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Integral Points on Conic Curves: x^2-Dy^2=±1(D \in [10000,19999])


[2016.10.10]x2-Dy2=±1 (D \in [10000,19999])の整点


■10000≦D≦19999かつ平方数でない整数Dについて、Pell方程式
     x2-Dy2 = ±1
の基本解を連分数を使って求める。

ここでは、文献[3],[4],[5]で紹介されているアルゴリズムを採用した。 √(D)の連分数を計算するために、多倍長精度の小数点演算を使う代わりに、整数演算で計算する。

■10000≦D≦19999について、x2-Dy2=1の基本解が最大のもの(D=17341)と2番目に大きいもの(D=18181)は、それぞれ、以下のようになる。

■10000≦D≦19999についての結果は、以下の通りである。
D fundamental solution [x,y] of x2-Dy2=-1 fundamental solution [x,y] of x2-Dy2=1
10001 [100, 1] [20001, 200]
10002 - [10001, 100]
10003 - [496414364019185554500601, 4963399186143800653020]
10004 - [5001, 50]
10005 - [4001, 40]
10006 - [15269651952212573820472108873564438500403664494778447320172213408587685, 152650731169997629164968862774767499011918640790944175264392200259198]
10007 - [537397546963366989364180781208, 5372095565111620250669012603]
10008 - [2501, 25]
10009 [5706343721431274132565526516441469915850743462805740526597162711889032, 57037775987595819178157681487331302741769138812605212984153607894765] [65124717334236245435952050187459435151945908477300755986826019749838645384323269086408783658507658517850400717088562998184654080723763794049, 650954309782441788031297341091747625782745331185606106908835855769544769212365549697731047424577358727886032883106619973973840883627434960]
10010 - [2001, 20]
10011 - [9140126426, 91351035]
10012 - [21515302964149300390546966065746496439421258296548378751, 215024053890283338664404673381644351570611185007760600]
10013 - [833626246305151, 8330849169840]
10014 - [727457010402928464499154333319677544744245, 7269483245535178608366681651835253119154]
10015 - [2757974827131643727248851124, 27559086701475664663993815]
10016 - [3130001, 31275]
10017 - [1386533457263359999, 13853564043547200]
10018 - [1784223875651310078415699902083, 17826202387485980222653020654]
10019 - [149713056477819301371324290, 1495710314278475869079889]
10020 - [1001, 10]
10021 - [40247399914231653160699096813143180551466878482149170429530062960850054181357127287370714551, 402052065871944527600045574298566015678262572343849924238474711251214047210952710554430860]
10022 - [744637800930488025067117637, 7438200483940512003126462]
10023 - [11671399, 116580]
10024 - [10240765556688399071935, 102284987139265722276]
10025 - [801, 8]
10026 - [74707907636890165020962560915699, 746109763321062134029542944190]
10027 - [8604793, 85932]
10028 - [82852996681927, 827372455074]
10029 - [51723618629539255, 516487821126716]
10030 - [24679965457214261, 246430285962198]
10031 - [75920302765304880, 758028991889273]
10032 - [783751, 7825]
10033 - [113771431256617279714535833457372680628392447596872765579590658989567, 1135841717351159307735204306226766006650816464350111832128863270944]
10034 [17783455, 177533] [632502543474051, 6314300233030]
10035 - [41589498271751, 415169071460]
10036 - [740669850038599793795048200429418021449, 7393402331989419806452337813456405310]
10037 [124399535870776507820198782018, 1241700334027587144334156585] [30950489049729222163529821505503667670605888551621360304649, 308933890487239997101727281146720846143157191224588577060]
10038 - [850122043, 8485114]
10039 - [309749798412788867876115158877080, 3091475473189322045436091358421]
10040 - [501, 5]
10041 - [859982773212648672075917568450063851156037828908375, 8582252111880255879524155234199074255135495679492]
10042 [342376048075164368979, 3416593152939246101] [234442716591134526417074624696022515004883, 2339519323168009465116878640611302201758]
10043 - [2205226, 22005]
10044 - [394649197502177907199, 3937838239555722320]
10045 - [39689, 396]
10046 - [10257277833911495811306876893334889880471825663279015209215, 102337671756429887632638524765164986837115421179219095088]
10047 - [5301284776, 52888705]
10048 - [4361542530341638623493917601, 43511123618917346614228695]
10049 [30749240394169348949360, 306741804879971629793] [1891031569636831956689425247924658671688819201, 18864154994391272725572285549708102848564960]
10050 - [401, 4]
10051 - [6681448801, 66644760]
10052 - [9325431659989887, 93012796896472]
10053 - [87897747594260774254246835664214545379849, 876657403855487870716413818542057396740]
10054 - [2681818435567761233220399012247196259768570725, 26746067201312624458055967660563436635919366]
10055 - [634676753639588889, 6329385593412932]
10056 - [200665583259905, 2001060684952]
10057 [16111065722483884834555164, 160653445506056927724989] [519132877428390364054057664420178588872547518133793, 5176596438183132955100234265229477776280683586392]
