寻找四元组
据说是一个很古老的题目，最终就是要寻找这样的四个正的有理数$(a,b,c,d)$，越多越好,使得其中任意的两个数$(X,Y)$，都满足$f(X,Y)$是平方数,其中$f(x,y)=x^2y^2+x^2+y^2$也就是说，要让下面的六个表达式都是平方数。
\ (0,0,0,0)
楼下继续:lol 四元组: ['7/26', '9/46', '15/26', '4/13']，互不相等的最小正有理数解。 考虑特殊解：
取任意 $u \in \mathbb{Q}\\{0}$，令 $a = b = c = d = (u^2-2)/(2u)$，
则六个表达式都等于 $((u^4-4)/(4u^2))^2$，为平方数。
例如取 $u = 2,3,4,5$，对应解为：
$(1/2,1/2,1/2,1/2)$
$(7/6,7/6,7/6,7/6)$
$(7/4,7/4,7/4,7/4)$
$(23/10,23/10,23/10,23/10)$
我找到了一个有理解:使得 $f(x,y)=x^2y^2+x^2+y^2=z^2$的有理解有三种表达式，分别是
1） $=[-\frac{(u-1) (u+1) v}{u (v^2+1)},-\frac{2 v}{(v-1) (v+1)}]$,
2） $=[\frac{(u-1) (u+1) (v-1) (v+1)}{2 u (v^2+1)},-\frac{(v-1) (v+1)}{2 v}]$,
3） $=[-\frac{u (v^2+1)}{(u-1) (u+1) v},\frac{2 u (v^2+1)}{(u-1) (u+1) (v-1) (v+1)}]$,
取这六个表达式的4个，比如$-\frac{u (v^2+1)}{(u-1) (u+1) v}=a$,或者$\frac{2 u (v^2+1)}{(u-1) (u+1) (v-1) (v+1)}=a$只要给定$a$，就都是关于$u,v$的椭圆曲线。 解方程$(1+x^2)(1+y^2)=1+z^2$
易知$(1+x^2)(1+y^2)=(x+y)^2+(1-xy)^2$
令$x+y=1,1-xy=z$ 解得：${x,y}=\frac{1 \pm \sqrt{4z-3}}{2}$
显然${x=k,y=k-1,z=k^2-k+1}$是一组通解 感觉除了前面给出的互不相等的最小正有理数解外，其他解可能不存在或者分子分母的数值将极其巨大。 我无意间发现了一个特殊解的情况。当存在两个数相等的时候，此数一定是形如$a=\frac{U^2-2}{2U},U$为有理数。 设另外一个数是b,那么就是求解$4 b + b U^4 - 8 U V + 4 U^3 V - 4 b U^2 V^2=0$的有理数解。
容易验证发现, $b=\frac{1}{2},b=\frac{4}{15}$有解,以$b=\frac{4}{15}$为例，此时$4 + U^4 - 30 U V + 15 U^3 V - 4 U^2 V^2=0$, 对应到秩为3的椭圆曲线，$y^2 = x^3 - 74956/81*x + 2622800/243=0$,计算得到a可以取如下值
17/6
31/90
791/120
553/696
20953/87804
241849/1211460
1350359/1414260
639919/18766830
4411273/41765964
361688383/715747506
868246831/1500902250
34890463/7528366134
34453087519/16419543510
136550126897/100560912246
178608061999/161131921350
7498226285401/10815099312000
2200220663743/18603644484774
11932801787263/40255883956866
22480302299833/46938280886856
3460494604679/53379631165320
186595261806839/108378056508420
2049249519408167/143585671323456
1281063059752049/571345492152030
5108437084951513/1340518393696884
6787856568179057/9825861425129874
2270434719315527/14889776838393936
433984132184261207/299708351381423676
18806031546180472337/718965249866471766
381902329533682097/918812470715100354
1713310062880868281/1456907591847849840
16129639869391109983/9323147096692268094
8746390845336294649/23980771770390146100
5512771604612832161/71745411060305178270
118612759905554022073/86256354204006058536
32647238326401553799/173721342834835472280
1504527435261223987649/179593275389032022670
40390574199053956249/542584175492261636820
160308592289309183777/568067870618281998186
456525908092106705921/756315717672626425290
2951580775096875450887/3556628264166398798316
150333178527623403287/4309059226234424087616
12170335775794274722481/7582069030146059616990
6621183932299153176799/43684147376649336449790
8787251733075302370031/47480880933151395489930
7902708558778338804782473/352018850139236386985436
1426332587540553920046401/377562260504873891900430
844789217888507044550711/2861681507785514946332400
541145943955591605613119151/128607656877183441281200830
85750819191141504244438679/149804986639578826740742500
1930588839779523620605565687/454442497329903650595467184
505521508056259951546904623/1271334381527406329297100114
378226614725954458061378849/1577811166648979352697475970
已经知道了b的一个值，比如b=17/6，结合a=4/15,我们已经有了三个值了，(4/15,17/6,17/6),那么第四个值也可以被一个椭圆曲线确定$-2 - 2 a^2 + U^2 + a^2 U^2 + 4 a U V + 2 V^2 - U^2 V^2=0$，可以转化成秩为2的椭圆曲线$ Y^2=X^3 -76204941766875*X +250884088221360093750,= [-4097*Y/((X + 10057350)*(X - 5613675)), -3250*Y/((X - 4443675)*(X - 5613675))]$
最终得到如下解，
(4/15, 17/6, 17/6, 17/6)
(4/15, 17/6, 17/6, 160/159)
