A^4 + B^4 + C^4 = D^4的正整数解

wayne · 发表于 2025-6-1 20:17:02

我已经统计过了，总共有$287$个=93(旧的)+186(新的)解，$437$个$u$值,构成246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}

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wayne · 发表于 2025-6-2 10:50:23

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.

------------------------------------------------
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}

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{-(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}}}

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wayne · 发表于 2025-6-2 19:54:42

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/a4b4c4 ... +%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[x_]:=Module[{p=x},p/Product[pp[[1]]^(2 Floor[pp[[2]]/2]),{pp,FactorInteger[p]}]];
ValidateU[x_]:=Module[{nm=x,m,n},If[(nm-2)^2>2,{m,n}=If[Mod[Denominator[x],2]==0,{Denominator[nm],Numerator[nm]},{Numerator[nm/2],Denominator[nm/2]}];
Union[Flatten[Map[Mod[FactorInteger[#][[All,1]],8]&, {RK[2 m^2+n^2],RK[2 m^2-2 m n+n^2],RK[2 m^2+2 m n+n^2],RK[2 m^2-4 m n+n^2]}]]]=={1},False]];
goodu={};Block[{mm=100,n,m},Monitor[Do[If[GCD[m,n]==1,If[ValidateU[n/m],AppendTo[goodu,n/m]];
If[ValidateU[-n/m],AppendTo[goodu,-n/m]];
If[ValidateU[m/n],AppendTo[goodu,m/n]];
If[ValidateU[-m/n],AppendTo[goodu,-m/n]]],{n,mm},{m,n+1,mm}],{n,m}]];goodu

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计算发现, 100以内的有98个,如下. 1000以内有6392个.
$-5/8,5/12,-15/8,-16/5,-16/15,-9/20,-1/20,24/5,8/25,25/4,-29/12,-24/29,-35/16,-35/24,-32/35,16/39,39/8,-40,-40/9,-39/40,-29/40,-11/40,3/40,-41/36,-15/44,-5/44,-45/4,-45/8,-45/32,-16/45,-8/45,-48/35,5/48,25/48,-55/12,-24/55,24/55,55/12,-27/56,-49/60,-31/60,-64/45,-63/64,-65/12,-65/32,-65/48,-64/65,-24/65,-72/41,-35/72,1/72,25/72,35/72,-75/44,-75/64,-65/76,-45/76,-15/76,-79/12,-79/60,-24/79,-80/11,-80/29,-80/39,-63/80,-23/80,-3/80,3/80,80/3,-53/84,-87/20,-40/87,40/87,87/20,-88/5,-88/15,-88/75,-3/88,-89/60,-91/20,-91/60,-40/91,40/91,91/20,-93/40,-93/80,-80/93,-95/36,-95/84,-72/95,-96/65,96/25,96/5,-99/4,-8/99,-99/100,-59/100,-33/100$

wayne · 发表于 2025-6-7 00:06:34

已知四次曲线的一个有理点,是可以拿到该曲线到Weierstrass 形式的双有理变换的, 苦于手头没有好的数学工具, 今天浏览帖子的时候发现Maple可以一条命令就能办到, 大大节省了手工来回拉扯的过程
我来示范一下:
就完全按照论文的符号来, 论文是求 $r^4+s^4+t^4=1 , r=x+y,s=x-y$的解,也就是下面两个圆锥曲线的有理参数解.
\[(2 m^2 + n^2) y^2 == -(3 n^2 - 8 n m + 6 m^2) x^2 - 2 (2 m^2 - n^2) x - 2 m n\]
\[ (2 m^2 + n^2) t^2 = 4 (2 m^2 - n^2) x^2 + 8 m n x + (n^2 - 2 m^2)\]
这个可以用eclib的solve_legendre 工具来做,直接返回$ ax^2 + by^2 + cz^2 = 0$的参数解

以$(m,n)=(8,-5)$为例,就是$72929 = (103 + 779 x)^2 + 119187 y^2,17009 == -15759 t^2 + 4 (-40 + 103 x)^2$

