素因数分解結果（追加：Wolstenholme 他）

    Date: 24 Jan 1998 01:23:06 -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９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
    http://www.loria.fr/~zimmerma/records/ecmnet.html
から手に入れることが出来ます。
楕円曲線を by^2 = x^3 + ax^2 + x mod p , 初期点を (x1:y1:z1) ,
楕円曲線番号を sigma とすると、
GMP-ECM では
        u=sigma^2-5, v=4*sigma
        x1=u^3, z1=v^3
        a=(v-u)^3*(3*u+v)/(4*u^3*v)-2
として計算しています。
UBASIC 添付の ECMX.UB では
        u=2*(sigma+1)/(3*(sigma+1)^2-1)
        x1=3*u^2+1, z1=4*u
        a=(-3*u^4-6*u+1)/(4*u^3)
として計算しています。
それ以外のパラメータとしては
        u=6*sigma/(sigma^2+6)
        x1=3*u, z1=4
        a=(-3*u^4-6*u+1)/(4*u^3)
が考えられます。

//Wolstenholme 182
 11279 * 44351 * 92551 *
 1154457118611904364667817129 *
 8419178767409756111696965536946504564873

//Wolstenholme 206
 6269 * 15597409 * 19277773297 *
 3324545201450680201600620923 *
 11776316386359286374577574969838282809183

//Wolstenholme 207
 313 * 117659 *
 17335258409108225219509478893 *
 347076934653761151662395805855749192111390894208149829

//Wolstenholme 228
 229 * 229 * 3761 * 225683 * 542837 *
 35204722042447321019 *
 8482831291695119488457801345827662001110972004187788202903

//Wolstenholme2 109
 1783 * 8023901 * 40151339 *
 1262562681454625037013437811981 *
 32279585330825825955838409602079534438584442841953

//!n 64
 2 * 157 *
 344230002001575753172255981669 *
 18638426270934935876693859505549209626410979849729246429

//!n 65
 2 * 991 * 7309321 *
 49317480549324296045142743 *
 180416599680773026188907102667108463982217781076933309

//!n 68
 2 * 1187 * 41993491453 * 106267395181967 *
 348468071084720079543731 *
 10028954143664092465446976705586446159068631

//Bernoulli 100
 263 * 379 * 28717943 *
 65677171692755556482181133 *
 503175397608024323584539371320514986481668897

GMP-ECM のログを次に示します。

Input number is 9719580860902391600489207457040970930855967836701267910870881109617
Using B1=1000000 and sigma=2116025677
A=2533431615868089791147486634095262024756962719673445965732868962953
starting point: x=4515817828036936847730847312434837920306774593123008774509088962858
Step 1 took 1002920ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=1865479715730500356814531086297154239710750395686511518057945937585
initialization of Step 2 took 18020ms
last interval is 3104609..3118753
********** Factor found during step 2: 1154457118611904364667817129
Found probable prime factor: 1154457118611904364667817129

Input number is 39150896133035780021503664127219904828769534428206405667369526335909
Using B1=1000000 and sigma=287234601
A=37804021295928920813792462795364231988480242431555961038682344632577
starting point: x=11123602101065984117934560910387594493652952725474071759615992440471
last prime is 652601
********** Factor found during step 1: 3324545201450680201600620923
Found probable prime factor: 3324545201450680201600620923

Input number is 6016668350064118985272651584812794193532727927128417087507000673299785837457059297
Using B1=1000000 and sigma=7647764
A=5817280236801966282102575757217927884075498359708153338292590296623311971839582689
starting point: x=3724987398687692038463636963403942983789557433620127219717111496786131761145548360
Step 1 took 1520845ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=1990977593273235131909072339686962253430403969644580986495996503723845409284340202
initialization of Step 2 took 24656ms
last interval is 61491041..61505185
********** Factor found during step 2: 17335258409108225219509478893
Found probable prime factor: 17335258409108225219509478893

Input number is 298635717757101053336540270713220059620352465768599282782960566174204748718157
Using B1=1000000 and sigma=219501388
A=195286212475026603870955831166816679286494553705073226841927165441913263668572
starting point: x=291768112655930256310830213698296287046709837365609940259531126786985662173356
Step 1 took 1021849ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=161712573781396153052141535598179121377097103473641099999228880646837718250066
initialization of Step 2 took 21277ms
last interval is 5452513..5466657
********** Factor found during step 2: 35204722042447321019
Found probable prime factor: 35204722042447321019

Input number is 40754999811530834438647845394206560640031153777962475321033979911962805712838893
Using B1=1000000 and sigma=1492966393
A=38262212704570445912128121675649230239011367116619486254804224737473891782199784
starting point: x=29224451123864788771788745202989268616216092409276242698943778544126340954706507
Step 1 took 1472377ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=33765438557802612998707437798587217096644730920285042603749262662730641642789664
initialization of Step 2 took 23881ms
last interval is 25395553..25409697
********** Factor found during step 2: 1262562681454625037013437811981
Found probable prime factor: 1262562681454625037013437811981

Input number is 6415905512550155078026483963060607761862786437523450235628801350503525484487007710001
Using B1=1000000 and sigma=642151313
A=3999704367722667544741132077089785849936013135796229193332993431046318974702091095530
starting point: x=4897479090887340124404872619761428518830261663127303860706805489295058388868454462236
Step 1 took 1533578ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=1862107859580946838808303121518764497646819670954661980823770420057764432763081064197
initialization of Step 2 took 25435ms
last interval is 3797665..3811809
********** Factor found during step 2: 344230002001575753172255981669
Found probable prime factor: 344230002001575753172255981669

Input number is 8897692145531751718109869796977044274860927893155067979383845666561727596326587
Using B1=1000000 and sigma=1662412343
A=786782056492522796420854894029935487966992275144504510214512873171853764547744
starting point: x=7371121214889539073131246250568569643792921921328302659131159301197589899278562
Step 1 took 1466150ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=133513899556948406902654168776819840984767628589088637393917347543451786919740
initialization of Step 2 took 23713ms
last interval is 17687073..17701217
********** Factor found during step 2: 49317480549324296045142743
Found probable prime factor: 49317480549324296045142743

Input number is 3494770305439736966191594654002237113934398552211565941258394802261
Using B1=1000000 and sigma=1619076789
A=3206041000893387074364162776066323899008891476277581788280069015421
starting point: x=3003211682511238277694132065740218351390122289987843430515153661084
Step 1 took 673762ms for 13872253 multiplications
start step 2 with B1=1000000, B2=100000000, D=7072
x=1212330700485115858090327879964062098361265494829526248432560307515
initialization of Step 2 took 12016ms
last interval is 26074465..26088609
********** Factor found during step 2: 348468071084720079543731
Found probable prime factor: 348468071084720079543731

Input number is 33047136980272757050310768909144101995804885526525029874352524486320301
Using B1=1000000 and sigma=693616517
A=25671342271804769421956662843609525130434111267528115367289567175657600
starting point: x=25010629758283541978561785145451743398067857426741744410415528379356036
last prime is 403993
********** Factor found during step 1: 65677171692755556482181133
Found probable prime factor: 65677171692755556482181133

　　　　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

この章の目次へ