X,Y,Z,X+Y,X+Z,Y+Z均为完全平方数?
求$(X,Y,Z)$,使得$X,Y,Z,X+Y,X+Z,Y+Z$均为完全平方数这题好像是经典的Euler Brick问题，相关网页链接如下：
https://mathworld.wolfram.com/EulerBrick.html
https://math.stackexchange.com/questions/4512726/euler-brick-saunderson-parametric-solution
存在参数公式，可以生成无穷多组满足条件的解；但是目前不存在覆盖所有可能解的单一通解公式。 我的思路:计算$b^2+c^2=x^2,a^2+c^2=y^2,a^2+b^2=z^2,a=s^2-1,b=2 s,z=s^2+1, a=r (t^2-1),c=2 r t,y=r (t^2+1)$
Solve[{b^2 + c^2 == x^2, a^2 + c^2 == y^2, a^2 + b^2 == z^2,
a == s^2 - 1, b == 2 s, z == s^2 + 1,
a == r (t^2 - 1), c == 2 r t, y == r (t^2 + 1)}, {a, b, c, x, y,
z, r}] // Factor
得到$=[(s-1) (s+1) (t-1) (t+1),2 s (t-1) (t+1),2 (s-1) (s+1) t]$
$=$
也就是说,我们需要选择合适的$s,t$,使得 $s^4 t^2+s^2 t^4-4 s^2 t^2+s^2+t^2 = U^2$,
锁定s为任意一个值,都是一个椭圆曲线.$-\frac{2}{27} \left(t^2+1\right)^2 \left(t^4-4 t^2+1\right) \left(t^4-10 t^2+1\right)+\frac{1}{3} \left(t^8-8 t^6+30 t^4-8 t^2+1\right) X-X^3+Y^2=0$
对应的变换是$=[\frac{18 t Y}{\left(-t^4-2 t^2+3 X-1\right) \left(-t^4+10 t^2+3 X-1\right)},\frac{t \left(-5 t^8+40 t^6+12 t^4 X-54 t^4-48 t^2 X+40 t^2+9 X^2+12 X-5\right)}{\left(-t^4-2 t^2+3 X-1\right) \left(-t^4+10 t^2+3 X-1\right)}]$
举个例子,拿$t=11$来说,得到的解是
(44, 117, 240)
(85, 132, 720)
(187, 1020, 1584)
(429, 880, 2340)
(1155, 6300, 6688)
(1575, 1672, 9120)
(134288, 574425, 732480)
(421245, 537152, 2297700)
(8430569, 38181000, 208260000)
(20931196, 114170160, 489336147)
(28170252, 153655920, 166102435)
(92736259, 505834140, 2290860000)
(102169320, 359216039, 557287200)
(338043024, 365425357, 1993229220)
(418623920, 1794232539, 9786722940)
(3951376429, 6130159200, 21552962340)
(329926825668, 1799600867280, 4546617155165)
(497624468880, 1006077873151, 2714315284800)
(2381010291027, 5108185261036, 27862828696560)
(3959121908016, 10002557741363, 54559405861980)
(8730371067099, 47620205820540, 102163705220720)
(11066856604661, 29857468132800, 60364672389060)
(25449277135418752, 138814238920465920, 210774942520039575)
(101797108541675008, 154568291181362355, 843099770080158300)
(450410684763544516, 709364905041339963, 2456785553255697360)
(699765977557726035, 3816905332133051100, 9973705892255971712)
(832942273137695613, 68328802703038504484, 372702560198391842640)
(954226333033262775, 2493426473063992928, 13600508034894506880)
(2601004651818246531, 9008213695270890320, 14187298100826799260)
(3054121668171550581, 16658845462753912260, 1366576054060770089680)
(16461419928508766772, 52694922192609886715, 89789563246411455120)
(115928828823741750773, 197537039142105201264, 632339066311318640580)
(376519973099289354720, 440893159037724679169, 2053745307814305571200)
(562555060429260952812, 728817478882353146765, 3068482147795968833520)
(1603398453541176922883, 6750660725151131433744, 8745809746588237761180)
(4849824749414971470859, 22591198385957361283200, 26453589542263480750140)
(286923578320379555581928, 481708525910523281368425, 1565037699929343030446880)
(353252919001050406336845, 1147694313281518222327712, 1926834103642093125473700)
(668307158733941522522408, 3645311774912408304667680, 23477285512996714162286775)
(2673228634935766090089632, 17216676042864257052343635, 93909142051986856649147100)
(1324995355029005375849735272, 7227247391067302050089465120, 11482930480042970474554827225)
(5299981420116021503398941088, 8420815685364845014673539965, 45931721920171881898219308900)
(306237842447673362869186018095764, 1670388231532763797468287371431440, 2311974598857045957291589993416573)
(6124756848953467257383720361915280, 8477240195809168510069163309194101, 46239491977140919145831799868331460)
(7161365367428812137443075752142148, 15855242762037856840729854140344435, 39061992913248066204234958648048080)
(34881534076483285049605679108757757, 85936384409145745649316909025705776, 190262913144454282088758249684133220)
(15219218300047526359487618925082850351361800835, 83013918000259234688114285045906456461973459100, 1288951653229768966262046683845729607443012097728)
