根据楼上 wanye 提供的相关资料及计算共有116组解（按照hyperellratpoints(f(u),v),1000000)计算结果）
[序号，\(\)]
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以上的值只依赖于下面24个u值得到：
[-9/20, -29/12, -93/80, -400/37, -136/133, 201/4, -5/8, 233/60, -56/165, -125/92, -361/540, -817/660, -865/592, 553/80, -12065/12396, 1744/495, -3168/1553, -1376/705, -1152/2345, 2265/184, -1245/5012, 1873/200, -97/400, 553/80]
要想得到较小的有理解（20位以内），需要找到新的u值，观察上面已知的u值（都不是太大，且绝大数分母含因子4；负数较多；分子分母的质因子都是2(指数不超过7）、3（指数不超过5）、5,7,13,17,19,23,29,31,37,41...(指数不超过1）；分子分母不同质因子总数在4~7；）
按以上的规则应该可以找到新的u值？
我已经统计过了，总共有$279$个=93(旧的)+186(新的)解，$428$个$u$值,构成240个$(u,v,w)$组合，(暂时不考虑单只椭圆曲线通过$n*P$产生的解，只是通过解超椭圆曲线，根据u,v互相关联产生的衍生解)，放在了服务器上，https://nestwhile.com/res/a4b4c4d4/u.50.3.txt
跟https://math.stackexchange.com/a/4857107/303309 完全一致， 而且信息比他更全，更丰富。该链接提到排行榜里的前19个解是GDRZ (Robert Gerbicz, Leonid Durman, Yuri Radaev, and Alexey Zubkov)通过暴力搜索$d<2*10^9$得到的。
完整的$(u,v,w)$对应关系可以查看文件https://nestwhile.com/res/a4b4c4d4/abcd-new.txt,
查看数据增长速度可以大胆的合理推测， 在排行第44-46之间极有可能有一个新的解。这个解源自于一个全新的$(u,v,w)$，从未出现过，而且这三个分数的分子分母都非常大，跟第9，第10个解类似。
下面是方便阅览，处理过的前50个解的前3个最小的u值。
{422481, 414560, 217519, 95800} -> {-9/20, 1000/47, -1041/320}
{2813001, 2767624, 1390400, 673865} -> {-9/20, -1425/412, 5728/215}
{8707481, 8332208, 5507880, 1705575} -> {-29/12, 1865/132, 6280/1359}
{12197457, 11289040, 8282543, 5870000} -> {-93/80, -400/37, -2433/920}
{16003017, 14173720, 12552200, 4479031} -> {-136/133, 201/4, -1005/568}
{16430513, 16281009, 7028600, 3642840} -> {12185/432, 22529/2988, 79416/247889}
{20615673, 18796760, 15365639, 2682440} -> {-5/8, -477/692, 20824/2003}
{44310257, 41084175, 31669120, 2164632} -> {-817/660, -1581/1520, -12065/12396}
{68711097, 65932985, 42878560, 10409096} -> {-21021/9788, 10498601/138604, -13104000/15069437}
{117112081, 106161120, 87865617, 34918520} -> {-93514757/75615072, -431691625/11587212, -936262392/509548501}
{145087793, 122055375, 121952168, 1841160} -> {-361/540, 1861/240, -7800/5509}
{156646737, 146627384, 108644015, 27450160} -> {-136/133, 201/4, -1005/568}
{589845921, 582665296, 260052385, 186668000} -> {-5/8, -1617/200, -34272/4885}
{638523249, 630662624, 275156240, 219076465} -> {-5/8, -1617/200, -34272/4885}
{873822121, 769321280, 606710871, 558424440} -> {-12285/4112, -214309/129780, -403496/89145}
{1259768473, 1166705840, 859396455, 588903336} -> {-41/36, 9360/2371, -4061/16308}
{1679142729, 1670617271, 632671960, 50237800} -> {-9/20, 1000/47, -1041/320}
{1787882337, 1662997663, 1237796960, 686398000} -> {-93/80, -400/37, -2433/920}
{1871713857, 1593513080, 1553556440, 92622401} -> {-865/592, -14177/20156, -230472/438737}
{3393603777, 3134081336, 2448718655, 664793200} -> {-5/8, -477/692, 36696/8687}
{5179020201, 4657804375, 3971389576, 24743080} -> {553/80, -33400/19537, -294473/635180}
{12558554489, 11988496761, 7813353720, 4707813440} -> {233/60, 7584/54605, -216285/23504}
{15434547801, 15355831360, 5821981400, 140976551} -> {-9/20, -1425/412, 5728/215}
{39871595729, 36295982895, 29676864960, 11262039896} -> {-1376/705, 14337/340, -81065/89412}
{46055390617, 41714673255, 34169217200, 18125123544} -> {-41/36, 9360/2371, -4061/16308}
