | 94497 | Lelio R Paula | 02/2014 |
9062 | 94*10^94493-3 | 94495 | Robert Price | 11/2016 |
9063 | (1000547^15749+1)/1000548 | 94492 | Faysal YAtim | 12/2008 |
9064 | (2^313859+2706)/2 | 94481 | Stefano Morozzi | 09/2020 |
9065 | (F(3*75345)^2+L(3*75345)^2)/4 | 94477 | Henri Lifchitz | 10/2017 |
9066 | 7^111792-32 | 94476 | Lelio R Paula | 04/2016 |
9067 | (2^313800+822)/2 | 94463 | Stefano Morozzi | 01/2021 |
9068 | 9*10^94462-47 | 94463 | Robert Price | 11/2021 |
9069 | 11*10^94460+3 | 94462 | Robert Price | 07/2021 |
9070 | (202*10^94441-1)/3 | 94443 | Robert Price | 05/2020 |
9071 | 11^90680-5850 | 94434 | Enzo Doria | 01/2013 |
9072 | (752^32833-1)/751 | 94432 | Michael Stocker | 05/2020 |
9073 | 2^313675-110125 | 94426 | Enzo Doria | 04/2014 |
9074 | (2999*29^64556-2427)/4 | 94410 | Curtis Bright | 05/2015 |
9075 | 2^313609-112225 | 94406 | Enzo Doria | 06/2014 |
9076 | 2^313584+72901 | 94399 | Enzo Doria | 04/2016 |
9077 | (2^313567+1310)/2 | 94393 | Stefano Morozzi | 02/2021 |
9078 | (2^313562+730)/2 | 94392 | Stefano Morozzi | 09/2020 |
9079 | 2^313510-112865 | 94376 | Enzo Doria | 06/2014 |
9080 | 7*10^94373+87 | 94374 | Robert Price | 11/2016 |
9081 | (9^98897+8^98897)/17 | 94371 | Robert Price | 12/2012 |
9082 | 2^313453-114703 | 94359 | Enzo Doria | 04/2015 |
9083 | 8*10^94357-51 | 94358 | Robert Price | 10/2017 |
9084 | 25903^4392+4392^25903 | 94356 | Hans Havermann | 09/2020 |
9085 | 31706^945+945^31706 | 94340 | Hans Havermann | 09/2020 |
9086 | (379*10^94337-1)/9 | 94339 | Robert Price | 08/2017 |
9087 | 2^313344-313344+1 | 94326 | Robert Price | 10/2014 |
9088 | (2^313313+3662)/2 | 94317 | Stefano Morozzi | 09/2020 |
9089 | 20^72481-49 | 94300 | Lelio R Paula | 05/2016 |
9090 | (34*10^94293-259)/9 | 94294 | Robert Price | 12/2021 |
9091 | 44725!2+128 | 94289 | Robert Price | 03/2015 |
9092 | 4*10^94286+9 | 94287 | Robert Price | 05/2015 |
9093 | 17^76621+24 | 94279 | Lelio R Paula | 07/2016 |
9094 | 26274^3875+3875^26274 | 94279 | Hans Havermann | 09/2020 |
9095 | 4*10^94270-33 | 94271 | Robert Price | 04/2021 |
9096 | 2^313156-112859 | 94270 | Enzo Doria | 06/2014 |
9097 | (49*10^94262-211)/9 | 94263 | Robert Price | 06/2022 |
9098 | 6^121130-59 | 94258 | Lelio R Paula | 05/2016 |
9099 | 3^197542+28 | 94252 | Robert Price | 11/2013 |
9100 | (2^313076+3550)/2 | 94245 | Stefano Morozzi | 09/2020 |
9101 | (3^197521+2)/51655 | 94237 | Lelio R Paula | 04/2013 |
9102 | 2^313045-5955 | 94236 | Lelio R Paula | 03/2016 |
9103 | 22099^18358+18358^22099 | 94227 | Hans Havermann | 08/2020 |
9104 | (67*10^94212+23)/9 | 94213 | Ray Chandler | 02/2011 |
9105 | 2^312909-1865203 | 94195 | Enzo Doria | 03/2016 |
9106 | 25013^5830+5830^25013 | 94191 | Hans Havermann | 09/2020 |
9107 | 2^312848-149 | 94177 | Lelio R Paula | 11/2012 |
9108 | (2^312819+3278)/2 | 94168 | Stefano Morozzi | 02/2021 |