(4/15, 17/6, 17/6, 63/466)
(4/15, 17/6, 17/6, 101343/266974)
(4/15, 17/6, 17/6, 6261520/1168491)
(4/15, 17/6, 17/6, 4108560/22615189)
(4/15, 17/6, 17/6, 92800506287/181532006934)
(4/15, 17/6, 17/6, 2191698790080/3023483730271)
(4/15, 17/6, 17/6, 2022877418800/4801099991211)
(4/15, 17/6, 17/6, 67262079744720/216818861155339)
(4/15, 17/6, 17/6, 16140420568857/411758724949174)
(4/15, 17/6, 17/6, 8658848705488457/10703681625160674)
(4/15, 17/6, 17/6, 26604947508338927/127722681059739786)
(4/15, 17/6, 17/6, 144645118366483474487/112679500822145722866)
(4/15, 17/6, 17/6, 64279713206107339971783/71924458362443416074106)
(4/15, 17/6, 17/6, 9934932660939479719745697/6225995415084505866137854)
(4/15, 17/6, 17/6, 34230976965794521434379440/15228763503231890295106699)
(4/15, 17/6, 17/6, 125181612321354222929236480/66113789562954680679406161)
(4/15, 17/6, 17/6, 173933922435451929058525360/661851513886991880809261589)
(4/15, 17/6, 17/6, 703642709949509861515724320/11646987219749462332452382881)
(4/15, 17/6, 17/6, 118430825412763365773722050720/256751204305934965691811729841)
(4/15, 17/6, 17/6, 693677755821778528696812881760/7524340034920961845223991597871)
(4/15, 17/6, 17/6, 12632858438543951789001789150777/21254147935580040602291190988186)
(4/15, 17/6, 17/6, 3944233467089453035513960891768257/395077393282017155328475818504974)
(4/15, 17/6, 17/6, 933666623549408820700039195326242807/44845452050381274976630035390003513774)
(4/15, 17/6, 17/6, 10648484467239068849528433394514731840/66786757170616348769472356042244785199)
(4/15, 17/6, 17/6, 11480893865874797132810749040727318980800/18492993390402622883369581973194118393361)
(4/15, 17/6, 17/6, 43313176690433952591257590784585834800617/185920345359532502330067653556076165497706)
(4/15, 17/6, 17/6, 112998023386513177561397924039831841580995527/2816880978047693797577813026915954504202814)
(4/15, 17/6, 17/6, 11709176636885136714954967057202968618340720/104611634345600947843142870720527578157555029)
(4/15, 17/6, 17/6, 44896251823171885248137540894123696799023622000/2725044685371844451743367456990620284759200149)
(4/15, 17/6, 17/6, 2141888712905279420428642102405364141221263737/6192991307739714658288795584976786536706037166)
(4/15, 17/6, 17/6, 4680573624492653631727222303285577116757474863/66579085873185255230430761347579105700965215834)
(4/15, 17/6, 17/6, 15044772936366074374178049446643636757051832079497/148200953290428342462935112229516955494927783883646)
(4/15, 17/6, 17/6, 456263727893207814583260335448156549728629446533760/307280161384345360292832651122050196197202430299519)
(4/15, 17/6, 17/6, 7448049046580916634843364391899112373004801186868297/29779786827985909016957848302526982428793081539956454)
(4/15, 17/6, 17/6, 271651365975292682226289051640386547247377454084218400/71583190331255473474211650519389955170665566508394449)
(4/15, 17/6, 17/6, 19990662491766869816645245880677607970465581958094569783/5484971646891155399972733574495185359154625166396951494)
(4/15, 17/6, 17/6, 800602431526948858799390765588554315874798941206570300057/701638580800947565699330146584701237793475782889451779626)
(4/15, 17/6, 17/6, 2259185329508549780909290752366832031329890270589705820640/1863271546940469349352458662457208024513044364069000616241)
(4/15, 17/6, 17/6, 4228827825521436615516763171112096050646057362199369284783/15255395706016149668588605379852652661515267938309650044006)
(4/15, 17/6, 17/6, 1268192030785592805316952944181855257421502940650878633766640/1921383096626292376686974263984207577280903835590933898496949)