{64244765937, 55479193841, 52289667920, 17111129720} -> {-125/92, -936/5281, 1717941/427352}
{76973733409, 71826977313, 49796687200, 39110088360} -> {-1245/5012, 248521/62784, -267904/221337}
{521084370137, 435210480720, 372623278887, 369168502640} -> {135/1208, 1744/495, -977657/480240}
{597385645737, 544848079888, 443873167360, 142485966505} -> {-1152/2345, -15461/13160, 19005/3688}
{820234293081, 814295112544, 337210257575, 78558599440} -> {2265/184, -68256/135125, 975499/3009200}
{1059621884297, 1041572957760, 535914713672, 187577183625} -> {-361/540, 1861/240, -7800/5509}
{1367141947873, 1226022682752, 1047978087905, 408600530760} -> {-56/165, -383021/380940, 2644685/570612}
{1682315502153, 1657554153472, 801719896720, 468405247415} -> {-1152/2345, -15461/13160, 19005/3688}
{2051764828361, 2032977944240, 894416022327, 125777308440} -> {-29/12, 1865/132, -30768/57253}
{5062297699257, 4987588419655, 2480452675600, 502038853976} -> {-5/8, 20824/2003, -124529/68084}
{6014017311081, 6010589044544, 1313903832425, 66822832760} -> {-97/400, 78065/484, -61583704/7959505}
{6382441853233, 6310500741600, 2927198165920, 613935345969} -> {-3168/1553, -857/3696, 980785/175296}
{7082388012473, 5819035124295, 5611660306848, 4408757988560} -> {135/1208, 1744/495, -977657/480240}
{25866132798297, 23449050222680, 18776929334105, 12035933588696} -> {-125/92, -936/5281, 1717941/427352}
{26969608212297, 26901926181047, 8528631804200, 487814048600} -> {1873/200, -51416/9425, -3599825/50036084}
{27497822498977, 25762744660064, 19031674138785, 2054845288320} -> {-3168/1553, -857/3696, 980785/175296}
{29999857938609, 27239791692640, 22495595284040, 7592431981391} -> {-9/20, -4209/3500, 30080/6007}
{45556888578449, 43940127884360, 27546142170735, 7908038161032} -> {-29/12, 6280/1359, -3333/107368}
{58844817090201, 56329979520665, 34511786481280, 26636493544576} -> {-1041/320, -2830405/222976, -2561104512/2803746965}
{230791363907489, 220093974949320, 148739531603136, 32467583677535} -> {-1376/705, 14337/340, -81065/89412}
{573646321871961, 514818101299289, 440804942580160, 130064300991400} -> {-9/20, 34225/6692, -41952/33865}
{5380742305932201, 5352683902805120, 1841841620201576, 1554532675059625} -> {553/80, -33400/19537, -294473/635180}
{20249506709579721, 18565945114216720, 14890026433468471, 3579087147375440} -> {-5/8, -176752/157345, 4718261/816880}
{62940516903410601, 56827813308111785, 47886740272114976, 8813425670440240} -> {-5/8, 398113/66200, -3589408/3049765}
{87486470529871881, 87375622888246360, 21794572772239369, 16306696482461560} -> {-97/400, 78065/484, -61583704/7959505}
PARI/Gp的hyperellratpoints还是太慢了,计算一个指定的u在高度为10^8的其他v值需要跑44个小时. 我现在换成了ratpoints.用C语言调用，速度要快好几倍. https://github.com/MichaelStollBayreuth/ratpoints
不过还好,我花了两天不间断的穷举所有1000以下高度为10^6的所有u值,没有发现新的u值.
倒是漏掉了$-5/44$, 因为搜索到高度需要提升到10^8,这个需要单独分析.
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根据文件https://nestwhile.com/res/a4b4c4d4/reverse.txt 的统计结果.分析u的情况. 分子分母都小于1000的所有u只有19个(多出3个).其中$-\frac{9}{20}$最多,有8组$(u,v,w)$,每个$(u,v,w)$对应2个$(a,b,c,d)$解.