10058 - [4370611370253, 43579914674]
10059 - [5090028108027449, 50750786440580]
10060 - [108714542430316391648568394018669588231, 1083898591494354526805546202039836466]
10061 [29538347658218, 294486657845] [1745027964755505609085871049, 17397298560264602336840420]
10062 - [4213, 42]
10063 - [13568659322127900137294075823643363954151, 135261189433572674773582100715480894540]
10064 - [123245001, 1228525]
10065 - [11262881279, 112264544]
10066 - [25885, 258]
10067 - [90602, 903]
10068 - [932070047201139239618399, 9289170800916886269180]
10069 [43824918, 436745] [3841246875413449, 38280627623820]
10070 - [460607311, 4590036]
10071 - [126905298453159868453735, 1264571695233064750012]
10072 - [640845364594942651248197, 6385507050359674236267]
10073 - [1800614961967275534207, 17940784828875974224]
10074 - [17665, 176]
10075 - [1202394930058086974, 11979111546028683]
10076 - [158961220199, 1583605890]
10077 - [91467060013173599, 911169325258920]
10078 - [26134762935629528090517290749874009639999, 260334297753443585051861820748515838200]
10079 - [261568763334583017801724808082376572488923878683520, 2605416483848458904573488106864583832312759464241]
10080 - [126001, 1255]
10081 [974829343949179912668, 9709051088076972985] [1900584499648777022421660281720864213756449, 18929335805118293944532700897928190547960]
10082 - [311363698964240484013304163, 3100949055675228873107018]
10083 - [98823482, 984159]
10084 - [157068221854053769919189294173186467900785347825382086892929936363934648449, 1564126624674642596622299283393112323759490949093588570869387151745111720]
10085 [34486200471455346458, 343405619104094497] [2378596045914813920619805807665626291529, 23685510046896077545572509299012483252]
10086 - [403988416827863757605, 4022623915540924122]
10087 - [138599, 1380]
10088 - [8478963, 84419]
10089 - [151985104364951, 1513132519540]
10090 - [58226310937313281, 579660480012288]
10091 - [6078626603228491684435396259050371007172793942895930, 60511561961786390115159255832626595587173992291917]
10092 - [72010600134783751, 716816196074550]
10093 [20578670078338146081677010909245617482, 204836415749712612438342681579141605] [846963324386179438780227279379396479583807710798328707589141100446928040649, 8430522039485287012109323292337173649086818908949879656286730979483077220]
10094 - [7088832449, 70557480]
10095 - [4005295564, 39864049]
10096 - [4794471714260877710395604276165927978529027199, 47716226324387994805902943414462097202672060]
10097 - [1131007986844765476755327, 11255621844506484822768]
10098 - [1202851, 11970]
10099 - [5548107895701060653518130, 55208470075434535994649]
10100 - [201, 2]
10101 - [26935, 268]
10102 - [80797239309417298752973956095104830434701959378301053667665399998314277, 803882991356111457018043722178323880564694015318644199627279712114642]
10103 - [5598103091045722827015624, 55694936791064879627125]
10104 - [7937097845, 78961441]
10105 - [1835337305704236281571065637649048361, 18257770066713357978783083143617828]
10106 - [3194799, 31780]
10107 - [16506645562632482, 164190374404933]
10108 - [2199950293420880308351, 21881659297350469320]
10109 - [8488019168370008728857233734382991750976271199, 84421342343580198597401748175346233051994920]
10110 - [238951774138571, 2376482833758]
10111 - [466626708408311621048474441584893473066023801588278962623778413463681466800, 4640582925691140871536698257536451306874152238862050012152123760042098497]
10112 - [7743, 77]
10113 - [3691523505515235992866777789486742302488023, 36708415134625262480703549918667456423116]
10114 [16392345, 162997] [537417949198051, 5343806115930]
10115 - [203253121, 2020944]
10116 - [10235103381188185123484801, 101762513025653726532840]
10117 - [1207, 12]
10118 - [2924101, 29070]
10119 - [17018902262638719700515498265, 169185346916776069376572036]
10120 - [1259389, 12519]
10121 - [116366307663299049, 1156686167014420]
10122 - [343341613, 3412662]
10123 - [17767774, 176595]
10124 - [249754993833237826132114536972634442241590546483968619712199, 2482207665321574142160428154853017678620753642386464306870]
10125 - [1730726404001, 17200097288]
10126 - [374062875987737748166760067230019036542268229685, 3717283184181977343850933021879391950381979418]
10127 - [29888, 297]
10128 - [440772247, 4379781]
10129 - [10276104790792781362477337963395494469275001623708782875797962884555028616415, 102104583656601146063034290519019681559764484405679578250677511708505331416]
