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 ? |
8251 | 12^87212-37 | 94118 | Lelio R Paula | 06/2016 |
8252 | 11^90376+76 | 94117 | Lelio R Paula | 05/2016 |
8253 | (6^120946-11)/5 | 94114 | Paul Bourdelais | 11/2011 |
8254 | (2^312622+534)/2 | 94109 | Stefano Morozzi | 09/2020 |
8255 | (10^94096+191)/3 | 94096 | Robert Price | 11/2017 |
8256 | 23794^9005+9005^23794 | 94093 | Hans Havermann | 09/2020 |
8257 | 2^312562-112685 | 94091 | Enzo Doria | 06/2014 |
8258 | 137^44026-2 | 94072 | Lelio R Paula | 10/2017 |
8259 | s(164441) | 94052 | Robert Price | 05/2020 |
8260 | 2^312404-869688105 | 94043 | Lelio R Paula | 09/2016 |
8261 | 2^312387+2793533 | 94038 | Enzo Doria | 03/2016 |
8263 | 10^94016+91 | 94017 | Robert Price | 04/2011 |
8262 | (15*2^312325-1)/7201 | 94017 | Lelio R Paula | 02/2014 |
8264 | 10^94016-188963 | 94016 | Patrick De Geest | 08/2013 |
8265 | (17*2^312315-1)/45 | 94016 | Lelio R Paula | 02/2014 |
8266 | (401*10^94010+13)/9 | 94012 | Robert Price | 09/2021 |
8267 | 23866^8691+8691^23866 | 94010 | Hans Havermann | 08/2020 |
8268 | 6^120802+7661 | 94003 | Mike Oakes | 04/2011 |
8269 | (67607*2^312254+1)/177 | 94001 | Lelio R Paula | 06/2014 |
8270 | (2^312245+702)/2 | 93995 | Stefano Morozzi | 09/2020 |
8271 | (56*10^93993+493)/9 | 93994 | Robert Price | 10/2021 |
8272 | (11*2^312207+1)/19 | 93984 | Lelio R Paula | 09/2013 |
8273 | 2^312201+51 | 93982 | Lelio R Paula | 11/2012 |
8274 | (4*6^120749+1)/5 | 93961 | Serge Batalov | 10/2014 |
8275 | (781*10^93942-7)/9 | 93944 | Robert Price | 12/2021 |
8276 | 3^196892+58 | 93942 | Lelio R Paula | 05/2016 |
8277 | 2^312060+18915 | 93940 | Lelio R Paula | 03/2016 |
8278 | 42^57867-41 | 93933 | Norbert Schneider | 08/2018 |
8279 | 8*10^93921+83 | 93922 | Robert Price | 11/2021 |
8280 | 2^311992-16209 | 93919 | Enzo Doria | 04/2013 |
8281 | (85^48677+1)/86 | 93917 | Paul Bourdelais | 08/2017 |
8282 | 2^311965-39 | 93911 | Lelio R Paula | 11/2012 |
8283 | 2^311920-111689 | 93898 | Enzo Doria | 05/2014 |
8284 | 130^44414+129 | 93889 | Norbert Schneider | 01/2019 |
8285 | (2^311892+2730)/2 | 93889 | Stefano Morozzi | 09/2020 |
8286 | 7*10^93887-47 | 93888 | Robert Price | 02/2017 |
8287 | 9*10^93878-37 | 93879 | Robert Price | 06/2021 |
8288 | (28*10^93874+323)/9 | 93875 | Robert Price | 10/2021 |
8289 | (2^311778+1530)/2 | 93855 | Stefano Morozzi | 09/2020 |
8290 | 12^86965-13 | 93851 | Lelio R Paula | 06/2016 |
8291 | 24862^5953+5953^24862 | 93848 | Hans Havermann | 09/2020 |