9109 | 4^156403-77597 | 94164 | Mike Oakes | 04/2011 |
9110 | 18^75004-85 | 94151 | Lelio R Paula | 06/2016 |
9111 | 19^73622-48 | 94145 | Lelio R Paula | 06/2016 |
9112 | (11^90397-3^90397)/8 | 94138 | Robert Price | 04/2011 |
9113 | (3^197287+5)/16 | 94129 | Lelio R Paula | 04/2013 |
9114 | (23*10^94123-29)/3 | 94124 | Robert Price | 03/2022 |
9115 | 24220^7691+7691^24220 | 94119 | Hans Havermann | 09/2020 |
9116 | 12^87212-37 | 94118 | Lelio R Paula | 06/2016 |
9117 | 11^90376+76 | 94117 | Lelio R Paula | 05/2016 |
9118 | (6^120946-11)/5 | 94114 | Paul Bourdelais | 11/2011 |
9119 | (2^312622+534)/2 | 94109 | Stefano Morozzi | 09/2020 |
9120 | (10^94096+191)/3 | 94096 | Robert Price | 11/2017 |
9121 | 23794^9005+9005^23794 | 94093 | Hans Havermann | 09/2020 |
9122 | 2^312562-112685 | 94091 | Enzo Doria | 06/2014 |
9123 | 137^44026-2 | 94072 | Lelio R Paula | 10/2017 |
9124 | s(164441) | 94052 | Robert Price | 05/2020 |
9125 | 2^312404-869688105 | 94043 | Lelio R Paula | 09/2016 |
9126 | 2^312387+2793533 | 94038 | Enzo Doria | 03/2016 |
9128 | 10^94016+91 | 94017 | Robert Price | 04/2011 |
9127 | (15*2^312325-1)/7201 | 94017 | Lelio R Paula | 02/2014 |
9129 | 10^94016-188963 | 94016 | Patrick De Geest | 08/2013 |
9130 | (17*2^312315-1)/45 | 94016 | Lelio R Paula | 02/2014 |
9131 | (401*10^94010+13)/9 | 94012 | Robert Price | 09/2021 |
9132 | 23866^8691+8691^23866 | 94010 | Hans Havermann | 08/2020 |
9133 | 6^120802+7661 | 94003 | Mike Oakes | 04/2011 |
9134 | (67607*2^312254+1)/177 | 94001 | Lelio R Paula | 06/2014 |
9135 | (2^312245+702)/2 | 93995 | Stefano Morozzi | 09/2020 |
9136 | (56*10^93993+493)/9 | 93994 | Robert Price | 10/2021 |
9137 | (11*2^312207+1)/19 | 93984 | Lelio R Paula | 09/2013 |
9138 | 2^312201+51 | 93982 | Lelio R Paula | 11/2012 |
9139 | (4*6^120749+1)/5 | 93961 | Serge Batalov | 10/2014 |
9140 | (781*10^93942-7)/9 | 93944 | Robert Price | 12/2021 |
9141 | 3^196892+58 | 93942 | Lelio R Paula | 05/2016 |
9142 | 2^312060+18915 | 93940 | Lelio R Paula | 03/2016 |
9143 | 42^57867-41 | 93933 | Norbert Schneider | 08/2018 |
9144 | 8*10^93921+83 | 93922 | Robert Price | 11/2021 |
9145 | 2^311992-16209 | 93919 | Enzo Doria | 04/2013 |
9146 | (85^48677+1)/86 | 93917 | Paul Bourdelais | 08/2017 |
9147 | 2^311965-39 | 93911 | Lelio R Paula | 11/2012 |
9148 | 2^311920-111689 | 93898 | Enzo Doria | 05/2014 |
9149 | 130^44414+129 | 93889 | Norbert Schneider | 01/2019 |
9150 | (2^311892+2730)/2 | 93889 | Stefano Morozzi | 09/2020 |
9151 | 7*10^93887-47 | 93888 | Robert Price | 02/2017 |
9152 | 9*10^93878-37 | 93879 | Robert Price | 06/2021 |
9153 | (28*10^93874+323)/9 | 93875 | Robert Price | 10/2021 |
9154 | (2^311778+1530)/2 | 93855 | Stefano Morozzi | 09/2020 |
9155 | 12^86965-13 | 93851 | Lelio R Paula | 06/2016 |
9156 | 24862^5953+5953^24862 | 93848 | Hans Havermann | 09/2020 |
