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 ? |
8751 | 3*10^97205-61 | 97206 | Robert Price | 04/2021 |
8752 | (7*2^322896-1)/57 | 97201 | Lelio R Paula | 02/2014 |
8753 | (35*10^97194-341)/9 | 97195 | Robert Price | 08/2021 |
8754 | (4872^26357-1)/4871 | 97194 | Serge Batalov | 06/2022 |
8755 | (2^322897+5)/3146107489 | 97193 | Lelio R Paula | 10/2013 |
8756 | (796*10^97183-31)/9 | 97185 | Robert Price | 07/2022 |
8757 | (2^322789+2214)/2 | 97169 | Stefano Morozzi | 09/2020 |
8758 | 2^322725+75761 | 97150 | Enzo Doria | 04/2016 |
8759 | (13*2^322686+1)/119 | 97138 | Lelio R Paula | 11/2013 |
8760 | (2175^29101-1)/2174 | 97121 | Serge Batalov | 06/2022 |
8761 | 22402^21639+21639^22402 | 97118 | Hans Havermann | 07/2020 |
8762 | 23399^14128+14128^23399 | 97108 | Hans Havermann | 06/2020 |
8763 | (2892^28057-1)/2891 | 97108 | Serge Batalov | 06/2022 |
8764 | (9578^24391-1)/9577 | 97104 | Serge Batalov | 06/2022 |
8765 | (2^322525+2606)/2 | 97090 | Stefano Morozzi | 09/2020 |
8766 | (9563^24391-1)/9562 | 97087 | Serge Batalov | 06/2022 |
8767 | (41*10^97085+373)/9 | 97086 | Robert Price | 04/2017 |
8768 | (2^322473+854)/2 | 97074 | Stefano Morozzi | 02/2021 |
8769 | 2^322413+77 | 97056 | Lelio R Paula | 12/2012 |
8770 | 2^322406+177 | 97054 | Lelio R Paula | 12/2012 |
8771 | (349^38167-1)/348 | 97050 | Serge Batalov | 06/2022 |
8772 | (14*10^97044+31)/9 | 97045 | Robert Price | 02/2015 |
8773 | 2^322302+255255 | 97023 | Lelio R Paula | 07/2016 |
8774 | (1302^31147-1)/1301 | 97008 | Serge Batalov | 06/2022 |
8775 | 2^322196-113093 | 96991 | Enzo Doria | 07/2014 |
8776 | (2^322169+2690)/2 | 96983 | Stefano Morozzi | 09/2020 |
8777 | (49*10^96980+221)/9 | 96981 | Robert Price | 02/2022 |
8778 | 9*10^96968-47 | 96969 | Robert Price | 11/2021 |
8779 | (5895^25717-1)/5894 | 96962 | Serge Batalov | 06/2022 |
8780 | (27473^21841-1)/27472 | 96946 | Serge Batalov | 06/2022 |
8781 | 23109^15668+15668^23109 | 96943 | Hans Havermann | 07/2020 |
8782 | 2^322000-1861469 | 96932 | Enzo Doria | 03/2016 |
8783 | 2^321971-112747 | 96923 | Enzo Doria | 06/2014 |
8784 | 23754^12023+12023^23754 | 96917 | Hans Havermann | 07/2020 |
8785 | 11^93062-92 | 96915 | Lelio R Paula | 06/2016 |
8786 | (37871^21169-1)/37870 | 96914 | Serge Batalov | 06/2022 |
8787 | (2^321922+2214)/2 | 96908 | Stefano Morozzi | 09/2020 |
8788 | (2^321900+2086)/2 | 96902 | Stefano Morozzi | 09/2020 |
8789 | (11*2^321841-7)/15 | 96884 | Renaud Lifchitz | 03/2011 |
8790 | 3^203051+34 | 96880 | Lelio R Paula | 05/2016 |
8791 | 2*10^96867-51 | 96868 | Robert Price | 11/2017 |