8292 | 22299^16148+16148^22299 | 93837 | Hans Havermann | 08/2020 |
8293 | 2^311705+16095 | 93833 | Lelio R Paula | 03/2016 |
8294 | (2^311701+3002)/2 | 93832 | Stefano Morozzi | 09/2020 |
8296 | 2^311665-81 | 93821 | Lelio R Paula | 11/2012 |
8295 | (2^311665+2654)/2 | 93821 | Stefano Morozzi | 09/2020 |
8297 | (2^311648+3058)/2 | 93816 | Stefano Morozzi | 09/2020 |
8298 | (2^311639+2738)/2 | 93813 | Stefano Morozzi | 01/2021 |
8299 | 730^32761+32761^730 | 93806 | Norbert Schneider | 09/2017 |
8300 | 11^90073+92 | 93802 | Lelio R Paula | 05/2016 |
8301 | 6*10^93792+41 | 93793 | Robert Price | 01/2017 |
8302 | 10^93790+64611 | 93791 | Patrick De Geest | 04/2020 |
8305 | (2^311580+7)/591329 | 93790 | Lelio R Paula | 06/2013 |
8303 | 10^93790-123456789 | 93790 | Robert Price | 02/2020 |
8304 | 10^93790-157517 | 93790 | Patrick De Geest | 04/2020 |
8306 | (49*10^93782+293)/9 | 93783 | Robert Price | 03/2021 |
8307 | 12^86900-77 | 93781 | Lelio R Paula | 06/2016 |
8308 | 2^311508-109769 | 93774 | Enzo Doria | 04/2014 |
8309 | 2^311486-1859621 | 93767 | Enzo Doria | 03/2016 |
8310 | Phi(93811,10)/968129521/2109997013/3280113810431/639637782545831 | 93765 | Kurt Beschorner | 04/2015 |
8311 | (10^93811-1)/(9*968129521*2109997013*3280113810431*639637782545831) | 93765 | Gord Palameta | 10/2018 |
8312 | 2^311471+311469 | 93763 | Renaud Lifchitz | 01/2003 |
8313 | (88*10^93760-1)/3 | 93762 | Robert Price | 10/2017 |
8314 | 2^311426+79459 | 93749 | Enzo Doria | 06/2016 |
8315 | 21537^21634+21634^21537 | 93745 | Norbert Schneider | 04/2020 |
8316 | 2^311409+111546435 | 93744 | Paul Bourdelais | 10/2021 |
8317 | (2^311360+3078)/2 | 93729 | Stefano Morozzi | 09/2020 |
8318 | (2^311341+2186)/2 | 93723 | Stefano Morozzi | 09/2020 |
8319 | (2^311313+774)/2 | 93715 | Stefano Morozzi | 09/2020 |
8321 | 2^311294+5775 | 93709 | Lelio R Paula | 03/2016 |
8320 | 25155^5312+5312^25155 | 93709 | Hans Havermann | 09/2020 |
8322 | 2^311252-1017 | 93697 | Lelio R Paula | 03/2016 |
8323 | 2^311247-112041 | 93695 | Enzo Doria | 06/2014 |
8324 | 7*10^93692-83 | 93693 | Robert Price | 09/2017 |
8325 | 2^311205-109645 | 93683 | Enzo Doria | 04/2014 |
8326 | 5^134027-84 | 93681 | Lelio R Paula | 04/2016 |
8327 | (2^311148-1017)/557 | 93663 | Lelio R Paula | 06/2013 |
8328 | (2^311027-2^155514+1)/5 | 93628 | | 11/2006 |
8329 | 26748^3161+3161^26748 | 93614 | Hans Havermann | 08/2020 |
8330 | 3^196192-28 | 93608 | Robert Price | 12/2013 |
8331 | 216091#+261433 | 93606 | Makoto Morimoto | 12/2015 |
8332 | 5^133917+68 | 93604 | Lelio R Paula | 04/2016 |
