求证：y＝p

太阳

1761位
109889010998901109890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989022087801209779121989000989001099001098890109990109879122076902307779230978021087910098911208789109899120888021986812218680131877912098901009890010989900000999990999979011199889110988001109000009988900109999010997801219778032997801999891009991009881009981109980021067802409778210988001109098999010008900109900108799121886813217681140768822090910978812189691310779210779122976904296571441857012088011097902107781229869030978921078822297480332778012987892000002088881209879110990100878102197881218969120188908100020088809209799110799110888012007802219669022000000890001987992108991008981008981130068810200898091020088810199800110899000890012087791210889001000090899101987993207870120179909079021287700309788211997999999110986912317679120899110888011996903217770030089908991029967822298700209789110899120876823208690110899999800211788012996793119880908982020968812289790200899091000099889112087791209989099899110998002007891020090799102007871031079910079910089911189790199989100000219867022197810209789110898110898011907891020000808892020869922078802200779100990110879012078891200000188790309778332974803217881118880129869032077811198801109791209659244065703398680320878111878113086992207880109091217868141076822307689220897910999010998910009899900911099790111778014197771119980011080010079010090010089000197800329778022097800109900010998890009999911099790110889002109880000090000099998990109989010098901208779230976922207690219789120899010887912108901008791119780131877813198681220879010990010988901099901099890099890220878012097791219890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989011098900109889011

太阳

合数和素数，怎么判断是合数？
\(\frac{\left( 10^{3795}+1\right)\times\left( 10^{253}+1\right)\times\left( 10^{115}+1\right)\times\left( 10^{69}+1\right)\times\left( 10^{55}+1\right)\times\left( 10^{33}+1\right)\times109889011}{\left( 10^{1265}+1\right)\times\left( 10^{759}+1\right)\times\left( 10^{345}+1\right)\times\left( 10^{165}+1\right)\times\left( 10^{23}+1\right)\times100000000001}\)

yangchuanju

太阳 发表于 2022-3-15 07:30
合数和素数，怎么判断是合数？
\(\frac{\left( 10^{3795}+1\right)\times\left( 10^{253}+1\right)\times\ ...

将大分式中的分子、分母中相同因子逐一相约（相除）后净剩一个1761位的未完全分解的复合因子1098890109...11<1761>，先生的这些分解式不知从哪里得来的，1761位大因子是什么颜色的数字，有没有特殊标记；
在我所下载复制的这个大因子是黄色的并放在方括号内（表示未完全分解的合数），无需另行判定其素合性。

yangchuanju

太阳 发表于 2022-3-15 07:00
11881位
(10^12650-10^6325+1)

11881位大合数来自(10^12650-10^6325+1)，
(10^12650-10^6325+1)=(10^18975+1)/(10^6325+1)，
(10^18975+1)  18976位，(10^6325+1)  6326位，相除后12650位，
12650-11881=769，既是再除以10位的9999900001还差759位，
不知太阳先生的11881位大数是如何构成的。

yangchuanju

(10^18975+1)/(10^6325+1)=7×13×211×241×2161×31051×176089×2501251×4015801×11313931×27581401×599144041×1122219451<10>×4124507971<10>×127688571001<12>×183411838171<12>×194656286401<12>×272168891611<12>×15334196149801<14>×143574021480139<15>×776471438412978001<18>×11153763122343284851<20>×124448981774280741575251<24>×19835636682880495867311241<26>×24649445347649059192745899<26>×697584995226846644351713949856887251<36>×781096971351185956306280104220477251<36>×2832165561296805799533565929552471103<37>×10000099999999989999899999000000000100001<41>×1112314101311286003379752617807870409611285281<46>×7386441591567953769105190285927615814299177561<46>×1863670848562011266258031940578523676121582933240001<52>×43810325613739522298108991741277641747137547565878057174130075470872858289958280148759277150273598401<101>×1677390810...<215>×[8634288537...<369>]×[9955571557...<867>]×[1098890109...<1761>][8911002202...<8791>]——12650位

9999900001=7*13*211*241*2161
(10^18975+1)/(10^6325+1)/9999900001=31051×176089×2501251×4015801×11313931×27581401×599144041×1122219451<10>×4124507971<10>×127688571001<12>×183411838171<12>×194656286401<12>×272168891611<12>×15334196149801<14>×143574021480139<15>×776471438412978001<18>×11153763122343284851<20>×124448981774280741575251<24>×19835636682880495867311241<26>×24649445347649059192745899<26>×697584995226846644351713949856887251<36>×781096971351185956306280104220477251<36>×2832165561296805799533565929552471103<37>×10000099999999989999899999000000000100001<41>×1112314101311286003379752617807870409611285281<46>×7386441591567953769105190285927615814299177561<46>×1863670848562011266258031940578523676121582933240001<52>×43810325613739522298108991741277641747137547565878057174130075470872858289958280148759277150273598401<101>×1677390810...<215>×[8634288537...<369>]×[9955571557...<867>]×[1098890109...<1761>][8911002202...<8791>]——12640位