$-\frac{5}{8}\to 7,-\frac{9}{20}\to 8,-\frac{29}{12}\to 4,-\frac{41}{36}\to 3,-\frac{5}{44}\to 4,-\frac{93}{80}\to 3,-\frac{125}{92}\to 5,-\frac{136}{133}\to 4,-\frac{56}{165}\to 3,\frac{201}{4}\to 3,\frac{233}{60}\to 3,-\frac{400}{37}\to 2,-\frac{97}{400}\to 2,-\frac{361}{540}\to 2,\frac{553}{80}\to 4,-\frac{477}{692}\to 2,-\frac{817}{660}\to 2,-\frac{865}{592}\to 4,\frac{1000}{47}\to 3,-\frac{1005}{568}\to 2$
{-(5/8),-(9/20),-(29/12),-(41/36),-(5/44),-(93/80),-(125/92),-(136/133),-(56/165),201/4,233/60,-(400/37),-(97/400),-(361/540),553/80,-(477/692),-(817/660),-(865/592),1000/47}
{-(5/8)->{{-(5/8),-(1617/200),-(34272/4885)},{-(5/8),-(477/692),36696/8687},{-(5/8),20824/2003,-(124529/68084)},{-(5/8),398113/66200,-(3589408/3049765)},{-(5/8),-(176752/157345),4718261/816880},{-(5/8),4037701/712772,-(34666792/31592857)},{-(5/8),541388136/3857219,-(4250103489/1384328092)}},-(9/20)->{{-(9/20),1000/47,-(1041/320)},{-(9/20),-(1425/412),5728/215},{-(9/20),-(4209/3500),30080/6007},{-(9/20),34225/6692,-(41952/33865)},{-(9/20),61008600/1159319,-(68433257/17513240)},{-(9/20),-(107014216/29258425),210232185/6094384},{-(9/20),14486729065/814087256,-(29393447736/9584944225)},{-(9/20),1469114228933808/12826625923015,-(1691398818144025/403973326391908)}},-(29/12)->{{-(29/12),1865/132,-(30768/57253)},{-(29/12),6280/1359,-(3333/107368)},{-(29/12),-(189274425/1076672236),118669843216/20351175339},{-(29/12),-(9943745400/11841832129),-(746942047637/2082322752)}},-(41/36)->{{-(41/36),9360/2371,-(4061/16308)},{-(41/36),-(2441565/942392),-(22869016/1897173)},{-(41/36),-(16622002576128560/24373901295740877),197214290538785133/31571140813884476}},-(5/44)->{{-(5/44),57878913/12642040,-(2741924904/1401894085)},{-(5/44),4669000304/944254963,-(117620301817/53219719132)},{-(5/44),-(43000836761/25579904000),8541935778968/2036027368195},{-(5/44),-(343268956016/144380152505),28531188247669/5494554320180}},-(93/80)->{{-(93/80),-(400/37),-(2433/920)},{-(93/80),-(84237/359800),11502160/2925527},{-(93/80),-(616293201485641717/29739528734286760),-(6685594928073633840/3091584890628164593)}},-(125/92)->{{-(125/92),-(936/5281),1717941/427352},{-(125/92),10490417/84724,-(1877877296/1326891341)},{-(125/92),-(4887278104/1856675737),-(1023342388301/134640713696)},{-(125/92),8204718073/2152051820,-(179700100672/1421567294005)},{-(125/92),-(10001951064/4247679185),-(2232712366053/234335456480)}},-(136/133)->{{-(136/133),201/4,-(1005/568)},{-(136/133),-(511289/1551044),27660845/6860848},{-(136/133),23685689/3885556,-(63528125/85096232)},{-(136/133),-(4599374556397238049/210488523233668004),-(27853328093630687925/11390499849797373712)}},-(56/165)->{{-(56/165),-(383021/380940),210241305/52130512},{-(56/165),2644685/570612,-(216116793/168202160)},{-(56/165),451525338984813/101494987368460,-(35324615808268585/29389382407370664)}},201/4->{{-(136/133),201/4,-(1005/568)},{201/4,4372152/935219,7919435/17426416},{201/4,6210699/13897628,216566800/45486779}},233/60->{{233/60,7584/54605,-(6625405/2392764)},{233