Solving ax^2 + by^2 + cz^2 = 0
Using method 4
Enter coefficients a b c: 1 119187 -72929
1 119187 -72929
Solution: (x:y:z) = (7953:20:39) --OK
x = [7953,119796,-641823] * [u^2,uv,v^2]
y = [-20,414,1504] * [u^2,uv,v^2]
z = [39,-12,3057] * [u^2,uv,v^2]
Disc(qx) = 34768754892
Disc(qy) = 291716
Disc(qz) = -476748
Parametric solution is OK
Enter coefficients a b c: 4 -15759 -17009
4 -15759 -17009
Solution: (x:y:z) = (3108:47:15) --OK
x = [12432,-9246,23280] * [u^2,uv,v^2]
y = [-188,304,239] * [u^2,uv,v^2]
z = [60,432,-273] * [u^2,uv,v^2]
Disc(qx) = -1072179324
Disc(qy) = 272144
Disc(qz) = 252144
Parametric solution is OK
Enter coefficients a b c:

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得到第一个方程的参数解是 $x\to \frac{2 (16 k^2+492 k-3889)}{19 (13 k^2-4 k+1019)},y\to -\frac{2 (10 k^2-207 k-752)}{3 (13 k^2-4 k+1019)}$
第一个方程的参数x代入第二个方程得到$t^2=(\frac{1}{57(1019 - 4 k + 13 k^2)})^2(2029662265 - 722880216 k - 20673106 k^2 - 2775528 k^3 - 493607 k^4)$

也就是说,我们需要解一个椭圆方程 $T^2=2029662265 - 722880216 k - 20673106 k^2 - 2775528 k^3 - 493607 k^4$ ,其中 $t=\frac{T}{57 (13 k^2-4 k+1019)}$

与此同时,其实我们已经得到了一个参数解.
\[(2(-2621 - 5409 k + 142 k^2))^4+(2 (-25955 - 2457 k + 238 k^2))^4 +t^4 = ( 57 (1019 - 4 k + 13 k^2))^4\]

我们知道,四次曲线如果有一个有理点,那么可以双有理变换到椭圆曲线. 所以,我们先找一个有理点看看, 这个可以通过很多工具,比如就用PARI/Gp, 找到了一个,$(77/358, 5546995679/128164)$,

(23:44:20) gp> f=-493607*k^4 - 2775528*k^3 - 20673106*k^2 - 722880216*k + 2029662265
%103 = -493607*k^4 - 2775528*k^3 - 20673106*k^2 - 722880216*k + 2029662265
(23:44:44) gp> hyperellratpoints(f,10^8,1)
%104 = [[77/358, 5546995679/128164]]

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于是接下来交给maple.

f := -493607*k^4 - 2775528*k^3 - 20673106*k^2 - 722880216*k + 2029662265;
Weierstrassform(-t^2 + f, k, t, K, T, [77/358, 5546995679/128164, 1]);

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得到对应的椭圆曲线Weierstrass形式是$K^3 + 17614011548453168/3*K + 7677736888784621012891264/27 + T^2=0$
其中$(K,T)=(722*(4011964193*k^2 + 388186326436*k - 46096917*t - 2078773529419)/(3*(128164*k^2 - 55132*k + 5929)), -27436*(231630938497595*k^3 + 3122275779742396*k^2 + 1224460941665*k*t + 157610415793579037*k - 12793960525688*t - 576376155229480472)/(45882712*k^3 - 29605884*k^2 + 6367746*k - 456533))$
逆变换就是$(u,t)=((124047*K^2 + 420405791530188*K + 49922961111*T - 9630079226516844064736)/(576738*K^2 - 8689914442038*K + 2165795195248168637816), (14396234372617959*K^4 + 216913130373890168792109*K^3 - 243541953212329054530*K^2*T + 91391074533383715206874049283748*K^2 + 11645386059055642374445745040*K*T + 15101318716970394594113412306361210828848*K + 826828278802105544855645036810617440*T - 440776135732756899795546344935759487514847376704)/(332626720644*K^4 - 10023607750944224088*K^3 + 2573707391644017172535041860*K^2 - 37641149891367141274456312293818016*K + 4690668827760052911724763861035448985249856))$

此时继续交给PARI/Gp,得知是一个 rank=3的曲线.