10130 [3566250337, 35432933] [25436282932305227139, 252725418504296842]
10131 - [1930278592606583570, 19177581477783127]
10132 - [1721249, 17100]
10133 [20126305635281992627339371982, 199937861294546148443268505] [810136357049567385668754469198498398125167853540333216649, 8048021008957307089984730774872292266757941764200053820]
10134 - [91205, 906]
10135 - [1273463424509462708655124, 12649536119627979028515]
10136 - [10645372091451, 105737136545]
10137 - [4928717527, 48952988]
10138 - [108478376210753, 1077375343704]
10139 - [4075736773881651520670765367850, 40477023262191431174766752421]
10140 - [1757602506271, 17454269836]
10141 [699360375405714525702329784478731286592726654721678063678082574333371084818, 6944814195172188813482697165567579888507641537162383597947719998585752925] [978209869375243901681722538518591599061984740576903294558903844151628911466117852611056329905581530718139372485528354123795235520519606513460300186249, 9713855725317114310906311760670685763797983642745605275282934481897359494297352296276426976809902376990137267099927671706738704980549656813133185300]
10142 - [826857899, 8210490]
10143 - [6099380351, 60562320]
10144 - [151792902880753863035506167194287475981250001, 1507116575169952975751638139046055969711875]
10145 [25270911932, 250896665] [1277237979749799945249, 12680775050495013560]
10146 - [1227665, 12188]
10147 - [79568596496463458853102244176075429482, 789901370718145011775768957624985247]
10148 - [8274617367, 82140566]
10149 - [575674656716782970853065, 5714332201595971391724]
10150 - [7072908645858049, 70204512445776]
10151 - [88709506006116850518620062031436077037329569164996903000, 880472400308260012235107437025026797930890643886857843]
10152 - [51977791151, 515872070]
10153 - [18122731068478321838255977, 179856630412224111501876]
10154 [90305875391444927759223085, 896184498048017768745937] [16310302260430357168891982981344259490998565593834451, 161861451216937825438982821597047438067411740711290]
10155 - [126087866, 1251219]
10156 - [374266066928903929679001933148375282821347630206083220077206942474078212694808199, 3713805090833516647572389831937520961452384438605066272313127983355811425870410]
10157 - [13572833172039156410788177526508749151, 134675248276645643473866843890227880]
10158 - [35605534962217008149, 353275411591147170]
10159 - [29800080807645474974385669891831317505715073757671685943021204802625482669589006560, 295659584047195307094751251372485751763641864993260233649740808502864404607602631]
10160 - [31751, 315]
10161 - [56449, 560]
10162 - [335724901254722949212039301377, 3330381301076245080163851888]
10163 - [535140026650739345082027562, 5308312393850134331957481]
10164 - [6049, 60]
10165 - [1802209718061420798000378492933889329298767871, 17875229883049526317247799368918478044234496]
10166 - [1306027340799, 12953204320]
10167 - [3015075970966491508, 29902110894214867]
10168 - [175827599, 1743690]
10169 [15523023797433452652670102824413548526438168, 153934937468776420347714516853363921027805] [481928535631370577787458889339447202974772909456970878228537208920184088274545454392449, 4779071395168493649028990005291318765484027166789225945667466590249107850628682522480]
10170 - [291754799441, 2893060612]
10171 - [786113565888326255743302788645305, 7794772849468842751106186964612]
10172 - [31054307327, 307906368]
10173 - [391464662212362445458870870958080602863351, 3881218044683192677475608924838445235980]
10174 - [59831479596579991562214626036169585345697823640198068204707081927239935, 593176416696204822748643748706056770068367725348831964143373957247024]
10175 - [75539384999, 748869660]
10176 - [3505951, 34755]
10177 [122037641124076745624123755965559360632, 1209717371865249501021284729698365505] [29786371702257895386490621089733193716574872255112998556781810448153262878849, 295262108978505227214145531373262314956131965138907671947700028446487598320]
10178 - [63520899, 629630]
10179 - [199644920441821945, 1978817341830084]
10180 - [219689611075957249, 2177387026852560]
10181 [119361023728194876977120182, 1182952526784130209067025] [28494107970885400897928407781311136983877428143426249, 282396849237617303261574288389321060318339036397100]
10182 - [7870496424479984474557157, 77998379456086805160742]
10183 - [19427641343, 192522816]
10184 - [164618455825, 1631245506]
10185 - [17059232471, 169035908]
10186 - [47823018756119413778837952241601824961894125, 473843743734322589602380116083441983159078]