8792 | (2^321779+2618)/2 | 96865 | Stefano Morozzi | 09/2020 |
8793 | (64*10^96863-1)/27 | 96864 | Serge Batalov | 05/2010 |
8794 | 2^321716-2786939 | 96847 | Enzo Doria | 03/2016 |
8795 | 22496^20181+20181^22496 | 96844 | Hans Havermann | 03/2020 |
8796 | 2^321701-113053 | 96842 | Enzo Doria | 07/2014 |
8797 | 2^321700-28635 | 96842 | Lelio R Paula | 06/2016 |
8798 | (5*2^321691-1)/27 | 96838 | Lelio R Paula | 02/2014 |
8799 | 6*10^96835+13 | 96836 | Robert Price | 10/2021 |
8800 | 6^124433-124433 | 96828 | Robert Price | 01/2014 |
8801 | 2^321631+69 | 96821 | Lelio R Paula | 12/2012 |
8802 | (27109^21841-1)/27108 | 96820 | Serge Batalov | 06/2022 |
8803 | (13*2^321606+1)/119 | 96813 | Lelio R Paula | 11/2013 |
8804 | (5052^26141-1)/5051 | 96809 | Serge Batalov | 06/2022 |
8805 | (46*10^96798-1)/9 | 96799 | Serge Batalov | 10/2014 |
8806 | (59*10^96798+31)/9 | 96799 | Robert Price | 09/2015 |
8807 | (2^321536+2062)/2 | 96792 | Stefano Morozzi | 09/2020 |
8808 | 223319#+231223 | 96790 | Makoto Morimoto | 02/2016 |
8809 | 2^321519-28635 | 96787 | Lelio R Paula | 06/2016 |
8810 | 2^321487-1867237 | 96778 | Enzo Doria | 03/2016 |
8811 | 2^321461-63 | 96770 | Lelio R Paula | 12/2012 |
8812 | (16053^23011-1)/16052 | 96770 | Serge Batalov | 06/2022 |
8813 | 2^321439+123 | 96763 | Lelio R Paula | 12/2012 |
8814 | 2^321434+709 | 96762 | Lelio R Paula | 02/2013 |
8815 | (2^321415+554)/2 | 96756 | Stefano Morozzi | 09/2020 |
8816 | 14^84413+89 | 96749 | Lelio R Paula | 07/2016 |
8817 | (28*10^96743+71)/9 | 96744 | Serge Batalov | 12/2008 |
8818 | 13^86847+96 | 96743 | Lelio R Paula | 06/2016 |
8819 | 9*2^321336-7 | 96733 | Lelio R Paula | 08/2014 |
8820 | (209*10^96722-17)/3 | 96724 | Robert Price | 05/2017 |
8821 | (54163^20431-1)/54162 | 96710 | Serge Batalov | 06/2022 |
8822 | 2^321237-4615 | 96702 | Yves Minet | 05/2019 |
8823 | (27273^21799-1)/27272 | 96691 | Serge Batalov | 06/2022 |
8824 | 2^321187-1159 | 96687 | Yves Minet | 11/2012 |
8825 | 2^321182-1850081 | 96686 | Enzo Doria | 08/2015 |
8827 | 23359^13768+13768^23359 | 96680 | Hans Havermann | 06/2020 |
8826 | (65630^20071-1)/65629 | 96680 | Serge Batalov | 06/2022 |
8828 | 2^321116-112829 | 96666 | Enzo Doria | 06/2014 |
8829 | 2^321108-109499 | 96664 | Enzo Doria | 04/2014 |
8830 | 2^321098+2793859 | 96661 | Enzo Doria | 03/2016 |
8831 | (2^321100+1170)/2 | 96661 | Stefano Morozzi | 09/2020 |
8832 | 2^321057+4097 | 96648 | Lelio R Paula | 06/2016 |
8833 | (4435*29^66081-4)/7 | 96640 | Curtis Bright | 05/2015 |
8834 | 22743^17750+17750^22743 | 96640 | Hans Havermann | 04/2020 |