(00:03:24) gp> E=ellinit([17614011548453168/3,-7677736888784621012891264/27])
%106 = [0, 0, 0, 17614011548453168/3, -7677736888784621012891264/27, 0, 35228023096906336/3, -30710947555138484051565056/27, -310253402829041569074573489236224/9, -281824184775250688, 245687580441107872412520448, -47885553668097585693739392779132998688955555643392, 9644765071369409460713207083952/20632952982319216650583523847, Vecsmall([1]), [Vecsmall([128, -1])], [0, 0, 0, 0, 0, 0, 0, 0]]
(00:17:07) gp> ellrank(E)
%107 = [3, 3, 0, [[2080126403/12, 19501061065057/8], [37232828108/75, 1397417614460784/125], [453368159597913604/2485122201, 324440344847642464196403184/123885826842051]]]
(00:17:19) gp> ellratpoints(E,10^8)
%108 = []
(00:17:37) gp> ellratpoints(E,10^10)
%109 = [[115862228/3, 0], [183658028/3, 551830693680], [183658028/3, -551830693680], [2080126403/12, 19501061065057/8], [2080126403/12, -19501061065057/8]]
(00:17:51) gp>

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代入ellratpoints的命令会得到5组解,如果再运用群加法,乘法运算.就可以产生无数多个

{20615673,18796760,15365639,2682440}
{589845921,582665296,260052385,186668000}
{638523249,630662624,275156240,219076465}
{3393603777,3134081336,2448718655,664793200}
{5062297699257,4987588419655,2480452675600,502038853976}
{20249506709579721,18565945114216720,14890026433468471,3579087147375440}
{62940516903410601,56827813308111785,47886740272114976,8813425670440240}
{108593344076382641697,107238802094189542120,46196947347028916440,38751631463616255521}
{19874054816411213708481009,19270755733101284410120384,11389900458885552539102735,5967420362778572362681840}
{133140691304639620846181457,129410861225043592041256520,74522041242387759937530799,41328162329293632574512440}
{1677479490238223823661446513,1507524066882038472584786800,1288056982586427591062203384,169218021322170204480680305}
{17644444348539480178025528601,17155429148630710395388779200,8825302955667506173409411815,8069533957326324451238272976}
{333129191568549251867199876057,303237986735307793676413527001,249130710645573788837883103720,52942590681106640544280040360}
{772094261800702773645712832721,760582529580318898681536071471,379334234716115837893323648560,66039836886902596560454161520}
{123946497886751603284917774006201,112194490587542543850845554026640,93833840730275457894522336118855,14274433969472603571500233839424}
{319280388704872568697808021788921,284600416391998689341341493583440,248781160855666365777930155696720,27240579159414963163833435822329}
{1396680473961341619647008005444513,1376543993038340844176274474873560,679990789464791197416063781213640,175424290963444842846031401841759}
{102597776485948701261490206779151828297,101347936969652305104608658289319870960,45723546709411652690271893833764888905,31058706146444468337573470216746671304}
{3169834808503132719068553468475898078793,3131108479640288781452874540720691424240,1381599778333161751111622711637871247945,1046738335539830238434807071109719218424}
{3209362476706814620541224955667144860121,3128294165330114744727797686810591297040,1739791427884216975168428630991483380185,1037354240745377010703026223898481485504}
{3205776133114443285786355494410044071015561,2942894144172212096782850563589047020019936,2351269049425087935841789662825005390318455,464024162889108246965771239428843936418960}
{31295958743027261374800168000606313319708062233,30542792711349255243090311315337919056425054695,16701371834728694354127181708107566951768007656,10301089588007086166857838935541176607862672400}
{119429377346393956325881294067788995970216498737,116155990279165125223764999207442763432438641384,57537351751926091192521443554403022567670623695,56848660621859333145298924209778474490790862960}
{5196008642592516875040620356115903270979245430874473,5011406205804748063883823789442281041594280799088848,3111552841717764104570537584587526845337953714233600,1453241756064660673368677836731233217930619882758505}
{492100246491184020435636988062827204434589033409534009,454460688173458759776298219620339662114310775063841735,355375216107651347984610759793140665567419635128329112,77827881217928043683340226848408595994367228820509200}
{5538946580537011600955153605791371848468489663912069617,5374800465439621405401868069617228231690678463718197745,2964373802228122968144593760590434631833690808975024080,2330404161205095051764270313005447210423215510409268992}
{31024765492981559366297173502703134823879900509853630321,30572141690679593197701890512129458786421585375454134415,15154562106530515844690398587562619929534804273729840304,3500095483049736783209480752013987513707527048270965440}
{268708082360732759200134338534795054472395664807408795249,251329254313301249758861141239301766208760919501589137807,186430306658192152586178723325107121291479804686544436280,62676171925266314595870430754611039090291189884712209880}