(20753479500064808672028571261476614115493364775, 322237913307442241565511670961432401860753024432, 1757661345313321317630063659789631282876834678720)
$t=\frac{2}{13}$的时候
(780, 2475, 2992)
(832, 855, 2640)
(2925, 3536, 11220)
(2964, 9152, 9405)
(4368, 4901, 13860)
(19604, 55440, 62205)
(3274128, 10389060, 73353371)
(41556240, 293413484, 931023555)
(641207248, 10522336500, 33388183125)
(757790384, 2404527180, 39458761875)
(974982112, 3093693240, 15200344305)
(10724803232, 52694526924, 167203787355)
(19067230399, 20933427840, 66423376800)
(28945943520, 39469328311, 91847705400)
(991495980748, 3146093015835, 3454015593600)
(2052405072172, 4776080680800, 6512439171315)
(18273366145271, 24450752595360, 77584118812200)
(950215039554092, 3015105413969715, 4034374178234400)
(309255575337828495, 477817636557388192, 1516152115999404840)
(351792607023337740, 889424695335183568, 1116265003054821675)
(1051138276305216944, 1319222276337516525, 3335342607506938380)
(1072085994504472116, 3401811328716113445, 5255994002131270112)
(1028542193853082650469, 1096223798575815207792, 3478402437788644409340)
(1684112432877209775349, 2931175709337150354768, 9300846000781342471860)
(4114168775412330601876, 13054573998904510563645, 13913609751154577637360)
(6736449731508839101396, 21375273186518431764045, 37203384003125369887440)
(82544279806466502584768, 261919349385903325509360, 3261247197030847729295145)
(211173996408679120470368, 258059939008029192410895, 670071334758308747646360)
(489315396603981439827660, 1049431022361001238752112, 1552635393070325722530075)
(894607788561167867024436, 2322913960495470325174048, 2838659329088321116519845)
(907987077871131528432448, 11305656949706938794889836, 35873719167339325022246595)
(1240236662790274191252496, 1834932737264930399353725, 3935366333853754645320420)
(2945625621031516782261232, 8401105528607180755129140, 26657354081157400473005925)
(3481193915764519833581456, 11046096078868187933479620, 31504145732276927831734275)
(15461229988351235341092359520, 49059672078422189063081525400, 171307448550806211637643299849)
(1828996758922822108095007716912, 5803547408120493227609159101740, 21796387911641088463138137635659)
(2551102948077953831280239320800, 8907987324641923005157451592148, 28265729010883024920211144475085)
(23214189632481972910436636406960, 87185551646564353852552550542636, 276646461955444584339830208452595)
(389494516942810044243885497979941, 538914396871985023575221312470512, 1710016836228414017113683010723740)
(1028303328280759575390997269698352, 1691858354188192712530275543758261, 3262885560890871729606049028850540)
(1557978067771240176975541991919764, 4943584253504896715403162089745405, 6840067344913656068454732042894960)
(6767433416752770850121102175033044, 13051542243563486918424196115402160, 21473586803157830582115035747701005)
(42148761733388425618986158742622816961008283872, 63981889943583855570213290482831386604403729295, 133741263192482504367936849856399323049353208440)
(44669119430854597165022738874864494927152982220, 141738552040211702542860613737550801211158501275, 293723694508722929573003847222620233466043100048)
(167509197865704739368835270780741855976823683325, 347128002601218007677186364899460275914414572784, 1101463854407710985898764427084825875497661625180)
(221803885137757365976739407007148806895266261556, 463636379067272681808847746168850986571091122592, 703800789379422411272346195311145252648441022245)
(296771915223720719005485462292189914999546187241078057846699236272, 941680115613729204536636563042525691825483094130343837398180268940, 2152294032682227110699913027097490281298037031430868099836606972171)
(3766720462454916818146546252170102767301932376521375349592721075760, 8609176130728908442799652108389961125192148125723472399346427888684, 27317578107120574866575819190083530493398162322007172036387703877555)
$t=5/39$的时候,秩为3.
