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 ? |
8752 | 2^257636-110235 | 77557 | Enzo Doria | 05/2014 |
8753 | (23*10^77556+97)/3 | 77557 | Robert Price | 02/2017 |
8751 | (2^257639+2630)/2 | 77557 | Stefano Morozzi | 01/2020 |
8754 | (2^257625+3354)/2 | 77553 | Stefano Morozzi | 01/2020 |
8755 | 2^257621-110071 | 77552 | Enzo Doria | 04/2014 |
8756 | 19580^9129+9129^19580 | 77546 | Hans Havermann | 03/2020 |
8757 | 2^257567-64875 | 77536 | Lelio R Paula | 06/2013 |
8758 | (18098^18211-1)/18097 | 77532 | Serge Batalov | 08/2020 |
8759 | (2^257429+2394)/2 | 77494 | Stefano Morozzi | 01/2020 |
8760 | (2^257373+1206)/2 | 77477 | Stefano Morozzi | 01/2020 |
8761 | 2*7^91674+9 | 77474 | Lelio R Paula | 10/2014 |
8762 | (2^257248+2442)/2 | 77440 | Stefano Morozzi | 01/2020 |
8763 | (17*10^77427-47)/3 | 77428 | Robert Price | 05/2017 |
8764 | 10^77418+841683 | 77419 | Patrick De Geest | 06/2013 |
8765 | 10^77418-33 | 77418 | Robert Price | 05/2011 |
8766 | 19^60539-36 | 77415 | Lelio R Paula | 03/2015 |
8767 | (9562^19447-1)/9561 | 77406 | Serge Batalov | 08/2020 |
8768 | 37401!!-1024 | 77397 | Robert Price | 11/2015 |
8769 | (2^257095+1410)/2 | 77394 | Stefano Morozzi | 01/2020 |
8770 | 11^74313-18750 | 77390 | Enzo Doria | 12/2012 |
8771 | 2^257044-112883 | 77378 | Enzo Doria | 06/2014 |
8772 | 8*10^77377+51 | 77378 | Robert Price | 05/2017 |
8773 | Phi(258121,2)/719 | 77375 | Jeppe Stig Nielsen | 09/2020 |
8774 | 2*7^91553+9 | 77372 | Lelio R Paula | 10/2014 |
8775 | 713^27118+27118^713 | 77371 | Norbert Schneider | 12/2016 |
8776 | (22711^17761-1)/22710 | 77367 | Serge Batalov | 08/2020 |
8777 | 18652^14043+14043^18652 | 77359 | Hans Havermann | 03/2020 |
8778 | (11391^19069-1)/11390 | 77351 | Serge Batalov | 08/2020 |
8779 | 6*10^77346-29 | 77347 | Robert Price | 08/2020 |
8780 | 2^256920-111155 | 77341 | Enzo Doria | 05/2014 |
8781 | 2^256896-6555 | 77334 | Lelio R Paula | 06/2013 |
8782 | (52*10^77322-7)/9 | 77323 | Robert Price | 12/2014 |
8783 | 2^256849-112939 | 77320 | Enzo Doria | 06/2014 |
8784 | 17275^18246+18246^17275 | 77316 | Norbert Schneider | 01/2020 |
8785 | (4*10^77309-31)/9 | 77309 | Erik Branger | 05/2011 |
8786 | 20936^4923+4923^20936 | 77301 | Hans Havermann | 04/2020 |
8787 | (25376^17551-1)/25375 | 77298 | Serge Batalov | 08/2020 |
8788 | (3^161988+7)/41864 | 77284 | Lelio R Paula | 10/2012 |
8789 | (14^67421-1)/13 | 77272 | Paul Bourdelais | 09/2008 |
8790 | (2^256679+3554)/2 | 77268 | Stefano Morozzi | 01/2020 |
8792 | 5*3^161923-4 | 77258 | Lelio R Paula | 06/2014 |