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我们可以看到, {{17644444348539480178025528601,17155429148630710395388779200,8825302955667506173409411815,8069533957326324451238272976}}
这组解只有29位数,前面却从未出现过

wayne · 发表于 2025-6-7 14:15:10

继续,用$-9/20$

{{422481,414560,217519,95800}}
{{2813001,2767624,1390400,673865}}
{{1679142729,1670617271,632671960,50237800}}
{{15434547801,15355831360,5821981400,140976551}}
{{29999857938609,27239791692640,22495595284040,7592431981391}}
{{573646321871961,514818101299289,440804942580160,130064300991400}}
{{101783028910511968041,99569174129827461335,54488888702794271560,21710111037730547416}}
{{120175486227071990769561,118508989446504950664160,56915898438422390129561,30248376090268690676600}}
{{1171867103503245199920081,1165970778032514255823760,440517744543240750721000,59421842165791512201169}}
{{6714012701109174954871521,6632467268281371571709360,3057432874236989781768479,1758067984180618846616200}}
{{21291952935426564624339201,20991236668646283695879935,10137374115207940432133560,5328636655728999148343576}}
{{227529118288906398066378489,226369052354324181334408840,85818832944459457142858489,2650718685573298353948640}}
{{10071617417759697610768412961,9242432327555089678593151711,7357036914591014461697267840,2816483748444464612495572520}}
{{17603812043504825450936682974289,15598034926522765354653857821400,13839507404456990114977553185711,3532227115076629388512728513640}}
{{3008737176230697819860120693587761,2627579709037322604373214365907761,2418374996832273585658338775101320,522826293046399380528268068283040}}
{{41012055337388705074554891139109121,36317909300592438507304859539931960,32273356417270580777406624317686504,8182525316629537977427894696507135}}
{{3395929891485334592370746651336421162321,3360388888883721642901019519945403343304,1475840549437269376782395216972838074000,926430612195290681941470814843189475665}}
{{433237181772556812898093275295697053740921,426119680281593866555377364468799176452640,215220663434500635984802398839688262060921,103127426991367923640113970772826280170680}}
{{8234762698423310578177015206662106449800564521,7620654701590271369073375691220522430753106729,5869555596785291641235700962130096065667529360,2496156394991961814239989060375233511277443800}}
{{38807308867109469306956130726405668929338587841,32920720418680293851782448531772814736860986760,32333025303302398783385143300102444875640407784,4906600101468927788661521948711588613529046335}}
{{354223669723977663139636004337272257099943181681,304748412501689376160113006993566248711444766080,290391283508845307152126235858671694598783125135,52564968895506015620475576200306150103474063944}}
{{3321572075009649001252611962106821513195063671209,3044379071273738585661144211705474716390151361000,2434252984408818343348925958795872446528017288791,918149918090947999552526356245666874385879372840}}
{{44640332089738322879347416308058725014037968637361,38477261102375419735275117572596404291785366402639,36511465811096491343440437522851763344890650557000,6760576369138924038289164408122673076473767214560}}
{{157341154218709859407116068479294717217914375135641,153381124820118869842963717744906937232275285497840,87435338364756041046905452942145613144742063455641,31337892518719453266175352452895421253414154076680}}
{{3588623560598142777526172807573639343778899922199721,3218876248203736317419479976410531013740491745640279,2760445716263703194599316420303453794834286756668600,809534089661784709276942569922420793182887541424560}}
{{7391353089627115257000084224457434424867964121346129,6470286811095237371333263370425876781155279154187440,5920933603962933496303983720083210169720252285431240,1315589250476794811585538444139374655093854779773871}}
{{32617684263565631089446501467507673912064014878981961,28947616610519179092032233980814017214560849022870711,25574300468084403875439819835101848119867580348399440,6647246502876352485677589335029454214703542712688920}}
{{1311616112859929755700825115296260387898060942012629929,1305103244791284518424986102510919043646182874464879240,491315051764821882627314642651532020371798053415509929,93781685901038260795253986106902196918876446390407440}}
{{8235271459752462911852981823274821485502436055819214201,8137050158107600479751960084169379745745742498491148935,3729480872463428815729575435665664894411364523133402040,2167607088774192831123139412231642285819649557442943576}}
{{624380409806176954969077437952432432851839431776907756201,608938552078229708809879754062278952791826092256533324424,345436871423933023677280495869318420176244540181898546800,125449033002908525332825565514009534715378298302955101865}}
{{5426643657694933001523533947641882548868116262351645507489,5027442226090570376951763739173405539273910350100601343720,3854508919681912946055428807967218248560524007797742012511,1662535960872050360795965715910917411778837772709047838440}}
{{21516758415285569567902156218168383559690924809793252600561,21407605685252770576127908200793552966431570463881880087040,8105387621700424289778533041762056144757341086230393214680,697900214749932448285980753035319890458151052558375639311}}
{{32765458201929412641416034051435520639258357113742031670969,30582196668976524718377935182763823727592320140313909866640,22671119786961601373534221417739977913860239743121836709640,10810393143175239019071959109997663051929470074089157449031}}