8835 | 22882^16679+16679^22882 | 96612 | Hans Havermann | 02/2020 |
8836 | 23452^13163+13163^23452 | 96608 | Hans Havermann | 02/2020 |
8837 | (1042^32009-1)/1041 | 96596 | Serge Batalov | 06/2022 |
8838 | (85*10^96594-373)/9 | 96595 | Robert Price | 05/2021 |
8839 | (2^320863+666)/2 | 96590 | Stefano Morozzi | 09/2020 |
8840 | p(19849)#/2-64 | 96581 | Robert Price | 04/2017 |
8841 | 2^320817+189 | 96576 | Lelio R Paula | 12/2012 |
8842 | 2^320816-4685 | 96576 | Yves Minet | 05/2019 |
8844 | (361*10^96567-163)/9 | 96569 | Robert Price | 10/2017 |
8843 | (19*10^96568+137)/3 | 96569 | Robert Price | 03/2022 |
8845 | 2^320788-110327 | 96567 | Enzo Doria | 05/2014 |
8846 | (3065^27697-1)/3064 | 96561 | Serge Batalov | 06/2022 |
8847 | 2^320764+793 | 96560 | Lelio R Paula | 02/2013 |
8848 | (2^320752+1842)/2 | 96556 | Stefano Morozzi | 02/2021 |
8849 | (48331^20611-1)/48330 | 96542 | Serge Batalov | 06/2022 |
8850 | (50595^20521-1)/50594 | 96529 | Serge Batalov | 06/2022 |
8851 | (9179*2^320652-1)/221 | 96528 | Lelio R Paula | 05/2015 |
8852 | 23288^13947+13947^23288 | 96517 | Hans Havermann | 07/2020 |
8854 | 2^320616-78795 | 96516 | Lelio R Paula | 03/2016 |
8853 | (19*10^96520+287)/9/109/409 | 96516 | Alfred Reich | 11/2022 |
8855 | 2^320612-461939 | 96514 | Enzo Doria | 07/2015 |
8856 | (21245^22303-1)/21244 | 96507 | Serge Batalov | 06/2022 |
8857 | 10^96504+987654321 | 96505 | Robert Price | 12/2019 |
8858 | 10^96504+71887 | 96505 | Patrick De Geest | 02/2020 |
8859 | 22654^18195+18195^22654 | 96505 | Hans Havermann | 07/2020 |
8860 | 10^96504-139277 | 96504 | Patrick De Geest | 02/2020 |
8861 | (218^41263-1)/217 | 96490 | Serge Batalov | 06/2022 |
8862 | (47220^20641-1)/47219 | 96474 | Serge Batalov | 06/2022 |
8863 | 5*2^320431+7 | 96461 | Lelio R Paula | 08/2014 |
8864 | 19^75431-90 | 96458 | Lelio R Paula | 06/2016 |
8865 | (11853^23677-1)/11852 | 96452 | Serge Batalov | 06/2022 |
8866 | (2^320386+2814)/2 | 96446 | Stefano Morozzi | 02/2021 |
8867 | (2^320370+3)/67 | 96440 | Lelio R Paula | 10/2013 |
8868 | 3*10^96431-17 | 96432 | Robert Price | 03/2021 |
8869 | 2^320309+1871477 | 96423 | Enzo Doria | 05/2016 |
8870 | (10412^24001-1)/10411 | 96421 | Serge Batalov | 06/2022 |
8871 | tau(809^6028) | 96410 | Nik Lygeros / Olivier Rozier / LR team | 10/2013 |
8872 | (29*10^96407-23)/3 | 96408 | Robert Price | 03/2022 |
8873 | (21*2^320252-1)/1895 | 96404 | Lelio R Paula | 02/2014 |
8874 | 2^320232-173 | 96400 | Lelio R Paula | 12/2012 |
8875 | 175^42975-2 | 96395 | Lelio R Paula | 10/2017 |