8791 | 20465^5958+5958^20465 | 77258 | Hans Havermann | 02/2020 |
8793 | (2^256591+954)/2 | 77242 | Stefano Morozzi | 01/2020 |
8794 | 24806^1299+1299^24806 | 77237 | Hans Havermann | 05/2020 |
8795 | 2^256552-1499 | 77230 | Yves Minet | 05/2013 |
8796 | 18339^16258+16258^18339 | 77227 | Hans Havermann | 04/2020 |
8797 | 21243^4316+4316^21243 | 77221 | Hans Havermann | 04/2020 |
8798 | (2^256493+3126)/2 | 77212 | Stefano Morozzi | 01/2020 |
8799 | 2^256460-112757 | 77203 | Enzo Doria | 06/2014 |
8800 | (2^256444+2578)/2 | 77198 | Stefano Morozzi | 01/2020 |
8801 | 2^256398+23299 | 77184 | Yves Minet | 02/2011 |
8802 | 2^256391-1155*256391 | 77182 | Henri Lifchitz | 03/2005 |
8803 | (2^256391-9)/56593 | 77177 | Lelio R Paula | 10/2012 |
8804 | (7665^19867-1)/7664 | 77170 | Serge Batalov | 08/2020 |
8805 | (2^256338+3834)/2 | 77166 | Stefano Morozzi | 01/2020 |
8806 | (2^256339+546)/2 | 77166 | Stefano Morozzi | 01/2020 |
8807 | (2^256308+9)/10585 | 77153 | Lelio R Paula | 10/2012 |
8808 | 21^58348-2 | 77149 | Lelio R Paula | 08/2013 |
8809 | (17*2^256291-1)/4785 | 77149 | Lelio R Paula | 02/2014 |
8810 | (5*2^256274-1)/2141 | 77144 | Lelio R Paula | 02/2014 |
8811 | (2^256244+2326)/2 | 77137 | Stefano Morozzi | 01/2020 |
8812 | 19^60321+82 | 77136 | Lelio R Paula | 03/2015 |
8813 | 5*2^256233-3 | 77135 | Lelio R Paula | 06/2014 |
8814 | (2^256155+1566)/2 | 77111 | Stefano Morozzi | 01/2020 |
8815 | 3*2^256149+7 | 77110 | Lelio R Paula | 06/2014 |
8816 | (2^256128+1890)/2 | 77102 | Stefano Morozzi | 01/2020 |
8817 | 3^161594-106 | 77100 | Lelio R Paula | 02/2015 |
8818 | 2^256114-185 | 77098 | Lelio R Paula | 03/2012 |
8819 | 8^85349-85349^8 | 77078 | Norbert Schneider | 05/2017 |
8820 | 2^256043-2786377 | 77077 | Enzo Doria | 03/2016 |
8821 | 2^255973+1869947 | 77056 | Enzo Doria | 05/2016 |
8822 | 12^71393-43 | 77046 | Lelio R Paula | 02/2015 |
8823 | (2^255882+3630)/2 | 77028 | Stefano Morozzi | 01/2020 |
8824 | 15^65490+44 | 77023 | Lelio R Paula | 02/2015 |
8825 | (3^161411-7)/940 | 77010 | Lelio R Paula | 10/2012 |
8826 | lucasU(1107,1,25297)-lucasU(1107,1,25296) | 77005 | Mike Oakes | 05/2009 |
8828 | (570^27943-1)/569 | 77005 | Serge Batalov | 12/2016 |
8827 | (570^27943-1)/(570-1) | 77005 | Michael Stocker | 04/2020 |
8829 | 2^255750-75 | 76989 | Lelio R Paula | 03/2012 |
8830 | 3*6^98926-5 | 76980 | Lelio R Paula | 10/2014 |
8831 | (2^255708+478)/2 | 76976 | Stefano Morozzi | 01/2020 |
8832 | 18^61320-1433 | 76974 | Enzo Doria | 03/2013 |
8833 | 6^98916-17093 | 76972 | Enzo Doria | 02/2013 |