10187 - [153630939705247, 1522143277272]
10188 - [38239443505885447, 378849828745454]
10189 - [64612672862457327382085804637673091439997180013306420625971316571044578024283188960861829383969310386015, 640106041124196227869354780890959872892828318640303654919005484142265511533978913816888516942421248104]
10190 - [650298835221, 6442076594]
10191 - [68180240, 675383]
10192 - [1419278889601, 14058469260]
10193 [558678579061880057900084023271044, 5533641348106089326776657349153] [624243509405202733271884947538114677345412614573610553866977699873, 6183053770795952746699506791450137348441237573052000394125651464]
10194 - [323325365515950150545, 3202340165781449236]
10195 - [9880337427714147540146625078335999, 97853905018477820733336193367520]
10196 - [100487356912123526021379257608724874758757013449, 995168245630392362985195989804467167614185190]
10197 - [514999, 5100]
10198 - [10581358553932237083429552841298868330861797, 104781334918818317027683665920107764781614]
10199 - [10200, 101]
10200 - [101, 1]
10202 [101, 1] [20403, 202]
10203 - [10202, 101]
10204 - [3257022829412129843871563832505700417263668897353200779822892671695869789443199, 32243009980491846306981528198345677830324793809826400747792533556298991385720]
10205 [515302, 5101] [531072302409, 5257111004]
10206 - [189868179946753234878336449, 1879422428600775582720040]
10207 - [30186624315177976975278880704652, 298789610701710209676764247177]
10208 - [3575704177, 35390871]
10209 - [130969496245430263159443178775, 1296219513663218157975941956]
10210 [581328972675417, 5753195036797] [675886748943711450037672247779, 6688997920685015829704638698]
10211 - [502706659600073170517061987730, 4974855840942648150864916497]
10212 - [207305303, 2051422]
10213 - [97120578503816114681310007968642604255761831606232221771773852599, 961024799025043326876406842213431627862312684141850372086592140]
10214 - [6114582598540541161524433993164947116832508925529765, 60501882700434383102001437748461543141271767060154]
10215 - [10614754956124, 105024539835]
10216 - [48613985299, 480973095]
10217 [1050172457008818416, 10389602382602789] [2205724378919877128390427976921498113, 21821748522965289630133911112324448]
10218 - [8998913, 89024]
10219 - [1019214003633664916890, 10082336372243359539]
10220 - [69950791, 691938]
10221 - [5418654390068391010655, 53597527749183755688]
10222 - [4445792303508509752354749125788043, 43972507470237181066508136653178]
10223 - [23204170620024488886288, 229496924427639847129]
10224 - [24220799, 239540]
10225 [111921796968, 1106835337] [25052977273092427986049, 247757999729443716432]
10226 - [2726206976200465, 26959133579268]
10227 - [395996257, 3915768]
10228 - [3776076546774119988466249, 37337516698575735082350]
10229 [10296170385862353370, 101802661345108313] [212022249229417845369883096469500713801, 2096355093886956690027666783461129620]
10230 - [167089, 1652]
10231 - [46989220757647076, 464557205588595]
10232 - [1626624753663, 16080780032]
10233 - [650468934487, 6430208764]
10234 - [19345707497127681807551, 191232591627292276920]
10235 - [285459228606, 2821630651]
10236 - [275238491898535, 2720470563022]
10237 - [182627240859007649, 1805008312091160]
10238 - [193822034717780924175453613843201599, 1915559233080154074026730363467560]
10239 - [90662008955680953132123124, 895976392248340175734425]
10240 - [9347391150304592810234881, 92372019423324448490697]
10241 - [46551, 460]
10242 - [243198723043, 2403083738]
10243 - [16167484016162, 159745581249]
10244 - [8138172863569276449, 80406679107115780]
10245 - [3224179361, 31853944]
10246 - [426030555983866534901538397517065226299, 4208851211620793674004304088893698130]
10247 - [28182602134510650506934408, 278408648099162441805827]
10248 - [61853, 611]
10249 [1518189822183575888019026013567215527680, 14996341753738029657372332188157694293] [4609800672363595547087788357927870065360071622949498058017233069042601692364801, 45534586841023347669272157415703364973327604605024301852013518848912239060480]
10250 - [55777922001, 550935044]
10251 - [36449, 360]
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19882 [141, 1] [39763, 282]
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19997 [2251584514, 15922301] [10139265647369232393, 71700812717693428]
19998 - [9899, 70]
19999 - [33384630690640035741763643209424, 236070889333608364601914958865]

■2≦D≦9999についての結果は、以下にようになる。


[参考文献]


Last Update: 2016.10.15
H.Nakao

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