8876 | (2^320170+2718)/2 | 96381 | Stefano Morozzi | 09/2020 |
8877 | 10^96377+199911 | 96378 | Patrick De Geest | 12/2016 |
8878 | 10^96377-20001 | 96377 | Robert Price | 11/2016 |
8879 | (18079^22639-1)/18078 | 96374 | Serge Batalov | 06/2022 |
8881 | 2^320124-2790449 | 96367 | Enzo Doria | 03/2016 |
8880 | 10^96366+19197 | 96367 | Patrick De Geest | 08/2021 |
8883 | 10^96366-10000000001 | 96366 | Robert Price | 04/2021 |
8882 | 10^96366-287367 | 96366 | Patrick De Geest | 08/2021 |
8884 | (26351^21799-1)/26350 | 96365 | Serge Batalov | 06/2022 |
8885 | 22488^19255+19255^22488 | 96351 | Hans Havermann | 11/2019 |
8886 | 23248^13939+13939^23248 | 96346 | Hans Havermann | 11/2019 |
8887 | (5928^25537-1)/5927 | 96345 | Serge Batalov | 06/2022 |
8889 | 2^320043-108775 | 96343 | Enzo Doria | 04/2014 |
8888 | 24989^7168+7168^24989 | 96343 | Hans Havermann | 11/2019 |
8890 | 10^96338+424939 | 96339 | Patrick De Geest | 03/2016 |
8891 | 10^96338-9281 | 96338 | Robert Price | 09/2015 |
8892 | (8527^24509-1)/8526 | 96336 | Serge Batalov | 06/2022 |
8893 | (3*2^320008+1)/7 | 96332 | Lelio R Paula | 09/2013 |
8894 | (2444^28433-1)/2443 | 96331 | Serge Batalov | 06/2022 |
8895 | (48690^20551-1)/48689 | 96327 | Serge Batalov | 06/2022 |
8896 | (61935^20101-1)/61934 | 96318 | Serge Batalov | 06/2022 |
8897 | 25699^5596+5596^25699 | 96317 | Hans Havermann | 07/2020 |
8898 | (7*10^96312+83)/9 | 96312 | Tyler Busby | 01/2023 |
8899 | (11*10^96298+529)/9 | 96299 | Robert Price | 09/2021 |
8901 | 26502^4297+4297^26502 | 96287 | Hans Havermann | 07/2020 |
8900 | (54180^20341-1)/54179 | 96287 | Serge Batalov | 06/2022 |
8902 | 2^319838+4257 | 96281 | Lelio R Paula | 03/2016 |
8903 | 2^319830-16995 | 96279 | Lelio R Paula | 03/2016 |
8904 | 6^123713+143 | 96268 | Lelio R Paula | 05/2016 |
8905 | (28*10^96265-97)/3 | 96266 | Robert Price | 07/2016 |
8906 | (3799^26893-1)/3798 | 96265 | Serge Batalov | 06/2022 |
8908 | 103^47825-102 | 96264 | Norbert Schneider | 10/2016 |
8907 | (54041^20341-1)/54040 | 96264 | Serge Batalov | 06/2022 |
8909 | (2^319777+2082)/2 | 96263 | Stefano Morozzi | 09/2020 |
8910 | (2^319771+3686)/2 | 96261 | Stefano Morozzi | 09/2020 |
8911 | 2^319713-81 | 96244 | Lelio R Paula | 11/2012 |
8912 | (55459*2^319706+1)/20167 | 96242 | Lelio R Paula | 07/2014 |
8913 | 22834^16393+16393^22834 | 96238 | Hans Havermann | 05/2020 |
8914 | 8*10^96236+27 | 96237 | Robert Price | 09/2021 |
8915 | 12^89153-43 | 96213 | Lelio R Paula | 06/2016 |
8916 | (25048^21871-1)/25047 | 96202 | Serge Batalov | 06/2022 |
8918 | (2*10^96195-59)/3 | 96195 | Robert Price | 05/2017 |