8834 | (2^255691+806)/2 | 76971 | Stefano Morozzi | 01/2020 |
8835 | 2^255659-110131 | 76962 | Enzo Doria | 04/2014 |
8836 | 69805!4-128 | 76956 | Robert Price | 11/2019 |
8837 | 12^71303+91 | 76949 | Lelio R Paula | 02/2015 |
8838 | 18^61298+59 | 76946 | Lelio R Paula | 02/2015 |
8839 | (223^32768+1)/131074 | 76945 | Lelio R Paula | 02/2009 |
8840 | (2^255613-5)/281517 | 76942 | Lelio R Paula | 10/2012 |
8841 | 18^61290-11 | 76936 | Lelio R Paula | 02/2015 |
8842 | (2^255567+1286)/2 | 76934 | Stefano Morozzi | 01/2020 |
8844 | 63^42752-2 | 76926 | Lelio R Paula | 08/2013 |
8843 | (11*2^255545-1)/117 | 76926 | Lelio R Paula | 02/2014 |
8845 | 2^255517+56387 | 76919 | Yves Minet | 02/2011 |
8846 | (7*10^76916+89)/3 | 76917 | Robert Price | 08/2020 |
8847 | 18376^15329+15329^18376 | 76914 | Hans Havermann | 03/2020 |
8848 | 2^255499-112567 | 76913 | Enzo Doria | 06/2014 |
8849 | 2^255492+345 | 76911 | Lelio R Paula | 06/2013 |
8850 | (2^255488+1678)/2 | 76910 | Stefano Morozzi | 01/2020 |
8851 | 10^76875+217591 | 76876 | Patrick De Geest | 05/2013 |
8853 | 10^76875-53 | 76875 | Robert Price | 05/2011 |
8852 | 11^73819-5880 | 76875 | Enzo Doria | 01/2013 |
8854 | 2^255313+18915 | 76857 | Lelio R Paula | 06/2013 |
8855 | (2^255320-1017)/1446667 | 76853 | Lelio R Paula | 10/2012 |
8856 | 2^255296+5775 | 76852 | Lelio R Paula | 06/2013 |
8857 | 14^67052-29 | 76851 | Lelio R Paula | 02/2015 |
8858 | (8*10^76840+43)/3 | 76841 | Robert Price | 10/2017 |
8859 | (2^255246+1642)/2 | 76837 | Stefano Morozzi | 01/2020 |
8860 | (2^255243+2250)/2 | 76836 | Stefano Morozzi | 01/2020 |
8861 | (2^255240+1218)/2 | 76835 | Stefano Morozzi | 01/2020 |
8862 | (7*10^76818-313)/9 | 76818 | Robert Price | 11/2016 |
8863 | 10^76801+69 | 76802 | Robert Price | 03/2011 |
8864 | 10^76801-80943 | 76801 | Patrick De Geest | 06/2013 |
8865 | 2^255105+57 | 76795 | Lelio R Paula | 03/2012 |
8866 | (lucasV(1,2,510187)+2)/(3*5*11) | 76789 | Mike Oakes | 04/2017 |
8867 | (2^255097-7)/20020055 | 76785 | Lelio R Paula | 10/2012 |
8868 | (2^255063+3098)/2 | 76782 | Stefano Morozzi | 01/2020 |
8869 | 2^255055+1869353 | 76780 | Enzo Doria | 05/2016 |
8870 | 17^62396-90 | 76776 | Lelio R Paula | 03/2015 |
8871 | 18797^12130+12130^18797 | 76765 | Hans Havermann | 03/2020 |
8872 | 7^90833-60 | 76763 | Lelio R Paula | 02/2015 |
8873 | 13^68901-56 | 76752 | Lelio R Paula | 03/2015 |
8874 | 2^254961+1869917 | 76751 | Enzo Doria | 05/2016 |
8875 | (2^254950+594)/2 | 76748 | Stefano Morozzi | 01/2020 |