复制代码

wayne · 发表于 2025-6-7 14:56:51

$-\frac{29}{12}$,秩为3

{8707481,8332208,5507880,1705575}
{2051764828361,2032977944240,894416022327,125777308440}
{45556888578449,43940127884360,27546142170735,7908038161032}
{1343040355606487633849060057,1202764577995511968722134745,979639439973824041716201624,699747942023521949377658320}
{12789647316029180654491744889,10580789876341193642008366840,10075555293380438521117740423,7911452233520044597538909640}
{40091546304694547841458375009,33576973488820972138923737112,30772298816749246194243925840,25392977662178650875395898015}
{70972506014932106900783284553,63561815312626498797940120905,53017144215667916424586256080,32741549977222806725829670104}
{8229646208430186915167706621881,7918174790330642885254291363640,5060779866902849712056608508592,425728077239657769125737790535}
{3762098874072146335431610024521161,3544724218928041607923371624299240,2511945393847796136912099490227552,1272617115110814812810988604290615}
{170375738319562822083708009930722214953,139183112313479824175110702256770883625,134644247306144612020738696242691402696,108519048033033937503228098331137860560}
{46256659821645525971234479881532515122129,41694683193231784023124900596067427074080,33484228917937394360883337727034561008728,23383022089376719961027087418506347644975}
{248547976067941028054155149128578217948081,196483973911659350887491261144061886959695,187898370812851843437574251710226474868008,181255963562210317551905541624169962744400}
{12400568517887174400198835901771772592151393,11583616867102513630776646503254029423530849,8460171659207818435298706355571331404595960,4773563898794384867094094946452997144841400}
{38762408830447155460522443500641472940343119507429601651937489,38668956374978631417687859822178380922200014722686929525080600,11205772656597819490920718538763914977416432397027037669104248,8773471186714106315848600742986728683910318773328287624689455}
{59930045536515867271057206845380289982799963744094964715489313,49606810546285944889094274770312452476847763586838039767269409,46252131124462025144807732526050905498214535075503896032508920,38805012981129581795079185441593754860322970040855049042235880}
{274687667868337000880285757506705416134068713460844041674185291281,248116570631013032284594444927467441430183967639491386748281417760,202857389500971407484926599660543831435174879804439737277143404015,120369925693356309106746198757370033598676512538272188516063485608}
{424142858725535740342032657766406881319966471838147332181167437481,388783243151642451735400039642077244737270346555959264581359388000,305709241345848240076328645039097379451955620634584935115546062167,167199238910635862794155398847314687259943277182503169264507864680}
{14381694310647980458407748516805869161149093111258509481058826760129,11401069285453185624252021596143668210475145327578208729051264654728,10744091360835112329909660904661527544579580382688682786390058570815,10586093929951345363040617510339895685758158725752268776323768007760}
{9530560210021360991085939380712294691011622383713725961442465119092817,8326028173071513267601107281206453461464237784288588702520703839701056,7119215763404631813174329362422535654575826391878743736466961221879160,5440250428548181551977375622800450079524282126080478814679210348575825}
{12372291259716855697152644820685124906513677563507209260899851847138233,11766767331858204467002867908126579071680051232853412870144428999858105,8035995976013946894499253447690948895658711168561032158148056202149656,3088728680097810544847206662453572589174378261195869657992226682508320}
{122403974227477951601716635423337067894952762555519091318580893332495692153,106271818364113415471829299407643113483067896670796920335046183195095652120,94207526107180885352171080770154281106370261695558274354310838945063863096,65286997243623564347015572811808416426418686803443772340104728004665469305}
{1333446782548670040782200662039509936775871678958654221183501027512052398424337,1058992261961681763094724607738520517203171012731052113165401894527775800577880,1041287216134791208004625033847761661148514807720136476571490033659505339497280,923775617214701627118609219882649898425581993938449634798318977559224911265041}