9157 | 22299^16148+16148^22299 | 93837 | Hans Havermann | 08/2020 |
9158 | 2^311705+16095 | 93833 | Lelio R Paula | 03/2016 |
9159 | (2^311701+3002)/2 | 93832 | Stefano Morozzi | 09/2020 |
9160 | norm(Mod(x,x^3-x-1)^768329-1) | 93831 | Paul Vanderveen | 09/2022 |
9162 | 2^311665-81 | 93821 | Lelio R Paula | 11/2012 |
9161 | (2^311665+2654)/2 | 93821 | Stefano Morozzi | 09/2020 |
9163 | (2^311648+3058)/2 | 93816 | Stefano Morozzi | 09/2020 |
9164 | (2^311639+2738)/2 | 93813 | Stefano Morozzi | 01/2021 |
9165 | 730^32761+32761^730 | 93806 | Norbert Schneider | 09/2017 |
9166 | 11^90073+92 | 93802 | Lelio R Paula | 05/2016 |
9167 | 6*10^93792+41 | 93793 | Robert Price | 01/2017 |
9168 | 10^93790+64611 | 93791 | Patrick De Geest | 04/2020 |
9171 | (2^311580+7)/591329 | 93790 | Lelio R Paula | 06/2013 |
9169 | 10^93790-123456789 | 93790 | Robert Price | 02/2020 |
9170 | 10^93790-157517 | 93790 | Patrick De Geest | 04/2020 |
9172 | (49*10^93782+293)/9 | 93783 | Robert Price | 03/2021 |
9173 | 12^86900-77 | 93781 | Lelio R Paula | 06/2016 |
9174 | 2^311508-109769 | 93774 | Enzo Doria | 04/2014 |
9175 | 2^311486-1859621 | 93767 | Enzo Doria | 03/2016 |
9176 | Phi(93811,10)/968129521/2109997013/3280113810431/639637782545831 | 93765 | Kurt Beschorner | 04/2015 |
9177 | (10^93811-1)/(9*968129521*2109997013*3280113810431*639637782545831) | 93765 | Gord Palameta | 10/2018 |
9178 | 2^311471+311469 | 93763 | Renaud Lifchitz | 01/2003 |
9179 | (88*10^93760-1)/3 | 93762 | Robert Price | 10/2017 |
9180 | 2^311426+79459 | 93749 | Enzo Doria | 06/2016 |
9181 | 21537^21634+21634^21537 | 93745 | Norbert Schneider | 04/2020 |
9182 | 2^311409+111546435 | 93744 | Paul Bourdelais | 10/2021 |
9183 | (2^311360+3078)/2 | 93729 | Stefano Morozzi | 09/2020 |
9184 | (2^311341+2186)/2 | 93723 | Stefano Morozzi | 09/2020 |
9185 | (2^311313+774)/2 | 93715 | Stefano Morozzi | 09/2020 |
9187 | 2^311294+5775 | 93709 | Lelio R Paula | 03/2016 |
9186 | 25155^5312+5312^25155 | 93709 | Hans Havermann | 09/2020 |
9188 | 2^311252-1017 | 93697 | Lelio R Paula | 03/2016 |
9189 | 2^311247-112041 | 93695 | Enzo Doria | 06/2014 |
9190 | 7*10^93692-83 | 93693 | Robert Price | 09/2017 |
9191 | 2^311205-109645 | 93683 | Enzo Doria | 04/2014 |
9192 | 5^134027-84 | 93681 | Lelio R Paula | 04/2016 |
9193 | (2^311148-1017)/557 | 93663 | Lelio R Paula | 06/2013 |
9194 | (2^311027-2^155514+1)/5 | 93628 | | 11/2006 |
9195 | 26748^3161+3161^26748 | 93614 | Hans Havermann | 08/2020 |
9196 | 3^196192-28 | 93608 | Robert Price | 12/2013 |
9197 | 216091#+261433 | 93606 | Makoto Morimoto | 12/2015 |
9198 | 5^133917+68 | 93604 | Lelio R Paula | 04/2016 |
9199 | 2^310924-1709 | 93598 | Yves Minet | 03/2013 |
9200 | 24555^6476+6476^24555 | 93587 | Hans Havermann | 08/2020 |
9201 | (2^310867+734)/2 | 93580 | Stefano Morozzi | 09/2020 |
9202 | (266*10^93577-11)/3 | 93579 | Robert Price | 09/2016 |
