素因数分解結果（追加：Wolstenholme 他（２））

    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

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