复制代码

wayne · 发表于 2025-6-7 17:50:19

u=-5/44的解.秩为3

{2778996090487120353,2556827383749699103,2024155336530384440,585715960903147640}
{20234461127553384633,19399184483902029008,12696186158476139705,2329747842666412840}
{3037467718844497770129,2877363855098380947880,2016612085130087009647,444897078221606141840}
{24504057146788194291849,23896480714429616100215,13623248018235893097232,1519814187310380835480}
{12137566662146665711886365252737,10460832656100181686547108582112,9872001147462059766910335739775,3898022937032042181537566499160}
{1166771203657741610801659018209393,1145400860459371360702253709703768,602860664444767185991355248844720,43015381264158179520049525808015}
{2985772999602784901636123486486649,2693763654464217358310238734024120,2269321665966462462485146483298632,739419769536770231012635302250375}
{6247103423292859699056869181299776761,6186967282345145998802943536561096200,2757331153546626033198927459840538873,28450385633108239635917565180981520}
{423066479438649765324959257878721918334214139854426777153,370406808781650641622782334159681517099042386660186241760,330968558202827741922514068162826245733445877829285021528,186603605088416118574786927633979678864769149731161710015}
{93462920614420217292227799597009282209218836114402628411161,78309748056632337038525961018304225042498242435643749858073,78195610807491710558856189636395708898435589339286372046040,33840741461223401964108133317521842712655096159022100155040}
{2123648872014321729434379401467859232555361713033258388885857,2109824630163430370972925020908893991236971439917181478285232,850991710247011297168221072329056123819031867337248807891615,50005113972779813078221763936693016498167960114724815630840}
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复制代码

wayne · 发表于 2025-6-7 17:59:01

u= -41/36,秩为3

{1259768473,1166705840,859396455,588903336}
{46055390617,41714673255,34169217200,18125123544}
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复制代码

wayne · 发表于 2025-6-7 18:43:46

-93/80, 这个比较特殊, 其他都是秩为3, 这个是秩为2.

{12197457,11289040,8282543,5870000}
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复制代码

wayne · 发表于 2025-6-7 18:49:19

u = -125/92, 秩为3,

{64244765937,55479193841,52289667920,17111129720}
{25866132798297,23449050222680,18776929334105,12035933588696}
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复制代码

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[讨论] A^4 + B^4 + C^4 = D^4的正整数解

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