(195, 748, 6336)
(780, 2475, 2992)
(1105, 9360, 35904)
(1560, 2295, 5984)
(1755, 4576, 6732)
(2925, 3536, 11220)
(6435, 24080, 24684)
(7800, 23751, 29920)
(16380, 62832, 98549)
(18525, 71060, 90576)
(65520, 102765, 394196)
(67925, 86580, 332112)
(118755, 149600, 455532)
(234780, 240669, 900592)
(472549, 3175380, 12180432)
(492765, 1890196, 12701520)
(3452085, 9364720, 13241844)
(8135400, 18244161, 31206560)
(37830780, 145114992, 4761596125)
(39576816, 1298617125, 4981362100)
(91220805, 156032800, 349913652)
(91306020, 129107979, 350240528)
(493712080, 1084859685, 4161410484)
(540637929, 1827134400, 7008700160)
(663289380, 2544310032, 9445339475)
(693902736, 2576001675, 9881278220)
(913516639, 1958439600, 7512373440)
(1178212889, 1429014600, 5481553440)
(2703189645, 10369158228, 35043500800)
(4640482275, 17800414060, 35560595376)
(4813692780, 18464831792, 40573752219)
(5708516580, 21897284112, 23194693291)
(5735527200, 22000894080, 92492619359)
(8456494464, 14064540555, 53950135052)
(12492548640, 47920135296, 79699063145)
(17015101675, 33991745580, 130388849712)
(20083482144, 20332546845, 77993564308)
(22595841840, 86675331776, 205583868495)
(22834066320, 24186979635, 92778773164)
(29668780440, 113806398816, 115217765455)
(58083025275, 98579216580, 378139764112)
(66281136064, 157211193555, 603046014252)
(68643575325, 263309714580, 446892448496)
(131435927220, 504174736208, 587240020125)
(153274149405, 549050531744, 587943916692)
(178135744605, 683310445972, 1464912820800)
(187176317640, 200435426145, 717989156896)
(229751513355, 881303240972, 1068902920800)
(595842870064, 694010932875, 2662154757900)
(4290174345600, 18036060775005, 69184479280532)
(5638326074160, 21628040530624, 123855387662745)
(9820954100925, 29949317353020, 114882509641328)
(11606582119275, 44521658590860, 135770238667024)
(16539089817536, 94712943506805, 363309137144052)
(20081922611595, 42772981150944, 77032195453708)
(63187358518440, 113797561465705, 242380226522016)
(173409425280881, 232634784236400, 892363172352960)
(233183323923420, 503134977428325, 894467314331888)
(244798780658445, 939023015038548, 957716445335936)
(320121482399505, 326494242728160, 1252398428516224)
(594614064233475, 1057097735119504, 2280878564341740)
(665928867403800, 777478031908471, 2554434834964320)
(858259262086200, 1575666469707911, 3292194502771680)
(988824875307165, 2478853932051280, 3793030803742356)
(1077964257825000, 2666987281571929, 4134960332580000)
(2096056397370000, 8040257360168000, 9830940311145681)
(2717955639821664, 4029323702829555, 15456072460084652)
(3707854843993344, 10945500868627845, 41985818716582708)
(4015161740645640, 15401748625656096, 22832834316034145)
(5477512837717440, 21011177449295616, 62024504922224455)
(12942091405045429, 148058726395195980, 567938088941572272)
(13495763764619565, 51768365620181716, 592234905580783920)
(24168825837499980, 36982050336487971, 92709137058717872)
(33814837929771795, 129710250110098988, 174010818608827200)
(38066316345344820, 88736332465853325, 146018485263168848)
(40201286800840000, 49154701555728405, 188552393659922292)
(104870211096008475, 172567300765563184, 402271373845201740)
(110301890868466440, 218927531697898545, 423106740356989216)
(151608216222151845, 498114792818042400, 581553567867536308)
(155287714193359839, 280445382344680800, 1075759723045237120)
(167415171298393005, 323552213214168224, 642187426313835732)