8917 | 9*10^96194-77 | 96195 | Robert Price | 08/2020 |
8919 | 2^319545+147 | 96193 | Lelio R Paula | 11/2012 |
8920 | 23357^13128+13128^23357 | 96189 | Hans Havermann | 02/2020 |
8921 | (19838^22381-1)/19837 | 96179 | Serge Batalov | 06/2022 |
8922 | (221^41023-1)/220 | 96172 | Serge Batalov | 06/2022 |
8923 | 25952^5073+5073^25952 | 96160 | Hans Havermann | 12/2019 |
8924 | (50891^20431-1)/50890 | 96157 | Serge Batalov | 06/2022 |
8925 | (50^56597+1)/51 | 96155 | Robert Price | 03/2014 |
8926 | 22313^20374+20374^22313 | 96149 | Hans Havermann | 12/2019 |
8927 | 22279^20668+20668^22279 | 96141 | Hans Havermann | 11/2019 |
8928 | (41*10^96137-59)/9 | 96138 | Robert Price | 10/2017 |
8929 | 2^319353-110343 | 96135 | Enzo Doria | 05/2014 |
8930 | 11^92309-42 | 96130 | Lelio R Paula | 05/2016 |
8931 | (21675^22171-1)/21674 | 96129 | Serge Batalov | 06/2022 |
8932 | 14^83869-13^83869 | 96125 | Jean-Louis Charton | 01/2009 |
8933 | (7*10^96112+143)/3 | 96113 | Robert Price | 04/2016 |
8934 | 26731^3940+3940^26731 | 96112 | Hans Havermann | 07/2020 |
8935 | 22640^17583+17583^22640 | 96109 | Hans Havermann | 07/2020 |
8936 | (19178^22441-1)/19177 | 96107 | Serge Batalov | 06/2022 |
8937 | 2^319237+65835 | 96100 | Lelio R Paula | 03/2016 |
8938 | 2^319194-1841 | 96087 | Yves Minet | 03/2013 |
8939 | 23949^10280+10280^23949 | 96084 | Hans Havermann | 07/2020 |
8940 | (1927^29251-1)/1926 | 96083 | Serge Batalov | 06/2022 |
8941 | 20^73847-31 | 96078 | Lelio R Paula | 05/2016 |
8942 | (60229^20101-1)/60228 | 96075 | Serge Batalov | 06/2022 |
8943 | (28044^21601-1)/28043 | 96074 | Serge Batalov | 06/2022 |
8944 | 50^56543-49 | 96065 | Norbert Schneider | 09/2018 |
8945 | 2^319107-109555 | 96061 | Enzo Doria | 04/2014 |
8946 | (3*2^319114-1)/2627 | 96060 | Lelio R Paula | 02/2014 |
8947 | 2^319098-112157 | 96059 | Enzo Doria | 06/2014 |
8948 | 15^81673+94 | 96055 | Lelio R Paula | 07/2016 |
8949 | (85761^19471-1)/85760 | 96052 | Serge Batalov | 06/2022 |
8950 | 2^319074+1871749 | 96051 | Enzo Doria | 05/2016 |
8951 | 25670^5517+5517^25670 | 96050 | Hans Havermann | 06/2020 |
8952 | 6*10^96037+19 | 96038 | Robert Price | 11/2021 |
8953 | 2^319007-112111 | 96031 | Enzo Doria | 06/2014 |
8954 | (57987^20161-1)/57986 | 96029 | Serge Batalov | 06/2022 |
8956 | 2^318946-1421 | 96013 | Yves Minet | 01/2013 |
8955 | (68310^19861-1)/68309 | 96013 | Serge Batalov | 06/2022 |
8957 | (20294^22291-1)/20293 | 96012 | Serge Batalov | 06/2022 |
8958 | (304^38669-1)/303 | 96008 | Serge Batalov | 06/2022 |
8959 | (1840^29401-1)/1839 | 95986 | Serge Batalov | 06/2022 |