8333 | 2^310924-1709 | 93598 | Yves Minet | 03/2013 |
8334 | 24555^6476+6476^24555 | 93587 | Hans Havermann | 08/2020 |
8335 | (2^310867+734)/2 | 93580 | Stefano Morozzi | 09/2020 |
8336 | (266*10^93577-11)/3 | 93579 | Robert Price | 09/2016 |
8337 | (16*10^93577+161)/3 | 93578 | Robert Price | 05/2016 |
8338 | 3^196113-56 | 93570 | Lelio R Paula | 04/2016 |
8339 | 2^310806-110345 | 93562 | Enzo Doria | 05/2014 |
8340 | 3^196093+86 | 93561 | Lelio R Paula | 04/2016 |
8341 | 2^310780-2327 | 93555 | Yves Minet | 10/2013 |
8342 | 30^63332+41 | 93550 | Lelio R Paula | 08/2012 |
8343 | (21181*2^310707+1)/21 | 93536 | Lelio R Paula | 06/2014 |
8344 | 8*5^133807+7 | 93528 | Lelio R Paula | 10/2014 |
8345 | 2^310686+8193 | 93526 | Lelio R Paula | 03/2016 |
8346 | 22681^13282+13282^22681 | 93520 | Hans Havermann | 09/2020 |
8347 | 2^310646+63 | 93514 | Lelio R Paula | 11/2012 |
8348 | (77*10^93508+31)/9 | 93509 | Robert Price | 10/2015 |
8349 | (1000907^15583+1)/1000908 | 93499 | Faysal YAtim | 04/2009 |
8350 | (2*10^93497+7)/9 | 93497 | Erik Branger | 03/2011 |
8351 | 3^195958-112852 | 93496 | Enzo Doria | 07/2014 |
8352 | 2^310583+16385 | 93495 | Lelio R Paula | 03/2016 |
8353 | (9913*2^310557-1)/35 | 93490 | Lelio R Paula | 04/2015 |
8354 | 9*7^110614+2 | 93481 | Lelio R Paula | 11/2014 |
8355 | (19*10^93467+83)/3 | 93468 | Robert Price | 08/2020 |
8356 | 2^310469-8295 | 93461 | Lelio R Paula | 03/2016 |
8357 | 2^310459-1159 | 93458 | Yves Minet | 11/2012 |
8358 | 6^120075-95 | 93437 | Lelio R Paula | 05/2016 |
8359 | (2^310362+1758)/2 | 93428 | Stefano Morozzi | 09/2020 |
8360 | 148^43041-147 | 93411 | Norbert Schneider | 05/2019 |
8361 | 2^310289+98327 | 93407 | Yves Minet | 11/2010 |
8362 | 21769^19510+19510^21769 | 93395 | Hans Havermann | 08/2020 |
8363 | 2^310217-113143 | 93385 | Enzo Doria | 07/2014 |
8364 | 7^110494-68 | 93379 | Lelio R Paula | 04/2016 |
8365 | 2^310184-113093 | 93375 | Enzo Doria | 07/2014 |
8366 | 2^310180-110417 | 93374 | Enzo Doria | 05/2014 |
8367 | gcd(V(1124905),5*F(224981)*(F(224981)-1)+1) | 93372 | Serge Batalov | 04/2015 |
8368 | (2^310173+3150)/2 | 93372 | Stefano Morozzi | 09/2020 |
8369 | 3^195669+68 | 93358 | Lelio R Paula | 04/2016 |
8370 | 10^93356+91 | 93357 | Robert Price | 04/2011 |
8371 | 10^93356-28313 | 93356 | Patrick De Geest | 08/2013 |
8372 | (85*10^93346+23)/9 | 93347 | Robert Price | 11/2015 |
8373 | (3^195648-5)/3158276 | 93342 | Lelio R Paula | 04/2013 |
8374 | 10^93333-4000001 | 93333 | Robert Price | 07/2021 |