(229503827581380011, 582551831849895180, 2234609078070367152)
(239322173146358835, 918015310325520044, 2330207327399580720)
(241786596089123120, 330108556021786035, 1266262563611774124)
(307254961593042645, 641977928040477600, 1178598519341517428)
(520062519906526155, 806317264853100000, 1994906486615802892)
(584249345070190800, 1932894611409551751, 2241120564679501120)
(776438570966799195, 2978338723503414348, 5378798615226185600)
(2018174391000155115, 3753344882305350320, 7741509971631364236)
(2357419311868950420, 9042818693733204688, 12346059995214797709)
(9664473057047758755, 11205602823397505600, 37071927418829351532)
(12104023642515709125, 18044349931208250864, 46429793254367950900)
(12534009417929646075, 27216595274431233456, 48079174587750642380)
(17248275669537298620, 44381420022557600125, 66162616414430253168)
(26015863129971031980, 45958034532408702275, 99794182672914522672)
(36595112602477165620, 75479722223405801301, 140375098598220101968)
(46910775973211990480, 326356572164282629515, 1251870338353248240396)
(162396855870605327439, 259712377234264663800, 996230041903743428320)
(346119289899282284580, 1327678096639298199312, 3576437048539378606891)
(457380065738816907180, 1754463021398128443952, 12205735798944170343861)
t=28/99的时候,秩为4
(1599565, 1929312, 3137916)
(6347880, 10324465, 12452832)
(7428960, 12082780, 13724763)
(10334016, 14468025, 16807688)
(27017760, 83287512, 135462391)
(34234200, 39770304, 55679975)
(81053280, 131828540, 406387173)
(175818720, 199711512, 324819391)
(328564215, 503927424, 819609232)
(407234520, 662343735, 1015853696)
(1418265849, 4176406080, 6792686440)
(4625072265, 133999455744, 217942476992)
(10741250520, 17470031735, 105842006688)
(34038380376, 55361485543, 163024474560)
(41599581408, 47230716435, 67659348044)
(98180386920, 140646203808, 159684803185)
(149418500960, 1149086601432, 1868923860951)
(259081897536, 421381939048, 823898981895)
(298471406688, 1775792347560, 2888225035705)
(366870666435, 2222682140448, 3615065811764)
(517787340840, 842151596745, 2927487236128)
(695346615985, 4524249147264, 7358433362352)
(777012140520, 1263765957985, 36614336134656)
(1390088589120, 1526758141871, 2260899857160)
(1523647856808, 2478126393369, 3454765899680)
(3283185473568, 5339914035924, 31770475392755)
(10752713477352, 11540181425760, 17488675581761)
(12515408025120, 18966591828107, 20355597792660)
(39036529002240, 63490689396320, 136040771029671)
(70792165760064, 138415028958360, 225124155143855)
(83373008077755, 289821236376672, 471377721574396)
(103547021165280, 168413327894540, 1295164235639043)
(119216156880744, 176541250818240, 193898284017617)
(550714519860120, 895705776619535, 5827879222982784)
(550848798142720, 1180297393722216, 1919686435640913)
(1717341590604717, 2394152768478240, 3893953014676820)
(2150417090700648, 3497530827353489, 17194948534246560)
(8849935527830400, 14549425601306904, 23663811444261247)
(10492592482060467, 51584845602739680, 83899811111093740)
(15018080725643688, 24426052291329209, 1728237569112660960)
(16097126826211200, 26181059269831600, 43042050737199591)
(23518216978729704, 25705794383624640, 38251039411472897)
(46756208682025920, 69574725228742041, 76046308384889560)
(99686618025421152, 148425708073242600, 241405954129947425)
(219694685409904320, 357320883539160760, 763707856824511599)