(4/15, 17/6, 17/6, 10549561424896004150114535996982247866304112481972928332976400/132914296972796339989271772562244712509617553562080072859075499)
(4/15, 17/6, 17/6, 1475143123219346287064825518354083663918022434374243027433683020080/1313486322375198437301357079810537123497862808949849198040370133109)
(4/15, 17/6, 17/6, 54477632239023327674047039094330868551390536775653733568121031267897/167747174697001499513233140396985918845653239314147551808125104568754)
(4/15, 17/6, 17/6, 354395366970671531668857985868191632810364547238245787638208600825040/207329301368196351611477345421056431494521831326516410702246466585269)
(4/15, 17/6, 17/6, 2335994376589475007654791205170048712903737540443666179705507615365520/4122081843781336069451610727181023807605451141342522817048516594605931)
(4/15, 17/6, 17/6, 291881750161364314180115843962755413966728455465480126686635644314749040/1320349358109378618780227871505030355475905530840613313199291166970883371)
(4/15, 17/6, 17/6, 1259250829330189169122989132625154100911462690463890029352490725305249273/8051262260574483728603614314489219997624885585764073942236985463888466786)
(4/15, 17/6, 17/6, 11366476777699382741349134114842029903529400803929942859324486263333484863/16473677407565118541040932448408240760052679647353991186248645996425152334)
(4/15, 17/6, 17/6, 44055266456847415470584860950782755777615610243736918600782566312069195623/21506814615096874364947275332328964569809228040716158242475247159602977386)
(4/15, 17/6, 17/6, 162879473174995771906522788711196871987343780226272072155124371405111212160/477193216214878394259319796265146761299920952351980902031715831375555048081)
(4/15, 17/6, 17/6, 8552604093762478733111426499710440132856910990992360539282599540048015558320/468407583039638071509319243554853198911892742619865054508961442389481238944011)
(4/15, 17/6, 17/6, 4044285854308477527724466949584006347989268034468683727257086618752829155583200/7545747491805619737544224043353581260483550254750861195453038775492353170134239)
同理，我跑一下其他的数据：
(4/15, 31/90, 31/90, 31/90)
(4/15, 31/90, 31/90, 1757275920/1048776661)
(4/15, 31/90, 31/90, 772771484523849063449/4583859849785493860670)
(4/15, 31/90, 31/90, 139809013793340582986341304184241403040/1353073795431179627686548719546716775759)
(4/15, 31/90, 31/90, 2071867363212711867038744580167662549052728188801231125307209/2653181080011688462616595103238961668356368421166008757657550)
(4/15, 31/90, 31/90, 3205001248345489444180449497431254807778700147245559182697170908774470685198994046047440/6253559609910387308812206254635673655062767468911226623018962833727862367259634929744459)
(4/15, 31/90, 31/90, 22019572118703141171680620383018339814907736232961912235312198218690843205946275026311482147998717813183221552074555151/409008563787975441754766240816727471972446623381046850280122691146263852435724590616950771889831208044112758146073073770)
\(a^2b^2+a^2+b^2\)是有理数的完全平方数，等价于
\(1+\frac1{a^2}+\frac1{b^2}\)是有理数的完全平方数，所以我们不妨记\(A=\frac1a\)等。
我们需要先寻找三个数A,B,C使得\(1+A^2+B^2, 1+A^2+C^2,1+B^2+C^2\)都是完全平方数
如果给定有理数A,B使得\(1+A^2+B^2\)是平方数, 那么我们现需要找到一个新的数C使得\(1+A^2+C^2,1+B^2+C^2\)也是完全平方数
那么相当于找C,使得\(1+A^2=(U-C)(U+C), 1+B^2=(V-C)(V+C)\)
设U-C=X, V-C=Y,于是我们得到\(\frac{1+A^2}X-X=2C=\frac{1+B^2}Y-Y\),记常数\(S=1+A^2,T=1+B^2\),我们得到方程\(SY-X^2Y=TX-XY^2\)。
所以我们的目标就是求这个三次曲线上的非零有理点(X,Y), 然后\(\frac S{2X}-\frac X2\), 就是符合条件的C.
同样，D可以通过这个三次曲线上另外一个有理点得到，然后我们需要找到这个曲线上两个点
\((X1,Y1),(X2,Y2)\),其中\(C=\frac S{2X1}-\frac {X1}2,D=\frac S{2X2}-\frac{X2}2\),使得\(1+C^2+D^2\)还是完全平方数。