太阳

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910188699311897991019089799111898001998891009991099879122087891109989999091008980120067921298798200909209787122996902117 880020000008890130876823208791008891119880030967822299699100910089700321777012997902108891008791231767913098891000001997 882119870032068801190910179790311877102097991009089999001108879122087900109001008890019979921089799092008979921289689200 899200899100888102098000108999997801329778021897902108891018891019880921089799991911079791300779211977992208990098891209 879211087999998112096902216670251967821188811198701309679221888011988902087903407559331977112096901107892217880128978920 100018977831208600220689021898901899902098791110790000891101878003087883318859120080020078911287700309800199789209888132 985703208791109791210769112088002097892097992207770231067911287910297800318788230886922107801097811218689131787922997803 008793108771131967812189801188801199791210868112088011087912296590330789011888001998012996793108881019981020858923278791 299779201089099988111087901198999198899219887121995912127779920001018879930978822188811098701210680232855814298791198890 209789221877013096902107892117770251966722198701210799999791111879012088900989912189780211878103087981109189997991119079 020088920088911198799219778033097799999910999801200789100899101899101886993118979910190897991118980019988910099910998791 220878911099899990910089801200679212987982009092097871229969021178800200000088901308768232087910088911198800309678222996 991009100897003217770129979021088910087912317679130988910000019978821198700320688011909101797903118771020979910090899990 011088791220879001090010088900199799210897990920089799212896892008992008991008881020980001089999978013297780218979021088 910188910198809210897999919110797913007792119779922089900988912098792110879999981120969022166702519678211888111987013096 792218880119889020879034075593429660131967912188812188691300779211989090878213185892317769231077910208900098900108999010 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878004087872220079998981120069021177811188901298789319767144185701208801109790210778122986903097892107882229748033277801 298789200000208888120987911099010087810219788121896912018890810002008880920979911079911088801200780221966902200000089000 198799210899100898100898113006881020089809102008881019980011089900089001208779121088900100009089910198799320787012017990 907902128770030978821199799999911098691231767912089911088801199690321777003008990899102996782229870020978911089912087682 320869011089999980021178801299679311988090898202096881228979020089909100009988911208779120998909989911099800200789102009 079910200787103107991007991008991118979019998910000021986702219781020978911089811089801190789102000080889202086992207880 220077910099011087901207889120000018879030977833297480321788111888012986903207781119880110979120965924406570339868032087 811187811308699220788010909121786814107682230768922089791099901099891000989990091109979011177801419777111998001108001007 901009001008900019780032977802209780010990001099889000999991109979011088900210988000009000009999899010998901009890120877 923097692220769021978912089901088791210890100879111978013187781319868122087901099001098890109990109989009989022087801209 7791219890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989011098900109889011=910018200100098361964518899984366143745719841905565988650229820714175067860251995724832958758014202457133068365971180155
377253947166366187127954488113086509243874706012763657126285797136243600813284599954661903812743746055358988902535027369
111089192989829987190979001152814537120990574374636417026798454390145966028917593482224963811416527693135187590743297682
174391892973473263616272374370298193562177702338086601540387734623298360431275386628929660550993544798629714636383475090
120735508252099880091690027990291891110873729535298889953551547347308309165549996392408097251640788492629847358219697470
251914681220793549821691663662649361773551981169654771240755202328856950237526681152967861472326118022955889656509148017
5473506634799989154601638010010018200091

太阳

(10^12650-10^6325+1)/9999900001含有因子109889010998901109890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989022087
801209779121989000989001099001098890109990109879122076902307779230978021087910098911208789109899120888021986812218680131
877912098901009890010989900000999990999979011199889110988001109000009988900109999010997801219778032997801999891009991009
881009981109980021067802409778210988001109098999010008900109900108799121886813217681140768822090910978812189691310779210
779122976904296571441857012088011097902107781229869030978921078822297480332778012987892000002088881209879110990100878102
197881218969120188908100020088809209799110799110888012007802219669022000000890001987992108991008981008981130068810200898
091020088810199800110899000890012087791210889001000090899101987993207870120179909079021287700309788211997999999110986912
317679120899110888011996903217770030089908991029967822298700209789110899120876823208690110899999800211788012996793119880
908982020968812289790200899091000099889112087791209989099899110998002007891020090799102007871031079910079910089911189790
199989100000219867022197810209789110898110898011907891020000808892020869922078802200779100990110879012078891200000188790
309778332974803217881118880129869032077811198801109791209659244066802288790309889210778211987102107989998102216879139977
931097810109810009790120888920999910889812199691211668114087991997992108871229969019980021079011186701429679120889020878
923088791199790210790200779222976893208871208970219968121176911307789219890218878131086821207700219699120689122886803218
781019881119880020878913188791198891208770340966921098811199800198700320689111898001897914107672120870022079891090001188
891209880100080110078011287800408787222007999898112006902117781118890129878931976714418570120880110979021077812298690309
789210788222974803327780129878920000020888812098791109901008781021978812189691201889081000200888092097991107991108880120
078022196690220000008900019879921089910089810089811300688102008980910200888101998001108990008900120877912108890010000908
991019879932078701201799090790212877003097882119979999991109869123176791208991108880119969032177700300899089910299678222
987002097891108991208768232086901108999998002117880129967931198809089820209688122897902008990910000998891120877912099890
998991109980020078910200907991020078710310799100799100899111897901999891000002198670221978102097891108981108980119078910
200008088920208699220788022007791009901108790120788912000001887903097783329748032178811188801298690320778111988011097912
096592440657033986803208781118781130869922078801090912178681410768223076892208979109990109989100098999009110997901117780
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yangchuanju