(308893573643143104, 502397791042608472, 1660013042978388105)
(799886263693626648, 1300969415534890239, 2887522554520084000)
(1533576914003009865, 3587299762943280384, 5834538593517236512)
(1880806026432349093, 133074292821674893920, 216437752231789776060)
(2093418978533844192, 3404826646724327756, 5069525036728895925)
(3628745373978952920, 5901947517526734935, 13805668784660503296)
(6260713196977631595, 21845435179134565152, 35530355160580154036)
(7378684261476165792, 12001009376935531556, 29151778124177425605)
(8575701204939858240, 18328988563788278376, 29811055173102255793)
(12698194436136323037, 28183846905386172000, 45839420553006333500)
(13014395937024368040, 21167173189781516345, 73858376081835910752)
(24946980122133703392, 60598971848683782360, 98560773657933200855)
(44211005611749545536, 146081147782098153240, 237592660452954373695)
(57534953693904086400, 93577322773797465200, 495385799250215922279)
(100104036981316702488, 162813510364453951359, 183998977322293800160)
(125049602812015981575, 246178780268891367744, 400395754272112817992)
(133236617780114160032, 198772862220143746920, 323292730634746783185)
(185780636059595800608, 302161615322758898644, 450788455392111711765)
(831117281862938761079, 1374729485558522163840, 2235919150663996095120)
(1589151587078402651997, 1795933680061262021280, 2920983764991414077540)
(2245855746571139164800, 11889259182005182134696, 19337202388914272602553)
(2486114627099258258289, 21477850567331183625600, 34932499741274401650800)
(6261728294240480565504, 10184344161105052896672, 12594492618786767501785)
(7335482614432278910080, 10523350276603732154664, 17115629409115413571177)
(11004365047457406378600, 17897972516761081045425, 35234826375945927983296)
(13342491999479184474240, 21700802734362158442320, 31131577901619374290881)
(15217720688088853753896, 24750755311056492478153, 215745944541736002265920)
(27602067458019432686616, 44893189442453323328863, 49773528439085041938240)
(45019090368929525688609, 392419788890874145853760, 638248419269302340036680)
(64897383248565246358056, 105551894796593222657033, 174590644665932314807680)
(81656116229974155031239, 90532953302587753446720, 147246688298959915733960)
(113605641909791575974144, 140490678226466476921320, 228500080369417067532385)
(188932431293137414687401, 400888436041401792644160, 652022191158968247523880)
(230284402790019835694976, 374544455259308957153968, 2485655729679842496495015)
(996647330802144832951008, 20479560880794057492329880, 33308838467193365153019215)
(1531446610293143087106024, 2490810621394890190554657, 21518419840625031416892800)
(2265032168331371496961568, 2409599699875428592732200, 3919076568141547550625225)
(3706626807702580453965504, 6028617230348875893471672, 7093734491895374146180175)
(3725912214055520850437280, 5987440250795466367953048, 9738230292464415627675439)
(4040778034961621203995360, 6572095155347932611187980, 10561179331264225399028293)
(12620192740308698215454784, 14590737581900929909537105, 20526024159336856386860712)
(15938625924976740527868480, 25923266588296405003569640, 37116886744165711849560441)
(20929593946845041491971168, 34040791597889924086057724, 699485607811060668213403515)
(52948694060970584012692192, 100921894614133424858398680, 164143709187525449485602615)
(62923468483563904801866624, 417590162586213539411162520, 679186597409792114875622735)
(67248800654032531093374144, 79130108698325863715136600, 128700611495816073794983175)