8876 | 20734^5027+5027^20734 | 76743 | Hans Havermann | 03/2020 |
8877 | 9*5^109791-8 | 76742 | Lelio R Paula | 10/2014 |
8878 | (469^28729-1)/468 | 76738 | Serge Batalov | 08/2020 |
8879 | (2^254908+982)/2 | 76735 | Stefano Morozzi | 01/2020 |
8880 | (2^254903+2766)/2 | 76734 | Stefano Morozzi | 01/2020 |
8881 | (4143^21211-1)/4142 | 76724 | Serge Batalov | 08/2020 |
8882 | 82^40089+40089^82 | 76723 | Serge Batalov | 02/2012 |
8883 | 2^254861-2790013 | 76721 | Enzo Doria | 03/2016 |
8884 | (10^278+1)*(10^115926+1)/(10^834+1)/(10^38642+1)/20459779741 | 76718 | Makoto Kamada | 03/2019 |
8885 | 10^76705+13 | 76706 | Robert Price | 01/2011 |
8886 | 10^76705-94077 | 76705 | Patrick De Geest | 06/2013 |
8887 | (13709^18541-1)/13708 | 76701 | Serge Batalov | 08/2020 |
8888 | (5*2^254785-1)/999 | 76696 | Lelio R Paula | 02/2014 |
8889 | 2^254758-110405 | 76690 | Enzo Doria | 05/2014 |
8890 | (2^254721+470)/2 | 76679 | Stefano Morozzi | 01/2020 |
8891 | 2^254716-3779 | 76678 | Yves Minet | 04/2017 |
8892 | 2^254708-110507 | 76675 | Enzo Doria | 05/2014 |
8893 | (47*10^76672+133)/9 | 76673 | Robert Price | 04/2017 |
8894 | 2^254670-112901 | 76664 | Enzo Doria | 06/2014 |
8895 | (3^160665+7)/981350 | 76651 | Lelio R Paula | 10/2012 |
8896 | 2^254598-113183 | 76642 | Enzo Doria | 07/2014 |
8897 | 19^59932+22 | 76639 | Lelio R Paula | 01/2015 |
8898 | 77*10^76635+3 | 76637 | Robert Price | 06/2020 |
8899 | 2^254569-112213 | 76633 | Enzo Doria | 06/2014 |
8900 | 2*5^109626+9 | 76626 | Lelio R Paula | 10/2014 |
8901 | 2^254521+1872527 | 76619 | Enzo Doria | 05/2016 |
8902 | (2^254515+2570)/2 | 76617 | Stefano Morozzi | 01/2020 |
8903 | (76*10^76615+167)/9 | 76616 | Robert Price | 08/2016 |
8904 | (2^254495+918)/2 | 76611 | Stefano Morozzi | 01/2020 |
8905 | 25^54799-24^54799 | 76606 | Robert Price | 07/2012 |
8906 | (24737*2^254464+1)/9 | 76605 | Lelio R Paula | 06/2014 |
8907 | 25863^916+916^25863 | 76604 | Hans Havermann | 01/2020 |
8908 | (2^254458+2398)/2 | 76600 | Stefano Morozzi | 01/2020 |
8909 | (2^254434+1354)/2 | 76592 | Stefano Morozzi | 01/2020 |
8910 | (2^254410+2842)/2 | 76585 | Stefano Morozzi | 01/2020 |
8911 | 19878^7121+7121^19878 | 76581 | Hans Havermann | 01/2020 |
8912 | 12^70961-85 | 76580 | Lelio R Paula | 02/2015 |
8913 | 18867^11452+11452^18867 | 76579 | Hans Havermann | 01/2020 |
8914 | (2^254333+1806)/2 | 76562 | Stefano Morozzi | 01/2020 |
8915 | 7^90590-38 | 76558 | Lelio R Paula | 02/2015 |
8916 | 2^254304+5227 | 76554 | Lelio R Paula | 06/2013 |
8917 | (76*10^76547-211)/9 | 76548 | Robert Price | 10/2017 |