8375 | 3^195556+8680 | 93304 | Lelio R Paula | 04/2016 |
8376 | 2^309942-109991 | 93302 | Enzo Doria | 04/2014 |
8377 | 2^309926-1867301 | 93298 | Enzo Doria | 03/2016 |
8378 | 2^309924+85951 | 93297 | Yves Minet | 03/2011 |
8379 | 2^309896-4229 | 93288 | Yves Minet | 08/2018 |
8380 | (86*10^93263-77)/9 | 93264 | Robert Price | 05/2016 |
8381 | 6^119836-125 | 93251 | Lelio R Paula | 05/2016 |
8382 | (77*10^93249-257)/9 | 93250 | Robert Price | 10/2017 |
8383 | 21822^18749+18749^21822 | 93245 | Hans Havermann | 09/2020 |
8384 | 3^195430-2 | 93244 | Theodore Burton | 02/2007 |
8385 | (13*10^93236+47)/3 | 93237 | Robert Price | 01/2017 |
8386 | (67*10^93226-7)/3 | 93228 | Robert Price | 10/2017 |
8387 | (86*10^93217-221)/9 | 93218 | Robert Price | 06/2017 |
8388 | 2^309637-16329 | 93211 | Enzo Doria | 04/2013 |
8389 | (41*10^93207+49)/9 | 93208 | Robert Price | 04/2016 |
8390 | 14^81312-57 | 93194 | Lelio R Paula | 06/2016 |
8391 | 41^57783-40 | 93192 | Theodore Burton | 02/2007 |
8392 | 3*10^93179-97 | 93180 | Robert Price | 08/2017 |
8393 | 5*7^110252-4 | 93175 | Lelio R Paula | 11/2014 |
8394 | 22022^16959+16959^22022 | 93140 | Hans Havermann | 08/2020 |
8395 | 83061!4-512 | 93137 | Robert Price | 11/2019 |
8396 | 25169^5016+5016^25169 | 93135 | Hans Havermann | 08/2020 |
8397 | (483641^16384+1)/2 | 93135 | Richard Fischer | 02/2021 |
8398 | 2^309380-27 | 93133 | Lelio R Paula | 11/2012 |
8399 | 25653^4268+4268^25653 | 93127 | Hans Havermann | 08/2020 |
8400 | (2^309355+1826)/2 | 93125 | Stefano Morozzi | 02/2021 |
8401 | (2^309334+2710)/2 | 93119 | Stefano Morozzi | 09/2020 |
8402 | 23*10^93116-7 | 93118 | Robert Price | 03/2016 |
8403 | (2^309270+3214)/2 | 93100 | Stefano Morozzi | 09/2020 |
8404 | (29*10^93082+1)/3 | 93083 | Robert Price | 11/2015 |
8405 | (2^309198+3738)/2 | 93078 | Stefano Morozzi | 09/2020 |
8406 | (58*10^93062-13)/9 | 93063 | Robert Price | 09/2015 |
8407 | (2^309138+1794)/2 | 93060 | Stefano Morozzi | 01/2021 |
8408 | 2^309131+1869383 | 93058 | Enzo Doria | 05/2016 |
8409 | (2^309114+3610)/2 | 93053 | Stefano Morozzi | 09/2020 |
8410 | 2^309066-93 | 93039 | Lelio R Paula | 11/2012 |
8411 | 2^309063-113269 | 93038 | Enzo Doria | 07/2014 |
8412 | 3^194965+16 | 93022 | Robert Price | 10/2013 |
8413 | (89*10^93019-737)/9 | 93020 | Robert Price | 01/2022 |
8414 | 2^308979+1869143 | 93012 | Enzo Doria | 05/2016 |
8415 | 5^133064+54 | 93008 | Lelio R Paula | 04/2016 |
8416 | (2^308947+2750)/2 | 93003 | Stefano Morozzi | 01/2021 |