/60,-(216285/23504),20642296/53214885},{233/60,-(7428842654432/6035445436845),-(236274775097565/1227586895452028)}},-(400/37)->{{-(93/80),-(400/37),-(2433/920)},{-(400/37),1867333/457280,8685847/22963880}},-(97/400)->{{-(97/400),78065/484,-(61583704/7959505)},{-(97/400),78855705/1344892,-(60671827752/8724916985)}},-(361/540)->{{-(361/540),1861/240,-(7800/5509)},{-(361/540),-(6393885/10250072),215657416/51845475}},553/80->{{553/80,-(33400/19537),-(294473/635180)},{553/80,878775/1856708,94116264/11111135},{553/80,-(3344476868232/81390104425),13857166965855/54779944848244},{553/80,-(5530784621247221929/4621511133840351100),-(161622483046997673880/223409745703774700161)}},-(477/692)->{{-(5/8),-(477/692),36696/8687},{-(477/692),2136525/340912,-(1688076136/1490944633)}},-(817/660)->{{-(817/660),-(1581/1520),8583400/714723},{-(817/660),21792/5035,-(3622925/12225132)}},-(865/592)->{{-(865/592),-(14177/20156),5846168/682447},{-(865/592),-(230472/438737),60906561/9414892},{-(865/592),1153296374048673/229783858565692,-(24932273792436648/74686203064345969)},{-(865/592),41460743268273973816/7028502539237212727,-(596668135995361389497/1299567350985701388812)}},1000/47->{{-(9/20),1000/47,-(1041/320)},{1000/47,330353/48940,3521543/9580960},{1000/47,-(81640949939264532129/32571198102287520620),-(1510140049031003568921/2491254192967045854920)}},-(1005/568)->{{-(136/133),201/4,-(1005/568)},{-(1005/568),-(7516797/66665212),163242376/37855247}}}
认真研读了Noam D. Elkies的文章,https://doi.org/10.2307/2008781，或者这里下载: https://nestwhile.com/res/a4b4c4d4/On%20A4%20+%20B4%20+%20C4%20=%20D4.pdf
他是第一个给出$a^4+b^4+c^4=d^4$的解的人。
然后代码验证了一下，发现u 确实满足一个同余关系，虽然结构简单，但是定义复杂。记$u=\frac{2m}{n}，GCD(m,n)=1$,且n是奇数, 如果u的分母是偶数，就做变换$u \to \frac{2}{u}$，再取m,n. 因为这种变换只是改变了最终结果的正负号。
定义一个关于整数k的函数$R(k) =\frac{k}{p^2}$,$p$是使得$p^2$整除$k$的最大因子，比如$R(23)=23,R(-24)=-6,R(25)=1$.
有这个定义之后，对于$\frac{u}{2}$的$(m,n)$，存在一个必要的条件, 就是$R(2 m^2+ n^2),R(2 m^2 - 2 m n + n^2),R(2 m^2 - 4 m n + n^2),R(2 m^2 +2 m n + n^2)$都是模8余1,的素数乘积。
RK:=Module[{p=x},p/Product]^(2 Floor]/2]),{pp,FactorInteger}]];
ValidateU:=Module[{nm=x,m,n},If[(nm-2)^2>2,{m,n}=If,2]==0,{Denominator,Numerator},{Numerator,Denominator}];
Union[],8]&, {RK,RK,RK,RK}]]]=={1},False]];
goodu={};Block[{mm=100,n,m},Monitor==1,If,AppendTo];
If,AppendTo];
If,AppendTo];
If,AppendTo]],{n,mm},{m,n+1,mm}],{n,m}]];goodu
计算发现, 100以内的有98个,如下. 1000以内有6392个.
$-5/8,5/12,-15/8,-16/5,-16/15,-9/20,-1/20,24/5,8/25,25/4,-29/12,-24/29,-35/16,-35/24,-32/35,16/39,39/8,-40,-40/9,-39/40,-29/40,-11/40,3/40,-41/36,-15/44,-5/44,-45/4,-45/8,-45/32,-16/45,-8/45,-48/35,5/48,25/48,-55/12,-24/55,24/55,55/12,-27/56,-49/60,-31/60,-64/45,-63/64,-65/12,-65/32,-65/48,-64/65,-24/65,-72/41,-35/72,1/72,25/72,35/72,-75/44,-75/64,-65/76,-45/76,-15/76,-79/12,-79/60,-24/79,-80/11,-80/29,-80/39,-63/80,-23/80,-3/80,3/80,80/3,-53/84,-87/20,-40/87,40/87,87/20,-88/5,-88/15,-88/75,-3/88,-89/60,-91/20,-91/60,-40/91,40/91,91/20,-93/40,-93/80,-80/93,-95/36,-95/84,-72/95,-96/65,96/25,96/5,-99/4,-8/99,-99/100,-59/100,-33/100$