8919 | 18^60979-5363 | 76546 | Enzo Doria | 03/2013 |
8918 | (2^254278+3250)/2 | 76546 | Stefano Morozzi | 01/2020 |
8920 | (19098^17881-1)/19097 | 76545 | Serge Batalov | 08/2020 |
8921 | (21181*2^254264+1)/55314379 | 76538 | Lelio R Paula | 06/2014 |
8923 | (34*10^76547-7)/9/61/94529/214807 | 76536 | Makoto Morimoto | 05/2016 |
8922 | (7*10^76536-313)/9 | 76536 | Robert Price | 11/2016 |
8925 | 2^254239-27 | 76534 | Lelio R Paula | 03/2012 |
8924 | 2^254238-1841 | 76534 | Yves Minet | 03/2013 |
8926 | (2^254235+3978)/2 | 76533 | Stefano Morozzi | 01/2020 |
8928 | (10^76537-1)/9/66127969 | 76529 | Jens K. Andersen | 04/2009 |
8927 | (2^254223+3938)/2 | 76529 | Stefano Morozzi | 01/2020 |
8929 | 2^254189-112225 | 76519 | Enzo Doria | 06/2014 |
8930 | 18^60938-1397 | 76494 | Enzo Doria | 03/2013 |
8932 | 2^254075-2788417 | 76485 | Enzo Doria | 03/2016 |
8931 | 18239^15612+15612^18239 | 76485 | Hans Havermann | 01/2020 |
8933 | 7*10^76483-23 | 76484 | Robert Price | 04/2016 |
8934 | 18471^13822+13822^18471 | 76481 | Hans Havermann | 01/2020 |
8935 | 18850^11407+11407^18850 | 76478 | Hans Havermann | 01/2020 |
8936 | 2^254042-112913 | 76475 | Enzo Doria | 06/2014 |
8937 | 14^66723-37 | 76474 | Lelio R Paula | 02/2015 |
8938 | (2^254040+2970)/2 | 76474 | Stefano Morozzi | 01/2020 |
8939 | 2^254037-110305 | 76473 | Enzo Doria | 05/2014 |
8940 | (2^254008+2290)/2 | 76464 | Stefano Morozzi | 01/2020 |
8941 | 15927^18196+18196^15927 | 76463 | Norbert Schneider | 12/2019 |
8942 | (2^253979+1110)/2 | 76455 | Stefano Morozzi | 01/2020 |
8943 | (1575^23911-1)/1574 | 76447 | Serge Batalov | 08/2020 |
8944 | 2^253931-111109 | 76441 | Enzo Doria | 05/2014 |
8945 | 2^253926-1167 | 76440 | Lelio R Paula | 05/2013 |
8946 | (49*10^76438-13)/9 | 76439 | Robert Price | 07/2015 |
8947 | (2^253876+1302)/2 | 76424 | Stefano Morozzi | 01/2020 |
8948 | (21*2^253845-1)/143 | 76415 | Lelio R Paula | 02/2014 |
8949 | 10^76374+31 | 76375 | Robert Price | 03/2011 |
8950 | 10^76374-222159 | 76374 | Patrick De Geest | 05/2013 |
8951 | (6300^20101-1)/6299 | 76367 | Serge Batalov | 08/2020 |
8952 | (2^253660+490)/2 | 76359 | Stefano Morozzi | 01/2020 |
8953 | 19227^9352+9352^19227 | 76349 | Hans Havermann | 01/2020 |
8954 | (2^253617+2874)/2 | 76347 | Stefano Morozzi | 01/2020 |
8955 | 2^253561-110533 | 76330 | Enzo Doria | 05/2014 |
8956 | (2^253546+954)/2 | 76325 | Stefano Morozzi | 01/2020 |
8957 | gcd(V(917285),5*F(183457)*(F(183457)+1)+1) | 76321 | Serge Batalov | 04/2015 |
8958 | 9*5^109182+2 | 76316 | Lelio R Paula | 10/2014 |