9203 | (16*10^93577+161)/3 | 93578 | Robert Price | 05/2016 |
9204 | 3^196113-56 | 93570 | Lelio R Paula | 04/2016 |
9205 | 2^310806-110345 | 93562 | Enzo Doria | 05/2014 |
9206 | 3^196093+86 | 93561 | Lelio R Paula | 04/2016 |
9207 | 2^310780-2327 | 93555 | Yves Minet | 10/2013 |
9208 | 30^63332+41 | 93550 | Lelio R Paula | 08/2012 |
9209 | (21181*2^310707+1)/21 | 93536 | Lelio R Paula | 06/2014 |
9210 | 8*5^133807+7 | 93528 | Lelio R Paula | 10/2014 |
9211 | 2^310686+8193 | 93526 | Lelio R Paula | 03/2016 |
9212 | 22681^13282+13282^22681 | 93520 | Hans Havermann | 09/2020 |
9213 | 2^310646+63 | 93514 | Lelio R Paula | 11/2012 |
9214 | (77*10^93508+31)/9 | 93509 | Robert Price | 10/2015 |
9215 | (1000907^15583+1)/1000908 | 93499 | Faysal YAtim | 04/2009 |
9216 | (2*10^93497+7)/9 | 93497 | Erik Branger | 03/2011 |
9217 | 3^195958-112852 | 93496 | Enzo Doria | 07/2014 |
9218 | 2^310583+16385 | 93495 | Lelio R Paula | 03/2016 |
9219 | (9913*2^310557-1)/35 | 93490 | Lelio R Paula | 04/2015 |
9220 | 9*7^110614+2 | 93481 | Lelio R Paula | 11/2014 |
9221 | (19*10^93467+83)/3 | 93468 | Robert Price | 08/2020 |
9222 | 2^310469-8295 | 93461 | Lelio R Paula | 03/2016 |
9223 | 2^310459-1159 | 93458 | Yves Minet | 11/2012 |
9224 | 6^120075-95 | 93437 | Lelio R Paula | 05/2016 |
9225 | (2^310362+1758)/2 | 93428 | Stefano Morozzi | 09/2020 |
9226 | 256*19^73047+9 | 93412 | Eric Chen | 12/2022 |
9227 | 148^43041-147 | 93411 | Norbert Schneider | 05/2019 |
9228 | 2^310289+98327 | 93407 | Yves Minet | 11/2010 |
9229 | 21769^19510+19510^21769 | 93395 | Hans Havermann | 08/2020 |
9230 | 2^310217-113143 | 93385 | Enzo Doria | 07/2014 |
9231 | 7^110494-68 | 93379 | Lelio R Paula | 04/2016 |
9232 | 2^310184-113093 | 93375 | Enzo Doria | 07/2014 |
9233 | 2^310180-110417 | 93374 | Enzo Doria | 05/2014 |
9234 | gcd(V(1124905),5*F(224981)*(F(224981)-1)+1) | 93372 | Serge Batalov | 04/2015 |
9235 | (2^310173+3150)/2 | 93372 | Stefano Morozzi | 09/2020 |
9236 | 3^195669+68 | 93358 | Lelio R Paula | 04/2016 |
9237 | 10^93356+91 | 93357 | Robert Price | 04/2011 |
9238 | 10^93356-28313 | 93356 | Patrick De Geest | 08/2013 |
9239 | (85*10^93346+23)/9 | 93347 | Robert Price | 11/2015 |
9240 | (3^195648-5)/3158276 | 93342 | Lelio R Paula | 04/2013 |
9241 | 10^93333+191061 | 93334 | Patrick De Geest | 09/2022 |
9243 | 10^93333-4000001 | 93333 | Robert Price | 07/2021 |
9242 | 10^93333-44807 | 93333 | Patrick De Geest | 09/2022 |
9244 | 3^195556+8680 | 93304 | Lelio R Paula | 04/2016 |
9245 | 2^309942-109991 | 93302 | Enzo Doria | 04/2014 |
9246 | 2^309926-1867301 | 93298 | Enzo Doria | 03/2016 |
9247 | 2^309924+85951 | 93297 | Yves Minet | 03/2011 |
9248 | 2^309896-4229 | 93288 | Yves Minet | 08/2018 |
9249 | (86*10^93263-77)/9 | 93264 | Robert Price | 05/2016 |
9250 | 6^119836-125 | 93251 | Lelio R Paula | 05/2016 |