(92122444083720870032268495, 141006353135285895151777344, 229338796215886168214930792)
(116382377676572647447439016, 189289303663357785359588313, 401645669753924440474710080)
(143285505241365521315487360, 233045707208043453409406480, 1962585647066945815028275161)
(162758650209373753357089960, 228966354002743524766108224, 264717667557345442644458905)
(224238184664805778199195232, 364710626104356728358972476, 387988580246013207511897275)
(301095017915845937763760704, 489713884658582759887415272, 1802383298112370062830664495)
(647846107959591999168702963, 2619803974191495261377407200, 4260961838976319042539697100)
(663386841972616075438087656, 1078960886375735777818404833, 1255886044671712475158682880)
(774602498250538864953232032, 1259846812179853705859179876, 1342568595912752947850986035) 简单跑了一下,前200个解 跟OEIS上的答案是一致的
{44, 117, 240}
{240, 252, 275}
{140, 480, 693}
{85, 132, 720}
{160, 231, 792}
{1008, 1100, 1155}
{187, 1020, 1584}
{429, 880, 2340}
{832, 855, 2640}
{780, 2475, 2992}
{828, 2035, 3120}
{1560, 2295, 5984}
{528, 5796, 6325}
{195, 748, 6336}
{1155, 6300, 6688}
{1755, 4576, 6732}
{495, 4888, 8160}
{1575, 1672, 9120}
{2964, 9152, 9405}
{7840, 9828, 10725}
{2925, 3536, 11220}
{1008, 1100, 12075}
{4368, 4901, 13860}
{1080, 1881, 14560}
{10296, 11753, 16800}
{7579, 8820, 17472}
{8789, 10560, 17748}
{6072, 16929, 18560}
{14112, 15400, 19305}
{5643, 14160, 21476}
{4599, 18368, 23760}
{4900, 17157, 23760}
{6435, 24080, 24684}
{935, 17472, 25704}
{7920, 15232, 26649}
{7800, 23751, 29920}
{4928, 10725, 30780}
{7560, 13728, 35321}
{23760, 35075, 35604}
{1105, 9360, 35904}
{2163, 15840, 37100}
{2964, 6160, 38475}
{21328, 25740, 38571}
{1188, 16016, 39195}
{5491, 41580, 46512}
{15939, 18460, 48720}
{36432, 51205, 51324}
{22304, 24225, 51480}
{9405, 23600, 53196}
{8532, 36960, 57275}
{42471, 54280, 59040}
{9504, 31372, 61845}
{27755, 42372, 62160}
{19604, 55440, 62205}
{2079, 44080, 65472}
{23936, 33120, 67575}
{24035, 30636, 70752}
{5643, 43680, 76076}
{35409, 54288, 79040}
{14500, 29568, 83475}
{41360, 69513, 83520}
{30080, 51129, 85800}
{7885, 16320, 85932}
{18525, 71060, 90576}
{16380, 62832, 98549}
{28083, 43056, 105820}
{47975, 84840, 107712}
{52611, 83952, 109340}
{3696, 9045, 121940}
{30195, 100100, 137904}
{28512, 69355, 138516}
{61215, 121264, 141120}
{30240, 77805, 141284}
{68172, 110979, 141680}
{4599, 23760, 144832}
{81840, 122636, 148005}
{22572, 56640, 151525}
{29601, 90480, 156032}
{47200, 106392, 156519}
{5720, 8415, 157248}
{58400, 154671, 165528}
{19635, 21964, 166320}
{26775, 50880, 176176}
{111159, 122760, 176800}
{38080, 47736, 177177}
{25344, 109140, 179333}
{117469, 161040, 194220}
{9856, 61560, 200583}
{9180, 72611, 206448}
{16016, 100035, 207900}
{100776, 166257, 209440}
{169312, 200385, 211200}
{46816, 122760, 214305}
{10395, 95004, 220400}
{48960, 181720, 227799}
{106227, 154660, 237120}
{118404, 134805, 241072}
{97812, 188859, 245440}
{28704, 128205, 247940}
{102828, 190405, 252000}
{186416, 201663, 262080}
{17325, 100320, 264404}
{14715, 148148, 267120}
{131157, 167440, 272580}
{25908, 95040, 273581}
{97152, 198220, 274275}
{53856, 66495, 277160}
{28644, 103075, 281808}
{108031, 212160, 289800}
{12915, 36720, 290444}
{209825, 223104, 293040}
{19175, 112320, 293832}
{34632, 66976, 299145}
{201300, 204336, 301645}
{19800, 52185, 302176}
{135700, 147600, 305877}
