Date: 30 Jan 1998 13:35:58 -0000 From: yamasaki@kusm.kyoto-u.ac.jp To: kc2h-msm@asahi-net.or.jp 三島 久典 様 素因数分解表で未分解の数を10個楕円曲線法で分解したので報告します。 いずれも残っていた合成数を二つの素数に分解しました。 使用プログラムは GMP-ECM です。 Wolstenholme 215 については大きい方の素因数が先に見つかりました。 //ΠPn+NextPrime 52 312589 * 4143330479 * 16400068545723686299224731 * 50017710989066719239301918797473310558572498351907079411 //Wolstenholme 187 717879787 * 2536425772544001721878050821 * 248286738633129588474724539063429455570481409 //Wolstenholme 215 251512486544718808969163 * 1763873102234883519255020898375271 * 106059863309948501166062480330621093 //Wolstenholme 221 1045233538553 * 343062552058014904688690797231807 * 395468892414751505262331550987097916404356538677 //Wolstenholme 223 30928577 * 784496413685081 * 26954923932485962435533886167347 * 48424751886108045240202553679269154700919 //Wolstenholme 225 439 * 1289 * 399347879251 * 2707227807593783 * 49124495525137459572020694191 * 1055383951243590910064643712605797 //Wolstenholme2 92 137 * 1103 * 220462248363066240176368147 * 25341393339654910669557256561688625560378379577 //!n 66 2 * 2877288626301631911991350133 * 1455632925044128835833353679033461512002967382137914395567491129 //!n 67 2 * 245820739 * 19999589196755376336101 * 1916994526395918672098449381 * 29323505547877103841335451286540423 //Euler 70 353 * 2586437056036336027701234101159 * 312210239910371909857727050224078527206101218811162523 GMP-ECM のログを次に示します。 Input number is 820293888720891073264191579947232719808170760505861562421772017012968119752113441 Using B1=1000000 and sigma=16446985 A=75401729813132369298645614871167100266089310869554066767720315392939822524448662 starting point: x=774915607202133885527292877825985340547537754671361413381038550120044013507107215 Step 1 took 1247997ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=142496042825945340987583038709653930539969051195236868102130055932862436583258998 initialization of Step 2 took 20237ms last interval is 10318049..10332193 ********** Factor found during step 2: 16400068545723686299224731 Found probable prime factor: 16400068545723686299224731 Input number is 629760882849966354558953549927806981023959379500398475833749113537686789 Using B1=1000000 and sigma=1856125005 A=587824234953291894089190194527144330975653989742526574181069480022702528 starting point: x=340392145506402020969688371863970410357053300409932544441253384726580505 Step 1 took 1239690ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=114750876473395176340818561568740862392817207829673634285598905562915561 initialization of Step 2 took 20580ms last interval is 1251745..1265889 ********** Factor found during step 2: 2536425772544001721878050821 Found probable prime factor: 2536425772544001721878050821 Input number is 187076140119126564157980417592108481715776036618578028090754022191203 Using B1=1000000 and sigma=325138945 A=88183924710685305171351582653379928948865220758532029299896013724921 starting point: x=60515087293244123162646290059879399532575196768844420934898983053196 Step 1 took 1162990ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=142468732336356663651255486429575841861834636859727016176252427656618 initialization of Step 2 took 19160ms last interval is 42000609..42014753 ********** Factor found during step 2: 106059863309948501166062480330621093 Found probable prime factor: 106059863309948501166062480330621093 Input number is 135670567491361183941920814910896516282299921043112984082340327609569131730099339 Using B1=1000000 and sigma=617758585 A=106628554944426824797573374482368238296870557539101471403200873545360053373982483 starting point: x=74838471072116975084536806027179995440682420581383670583724080750873802986138198 Step 1 took 1483241ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=104731245272331046737959867744012732635106693100233648246700038829230549464986736 initialization of Step 2 took 25232ms last interval is 3868385..3882529 ********** Factor found during step 2: 343062552058014904688690797231807 Found probable prime factor: 343062552058014904688690797231807 Input number is 1305285503539548497349571716224069416922041530318145336469075072968691893 Using B1=1000000 and sigma=1221926294 A=908404430313601800311029634607634478507641613938003437565982125378260114 starting point: x=8637313722922084248074826615918556316038946463733890725059108555597090 Step 1 took 1246147ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=282366676744924585792736228956393187519283349848055209967177726974177428 initialization of Step 2 took 21555ms last interval is 5268641..5282785 ********** Factor found during step 2: 26954923932485962435533886167347 Found probable prime factor: 26954923932485962435533886167347 Input number is 51845204190167672472656720711928806790444450419674497470825227 Using B1=1000000 and sigma=1409716990 A=41773089993934934012549860017776862818026352004607146350059022 starting point: x=7255207511116378631136330064839968759083388253311256247527895 Step 1 took 967352ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=15575624791540362839600866971653174657581047449122522754489428 initialization of Step 2 took 16759ms last interval is 28705249..28719393 ********** Factor found during step 2: 49124495525137459572020694191 Found probable prime factor: 49124495525137459572020694191 Input number is 5586820552313153551108743846347738885879237275305794427374951904858133819 Using B1=1000000 and sigma=1338595355 A=2236625713477239544060917238249353581269805957929636426769847922524974428 starting point: x=5193280270445339193936948248065375210040970628295449186356264285776711452 Step 1 took 1250530ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=1319309191751923836227849916346365834507912160388632120308087250467790681 initialization of Step 2 took 20860ms last interval is 2001377..2015521 ********** Factor found during step 2: 220462248363066240176368147 Found probable prime factor: 220462248363066240176368147 Input number is 4188276059299647789760552006042529381913174491230682524247456236194114293104264460210470157 Using B1=1000000 and sigma=2044072316 A=1712824386589410120320618518061684357637845302515666052250182528132829803896925998962181572 starting point: x=3096067446634139066350332297831472123247074819910045924828969442692065732422125268582030648 Step 1 took 1325572ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=880548686177656359401900634453146748767029806864166405781272174520575819691858064786894907 initialization of Step 2 took 26161ms last interval is 1605345..1619489 ********** Factor found during step 2: 2877288626301631911991350133 Found probable prime factor: 2877288626301631911991350133 Input number is 56212999630020762362360648988043845685864471140993814275828163 Using B1=1000000 and sigma=870357984 A=51464136564316806990116008495947664177771856628890484685413801 starting point: x=4383482295565025003535275719266332928855103908233607266029372 Step 1 took 812970ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=15868539013060321643449844856989458404910198035378760445935627 initialization of Step 2 took 14020ms last interval is 92409825..92423969 ********** Factor found during step 2: 1916994526395918672098449381 Found probable prime factor: 1916994526395918672098449381 Input number is 807512133778180506323509148885947137405822214669278299941510513050936055371771664157 Using B1=1000000 and sigma=1441778847 A=40625472412661531377969149073881344921913968938586948262682302772773401678999415004 starting point: x=42265762030026479671095926772777618289183164142749297488695828030939563076683447599 Step 1 took 1539144ms for 13872253 multiplications start step 2 with B1=1000000, B2=100000000, D=7072 x=705806860301947731188096424954491439270419034637105575589067327660670633968420719656 initialization of Step 2 took 25383ms last interval is 6032417..6046561 ********** Factor found during step 2: 2586437056036336027701234101159 Found probable prime factor: 2586437056036336027701234101159 yamasaki@kusm.kyoto-u.ac.jp URL: http://www.kusm.kyoto-u.ac.jp 京都大学大学院理学研究科 数学教室助手 山崎愛一