8417 | (2^308905-509203)/119 | 92988 | Lelio R Paula | 06/2013 |
8418 | (94*10^92981-49)/99 | 92981 | Ray Chandler | 07/2011 |
8419 | (2*10^92976-17)/3 | 92976 | Robert Price | 08/2014 |
8420 | 24783^5642+5642^24783 | 92972 | Hans Havermann | 08/2020 |
8421 | (472489^16384+1)/2 | 92969 | Richard Fischer | 01/2021 |
8422 | 2^308809+37967 | 92961 | Phil Moore | 03/2008 |
8423 | (3^194825+5)/11656 | 92952 | Lelio R Paula | 04/2013 |
8424 | (2^308679+5)/1573 | 92919 | Lelio R Paula | 06/2013 |
8425 | 2^308650+513 | 92913 | Lelio R Paula | 02/2016 |
8426 | 5*7^109936+6 | 92908 | Lelio R Paula | 10/2014 |
8427 | 2*10^92897+89 | 92898 | Robert Price | 07/2016 |
8428 | (2^308585+3182)/2 | 92894 | Stefano Morozzi | 09/2020 |
8429 | (467307^16384+1)/2 | 92891 | Richard Fischer | 01/2021 |
8430 | 12^86073-29 | 92889 | Lelio R Paula | 06/2016 |
8431 | 2^308566-117 | 92888 | Lelio R Paula | 11/2012 |
8432 | 11594^22851-22851^11594 | 92872 | Norbert Schneider | 03/2022 |
8433 | (17*10^92869-11)/3 | 92870 | Robert Price | 07/2015 |
8434 | 2^308459+75431 | 92856 | Enzo Doria | 04/2016 |
8435 | 2^308426-1866377 | 92846 | Enzo Doria | 03/2016 |
8436 | 24810^5521+5521^24810 | 92840 | Hans Havermann | 09/2020 |
8437 | (96^46831-1)/95 | 92830 | Paul Bourdelais | 09/2017 |
8438 | 2^308349-112111 | 92823 | Enzo Doria | 06/2014 |
8439 | 9*10^92821+19 | 92822 | Robert Price | 05/2016 |
8440 | 2^308337-75 | 92819 | Lelio R Paula | 11/2012 |
8441 | 4*3^194508+7 | 92805 | Lelio R Paula | 08/2014 |
8442 | 22*25^66379-113 | 92796 | Curtis Bright | 05/2015 |
8443 | 24538^6047+6047^24538 | 92792 | Hans Havermann | 09/2020 |
8444 | 11^89101-1080 | 92790 | Enzo Doria | 12/2012 |
8445 | 2^308236-1867859 | 92789 | Enzo Doria | 04/2016 |
8446 | 12^85963+25 | 92770 | Lelio R Paula | 06/2016 |
8447 | 23692^8221+8221^23692 | 92753 | Hans Havermann | 08/2020 |
8448 | 7^109750-66 | 92750 | Lelio R Paula | 04/2016 |
8449 | 9*10^92742+11 | 92743 | Robert Kinser | 01/2017 |
8450 | (2^308054+778)/2 | 92734 | Stefano Morozzi | 09/2020 |
8451 | (21*2^308046-1)/8857 | 92729 | Lelio R Paula | 02/2014 |
8452 | 2^307997-110655 | 92717 | Enzo Doria | 05/2014 |
8453 | L(443609) | 92709 | Renaud Lifchitz | 11/2003 |
8454 | 10^92691+9 | 92692 | Robert Price | 03/2011 |
8455 | 10^92691-252291 | 92691 | Patrick De Geest | 08/2013 |
8456 | (2^307883+3506)/2 | 92682 | Stefano Morozzi | 09/2020 |
8457 | 2^307879-2521 | 92681 | Yves Minet | 07/2014 |
8458 | 21269^21412+21412^21269 | 92666 | Norbert Schneider | 04/2020 |