{15225, 17792, 308880}
{66495, 146160, 313472}
{42653, 240240, 315180}
{120320, 204516, 316635}
{112320, 300900, 323323}
{10395, 63364, 327360}
{67925, 86580, 332112}
{20163, 33660, 332384}
{32175, 169600, 339552}
{5320, 63063, 353760}
{34965, 62900, 358512}
{49088, 169575, 360360}
{57904, 116928, 361665}
{33201, 59400, 362080}
{62415, 145464, 362848}
{54087, 54784, 364320}
{216720, 287287, 369984}
{160185, 219600, 375232}
{96900, 205920, 376363}
{217217, 279744, 378000}
{331485, 350020, 379008}
{65520, 102765, 394196}
{119680, 209385, 402696}
{144837, 363440, 404700}
{34452, 134064, 406315}
{115115, 330372, 408096}
{294492, 327600, 414869}
{206625, 222200, 421344}
{314160, 332469, 422300}
{61975, 412920, 425568}
{32375, 49896, 427200}
{112860, 171171, 429520}
{83804, 108405, 432960}
{215072, 224025, 434304}
{66528, 103095, 446600}
{5368, 163680, 450225}
{117711, 255200, 450360}
{118755, 149600, 455532}
{182457, 237120, 457840}
{109600, 130152, 462825}
{169312, 200385, 467016}
{75152, 413364, 469395}
{72864, 143640, 474145}
{7336, 274527, 480480}
{108108, 250800, 486875}
{180180, 215760, 495349}
{80080, 229824, 500175}
{102080, 343476, 503685}
{72765, 326128, 511980}
{230100, 271040, 518661}
{157760, 274911, 526680}
{168245, 495264, 533052}
{127136, 452100, 536877}
{129888, 141075, 536900}
{42240, 63296, 539847}
{218595, 431460, 544544}
{272987, 371280, 550116}
{84609, 187488, 556160}
{96075, 164164, 556320}
{128520, 459360, 564311}
{164808, 411840, 570505}
{27027, 62700, 573040}
{59675, 163152, 587100}
{69165, 566720, 599748}
{153076, 570960, 600357}
{273840, 524349, 609620}
{264860, 562848, 611325}
{70700, 134064, 616605}
{497904, 508635, 627628}
{86508, 102256, 638685}
{8415, 157248, 643720}
{209300, 631125, 644688}
{211640, 326895, 645696}
{230112, 256360, 645975}
{233772, 460845, 653200}
{99603, 295460, 654720}
{251160, 465120, 657041}
{77268, 99360, 673475}
{101952, 272745, 684400}
{263120, 477369, 691008}
{100125, 199056, 691708}
{102340, 643104, 712725}
{242535, 560120, 713952}
{58425, 300608, 729144}
如果还要求 $X+Y+Z$也是平方数,
Solve[{a == s^2 - 1, b == 2 s, z == s^2 + 1,
c == p (t^2 - 1), a == 2 p t, y == p (t^2 + 1),
c == q (r^2 - 1), b == 2 q r, x == q (r^2 + 1),
d^2 == a^2 + b^2 + c^2
}, {a, b, c, d, x, y, z, p, q, t}]
解得需要满足$r^2 + s^2 + r^4 s^2 + r^2 s^4 $跟$r^2 + s^2 - 4 r^2 s^2 + r^4 s^2 + r^2 s^4$都是平方数, 不妨设$r^2 + s^2 + r^4 s^2 + r^2 s^4 =(v(1+u^2))^2, 2 r s = 2 v u$
消元r,得到$-r^2 s^2 + r^2 u^2 + s^2 u^2 - 2 r^2 s^2 u^2 + r^4 s^2 u^2 +r^2 s^4 u^2 - r^2 s^2 u^4=0$,这个是genus=5的曲线, 所以 只有有限个解,或者无解. 继续,$-r^2 s^2 + r^2 u^2 + s^2 u^2 - 2 r^2 s^2 u^2 + r^4 s^2 u^2 +r^2 s^4 u^2 - r^2 s^2 u^4=0$, 换种表达形式, 就是$(r+\frac{1}{r})^2+(s+\frac{1}{s})^2=(u+\frac{1}{u})^2+4$.我们就是要寻找这个方程的有理数解.
设$f(x)=\frac{x^2+1}{2 x}$,那么存在$\frac{f(r)-f(u)}{f(s)+1}=\frac{1-f(s)}{f(r)+f(u)}=t$,联立方程解r,s, 得到 $\left(t^2 u^2+t^2-4 t u-u^2-1\right)^2-4 \left(t^2 u+u\right)^2$必须是平方数, 也就是说可以继续设
$t^2 u^2+t^2-4 t u-u^2-1 = m (1 + n^2), 2 (u + t^2 u) = 2 m (1 - n^2)$,代入消元,得到$1-n^2-t^2+n^2 t^2+2 u+2 n^2 u+4 t u-4 n^2 t u+2 t^2 u+2 n^2 t^2 u+u^2-n^2 u^2-t^2 u^2+n^2 t^2 u^2=0$
f:=(1+x^2)/(2x);
Solve[{(f-f)/(1+f)==t,(1-f)/(f+f)==t},{r,s}]
GroebnerBasis[{(f-f)/(1+f)==t,(1-f)/(f+f)==t,-1+t^2-4 t u-u^2+t^2 u^2==m(1+n^2),2(u+t^2 u)==m(1-n^2)},{},{r,s,m}]//Factor
这个是椭圆曲线, 秩为0,没有有理数解.
所以, 欧拉方块问题是 无解的.
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