太阳 发表于 2022-3-15 10:07
(10^12650-10^6325+1)/9999900001含有因子1098890109989011098900098900109900109889010999010998901108790 ...

已查清760位大数等于
31051×176089×11313931×599144041×4124507971<10>×127688571001<12>×183411838171<12>×194656286401<12>×272168891611<12>×143574021480139<15>×776471438412978001<18>×19835636682880495867311241<26>×24649445347649059192745899<26>×2832165561296805799533565929552471103<37>×1112314101311286003379752617807870409611285281<46>×7386441591567953769105190285927615814299177561<46>×43810325613739522298108991741277641747137547565878057174130075470872858289958280148759277150273598401<101>×[8634288537...<369>]
因子位数和等于5+6+8+9+10+12+12+12+12+15+18+26+26+37+46+46+101+369=770（积为760位数）

剩余因子积等于
2501251×4015801×27581401×1122219451<10>×15334196149801<14>×11153763122343284851<20>×124448981774280741575251<24>×697584995226846644351713949856887251<36>×781096971351185956306280104220477251<36>×10000099999999989999899999000000000100001<41>×1863670848562011266258031940578523676121582933240001<52>×1677390810...<215>×[9955571557...<867>]×[1098890109...<1761>][8911002202...<8791>]
剩余因子位数和等于7+7+8+10+14+20+24+36+36+41+52+215+867+1761+8791=11889（积应该是11881位）

11881位大合数不但含有素因子2501251，还含有更多的其它素因子。

yangchuanju

两个日本分解网站的分解结果不完全相同，《factordb.com》的分解能力差一些，如对( 10^246362-1)/9的分解：
《Factorization of 11...11 (Repunit)》的分解结果是：
R246362 =( 10^246362-1)/9 = 1111111111...<246362> = 11 × 797 × 2477 × 13619 × 29453 × 221603 × 2253079 × 634201866251<12> × 3545105703277<13> × 10765760180213<14> × 12277415800813<14> × 592119999726203887006567<24> × 168551725030519992504486587<27> × 180724074168604579796294647309331<33> × 314293140427626491056008875724151058487007<42> × 34156671999167558280316457548021397439888515117317065783422183070342102287109965498971<86> × 21008411332176290590182609174580040662121810909787962004744874106431821734447340411991127851251006881567865355288584910608794705098365152866127635471781892711877278779909777694559002649<185> × [1777562374...<579>] × [1200943696...<584>] × [9000000000...<122364>] × 1099999999...<122365> (49.86%)——共21个因子，最后一个大因子是122365位的素数。
而《factordb.com》的分解结果是：
(10^246362-1)/9<246362> = 11×797×2477×29453×221603×2253079×168551725030519992504486587<27>× 2100841133...49<185> ×1200943696...63<584>×8181818181...81<245544>——仅给出10个因子。

太阳

1761位数，不采用试除法，准确判断是否为素数，有这样方法吗？
109889010998901109890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989022087801209779121989000989001099001098890109990109879122076902307779230978021087910098911208789109899120888021986812218680131877912098901009890010989900000999990999979011199889110988001109000009988900109999010997801219778032997801999891009991009881009981109980021067802409778210988001109098999010008900109900108799121886813217681140768822090910978812189691310779210779122976904296571441857012088011097902107781229869030978921078822297480332778012987892000002088881209879110990100878102197881218969120188908100020088809209799110799110888012007802219669022000000890001987992108991008981008981130068810200898091020088810199800110899000890012087791210889001000090899101987993207870120179909079021287700309788211997999999110986912317679120899110888011996903217770030089908991029967822298700209789110899120876823208690110899999800211788012996793119880908982020968812289790200899091000099889112087791209989099899110998002007891020090799102007871031079910079910089911189790199989100000219867022197810209789110898110898011907891020000808892020869922078802200779100990110879012078891200000188790309778332974803217881118880129869032077811198801109791209659244065703398680320878111878113086992207880109091217868141076822307689220897910999010998910009899900911099790111778014197771119980011080010079010090010089000197800329778022097800109900010998890009999911099790110889002109880000090000099998990109989010098901208779230976922207690219789120899010887912108901008791119780131877813198681220879010990010988901099901099890099890220878012097791219890009890010990010988901099901099890110879012088891209879110989001099001098890109990109989009989011098900109889011