8959 | 11^73279-3690 | 76313 | Enzo Doria | 12/2012 |
8960 | (598*10^76309-31)/9 | 76311 | Robert Price | 02/2017 |
8961 | 11^73274-24000 | 76308 | Enzo Doria | 12/2012 |
8962 | 7^90291-6 | 76305 | Theodore Burton | 11/2005 |
8963 | (2^253479+3650)/2 | 76305 | Stefano Morozzi | 01/2020 |
8964 | (2^253460+2902)/2 | 76299 | Stefano Morozzi | 01/2020 |
8965 | 6*10^76295-29 | 76296 | Robert Price | 08/2020 |
8966 | (21181*2^253432+1)/21437 | 76291 | Lelio R Paula | 06/2014 |
8967 | (2^253399+3254)/2 | 76281 | Stefano Morozzi | 01/2020 |
8968 | 69247!4+4 | 76280 | Robert Price | 09/2019 |
8969 | 6^98025+145 | 76279 | Lelio R Paula | 01/2015 |
8970 | (2^253382+1978)/2 | 76276 | Stefano Morozzi | 01/2020 |
8971 | 2^253355+1155 | 76268 | Lelio R Paula | 05/2013 |
8972 | (73*10^76262+413)/9 | 76263 | Robert Price | 10/2017 |
8973 | 2^253292-71219 | 76249 | Enzo Doria | 04/2016 |
8974 | 7^90221-36 | 76246 | Lelio R Paula | 02/2015 |
8975 | (2^253273+3486)/2 | 76243 | Stefano Morozzi | 01/2020 |
8976 | (2^253238+894)/2 | 76232 | Stefano Morozzi | 01/2020 |
8977 | 18^60727-3473 | 76229 | Enzo Doria | 03/2013 |
8978 | (2^253210+1170)/2 | 76224 | Stefano Morozzi | 01/2020 |
8979 | (78557*2^253184+1)/3 | 76221 | Lelio R Paula | 06/2014 |
8980 | (8*10^76220-83)/3 | 76221 | Robert Price | 10/2017 |
8981 | 18568^12717+12717^18568 | 76211 | Hans Havermann | 01/2020 |
8982 | (44791^20480+1)/(44791^4096+1) | 76206 | Serge Batalov | 01/2017 |
8983 | (5168^20521-1)/5167 | 76198 | Serge Batalov | 08/2020 |
8984 | 19443^8294+8294^19443 | 76193 | Hans Havermann | 12/2019 |
8985 | 11^73163-2010 | 76192 | Enzo Doria | 12/2012 |
8986 | 19006^10201+10201^19006 | 76189 | Hans Havermann | 12/2019 |
8987 | 2^253060+54615 | 76179 | Lelio R Paula | 05/2013 |
8988 | (2^253049+3966)/2 | 76176 | Stefano Morozzi | 01/2020 |
8989 | 2^253029+1869761 | 76170 | Enzo Doria | 05/2016 |
8990 | (2^253028+1906)/2 | 76169 | Stefano Morozzi | 01/2020 |
8991 | (1078^25111-1)/1077 | 76150 | Serge Batalov | 08/2020 |
8992 | 2^252955-111007 | 76148 | Enzo Doria | 05/2014 |
8993 | 2*5^108931+7 | 76140 | Lelio R Paula | 10/2014 |
8994 | (86*10^76128+121)/9 | 76129 | Robert Price | 03/2017 |
8995 | 22154*29^52050+9 | 76123 | Curtis Bright | 05/2015 |
8996 | 18915^10574+10574^18915 | 76119 | Hans Havermann | 01/2020 |
8997 | 3^159530-112358 | 76116 | Enzo Doria | 06/2014 |
8998 | 11^73089-48 | 76115 | Lelio R Paula | 02/2015 |
8999 | 86^39337-85 | 76098 | Norbert Schneider | 10/2015 |
9000 | (3^159505-7)/313297316 | 76095 | Lelio R Paula | 06/2012 |