8459 | (2^307780-7)/19959669 | 92644 | Lelio R Paula | 06/2013 |
8460 | 14^80831+47 | 92643 | Lelio R Paula | 06/2016 |
8461 | 15^78744+92 | 92611 | Lelio R Paula | 07/2016 |
8462 | 65^51083-64 | 92610 | Norbert Schneider | 11/2014 |
8463 | (2^307639+506)/2 | 92609 | Stefano Morozzi | 09/2020 |
8464 | 131^43737-130 | 92604 | Norbert Schneider | 02/2019 |
8465 | 6^118996-16943 | 92597 | Enzo Doria | 02/2013 |
8466 | (16*10^92591+41)/3 | 92592 | Robert Price | 10/2017 |
8467 | (2^307566+3094)/2 | 92587 | Stefano Morozzi | 09/2020 |
8468 | (2^307528+1578)/2 | 92575 | Stefano Morozzi | 09/2020 |
8469 | 9*10^92573+83 | 92574 | Robert Price | 11/2020 |
8470 | (3^194014-7)/2 | 92568 | Lelio R Paula | 04/2013 |
8471 | (2^307498+570)/2 | 92566 | Stefano Morozzi | 09/2020 |
8472 | (63^51439+1)/64 | 92555 | Paul Bourdelais | 08/2017 |
8473 | (26*10^92546-77)/3 | 92547 | Robert Price | 02/2017 |
8474 | 3^193954+340 | 92540 | Lelio R Paula | 04/2016 |
8475 | 2^307407-307407^2 | 92539 | Henri Lifchitz | 05/2005 |
8476 | 24299^6430+6430^24299 | 92536 | Hans Havermann | 08/2020 |
8478 | (17*10^92513-11)/3 | 92514 | Robert Price | 07/2015 |
8477 | 2^307322-399071 | 92514 | Harrison Mesh | 06/2022 |
8479 | 2^307318-4211 | 92512 | Yves Minet | 06/2011 |
8480 | 2^307280+193 | 92501 | Lelio R Paula | 11/2012 |
8481 | 20^71075-33 | 92471 | Lelio R Paula | 05/2016 |
8482 | 2^307126-78795 | 92455 | Lelio R Paula | 02/2016 |
8483 | 2^307125-49789 | 92454 | Henri Lifchitz | 08/2005 |
8484 | 2^307099+23 | 92447 | Lelio R Paula | 11/2012 |
8485 | 129^43798+128 | 92440 | Norbert Schneider | 01/2019 |
8486 | 21826^17181+17181^21826 | 92435 | Hans Havermann | 09/2020 |
8487 | Phi(277314,10)/3327769 | 92430 | Makoto Kamada | 05/2019 |
8488 | (2^307022+1038)/2 | 92423 | Stefano Morozzi | 09/2020 |
8489 | 7*10^92414-69 | 92415 | Robert Price | 01/2017 |
8490 | (9787*2^306975-1)/2129155 | 92407 | Lelio R Paula | 04/2015 |
8491 | 22280^14037+14037^22280 | 92402 | Hans Havermann | 09/2020 |
8492 | 2^306915+1873373 | 92391 | Enzo Doria | 06/2016 |
8493 | 2^306908+12915 | 92389 | Lelio R Paula | 02/2016 |
8494 | 21620^18759+18759^21620 | 92387 | Hans Havermann | 09/2020 |
8495 | (2^306897+1142)/2 | 92385 | Stefano Morozzi | 09/2020 |
8496 | 19^72244-80 | 92383 | Lelio R Paula | 06/2016 |
8498 | 2^306849+75 | 92371 | Lelio R Paula | 11/2012 |
8497 | (397*10^92369+53)/9 | 92371 | Robert Price | 11/2017 |
8499 | 2^306813-869688105 | 92360 | Lelio R Paula | 08/2016 |
8500 | (2^306797+2694)/2 | 92355 | Stefano Morozzi | 09/2020 |