8960 | 22830^16003+16003^22830 | 95982 | Hans Havermann | 02/2020 |
8961 | 2^318834-110411 | 95979 | Enzo Doria | 05/2014 |
8962 | 2^318831-1965 | 95978 | Lelio R Paula | 03/2016 |
8963 | (5*10^95976+7)/3 | 95977 | Erik Branger | 03/2014 |
8964 | (12797^23369-1)/12796 | 95975 | Serge Batalov | 06/2022 |
8965 | (10534^23857-1)/10533 | 95963 | Serge Batalov | 06/2022 |
8966 | 22871^15696+15696^22871 | 95962 | Hans Havermann | 01/2020 |
8967 | (2^318760+858)/2 | 95957 | Stefano Morozzi | 02/2021 |
8968 | 23167^13860+13860^23167 | 95953 | Hans Havermann | 01/2020 |
8969 | (823^32911-1)/822 | 95946 | Serge Batalov | 06/2022 |
8971 | 5*3^201089-8 | 95945 | Lelio R Paula | 08/2014 |
8970 | (81783^19531-1)/81782 | 95945 | Serge Batalov | 06/2022 |
8972 | 2^318711-139 | 95942 | Lelio R Paula | 11/2012 |
8973 | 2^318635-1167 | 95919 | Lelio R Paula | 03/2016 |
8974 | 15^81555+16 | 95917 | Lelio R Paula | 07/2016 |
8975 | (37*10^95895+377)/9 | 95896 | Robert Price | 10/2017 |
8976 | (22699*2^318527+1)/21 | 95890 | Lelio R Paula | 07/2014 |
8977 | 2^318496-97305 | 95877 | Lelio R Paula | 03/2016 |
8978 | 22129^21496+21496^22129 | 95871 | Hans Havermann | 07/2020 |
8980 | 2^318466+15015 | 95868 | Lelio R Paula | 03/2016 |
8979 | 24847^7216+7216^24847 | 95868 | Hans Havermann | 07/2020 |
8981 | 23706^11065+11065^23706 | 95866 | Hans Havermann | 07/2020 |
8982 | (2863^27733-1)/2862 | 95865 | Serge Batalov | 06/2022 |
8983 | 6^123187-173129 | 95859 | Mike Oakes | 04/2011 |
8984 | 2^318423-487 | 95855 | Yves Minet | 03/2012 |
8985 | 10^95848+443709 | 95849 | Patrick De Geest | 10/2022 |
8986 | 10^95848-5000000001 | 95848 | Robert Price | 09/2022 |
8987 | 10^95848-61971 | 95848 | Patrick De Geest | 10/2022 |
8988 | (2^318359+678)/2 | 95836 | Stefano Morozzi | 09/2020 |
8989 | (10223*2^318350+1)/1563 | 95834 | Lelio R Paula | 07/2014 |
8990 | (8060^24533-1)/8059 | 95831 | Serge Batalov | 06/2022 |
8991 | (9853^23993-1)/9852 | 95814 | Serge Batalov | 06/2022 |
8993 | (2^318267+2550)/2 | 95808 | Stefano Morozzi | 09/2020 |
8992 | (5048^25873-1)/5047 | 95808 | Serge Batalov | 06/2022 |
8995 | 4*10^95799+13 | 95800 | Robert Price | 02/2021 |
8994 | (65533^19891-1)/65532 | 95800 | Serge Batalov | 06/2022 |
8996 | 45369!2+1024 | 95787 | Robert Price | 02/2015 |
8997 | 3^200758-86 | 95786 | Lelio R Paula | 05/2016 |
8998 | (2^318271+1)/(3*46031404238891*654090604244843) | 95781 | Gord Palameta | 12/2018 |
8999 | (2^318171+2030)/2 | 95779 | Stefano Morozzi | 09/2020 |
9000 | 3*10^95776-59 | 95777 | Robert Price | 10/2017 |