A^4 + B^4 + C^4 = D^4的正整数解

wayne

太神奇了, 真没想到这题还有后续. 我也看懂思路了,问题是求$x^4+y^4+z^4 = 1$ 的有理解.
记$f(x,y,z)=\frac{(x-y)^2-z^2-1}{(x-y)+x^2-x y+y^2}$, 那么,可以设置三个新的变量$u=f(x, y, z),v=f(y, z,x),w=f(z,x, y, )$,
于是存在关系 \[u v w-2 u-2 v-2 w+4 = \frac{(1 + x^2 + y^2 + z^2) (-1+x^4 + y^4 + z^4)}{(x - y + x^2 - x y + y^2) (y - z + y^2 - y z + z^2) (z - x + x^2 - x z + z^2)}=0\]

用$u,v$表达$x,y,z$,发现,只需要解一个方程就行

1. func=Function[{x,y,z},((x-y)^2-z^2-1)/(x^2-x y+y^2+(x-y))];
   
2. First@Solve[{func[x,y,z]==u,func[y,z,x]==v,1==x^4+y^4+z^4},{x,y,z}]//Factor
   
3. Collect[-48+32 u-16 u^3+4 u^4+32 v+32 u v-64 u^2 v+48 u^3 v-8 u^4 v-64 u v^2+48 u^2 v^2-16 u^3 v^2-16 v^3+48 u v^3-16 u^2 v^3+8 u^3 v^3-4 u^4 v^3+4 v^4-8 u v^4-4 u^3 v^4+u^4 v^4,v,Factor]
   

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\[
\begin{align}
D^2 = 4(-6 - 2u + u^2)(2 - 2u + u^2) - 8(-2 - 4u + u^2)(2 - 2u + u^2)v - 16u(4 - 3u + u^2)v^2 - 4(4 - 12u + 4u^2 - 2u^3 + u^4)v^3 + (4 - 8u - 4u^3 + u^4)v^4
\end{align}\]

\[\begin{align}
x &= \frac{P_1\pm(u^2-2u+2)(v^2-2v)\sqrt{D^2}}{P_0+D^2}\\
y &= \frac{-(P_1+P_2+P_3)\pm 2(u+v-2)(u+v-uv)\sqrt{D^2}}{P_0+D^2 \quad}\\
z &= \frac{P_3\pm(v^2-2v+2)(u^2-2u)\sqrt{D^2}}{P_0+D^2}
\end{align}
\]
\[\begin{align}
P_0 &= (2 + u^2)(2 + v^2)(12 - 8u + 2u^2 - 8v + 2v^2 + u^2v^2)\\
P_1 &= (-4 + 4u + 2v - u^2v)(8u - 4u^2 + 4v - 8 u v + 2u^2v - 4v^2 + 2v^3 + u^2v^3)\\
P_2 &= 2(- 4 + 4u + 2v - u^2v)(4 - 2u - 4v + u v^2)(-u + v)\\
P_3 &= (4 - 2u - 4v + u v^2)(4u - 4u^2 + 2u^3 + 8v - 8 u v - 4v^2 + 2 u v^2 + u^3v^2)\end{align}\]

最终,我们只需要找到方程$(1)$的有理解就行. 对于每一组有理解$(u,v,D)$,原题都可以双有理变换到一个椭圆曲线,每个椭圆曲线构成一族无穷组有理解.

有人根据已知的解逆向反推,再结合暴力搜索,找到了16个较小的u的解

\[\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{#} & u & v & &\text{#} & u & v\\
\hline
1 & -\dfrac{9}{20} & \; -\dfrac{1041}{320} & & 9 & -\dfrac{41}{36} & -\dfrac{4061}{16308}\\
\hline
2 & -\dfrac{29}{12} & \;\dfrac{1865}{132} & & \color{blue}{10} & -\dfrac{5}{44} & \dfrac{57878913}{12642040}\\
\hline
3 & -\dfrac{93}{80} & -\dfrac{400}{37} & & \color{blue}{11} & +\dfrac{233}{60} & \;\dfrac{7584}{54605}\\
\hline
\color{red}4 & -\dfrac{400}{37} & -\dfrac{93}{80} & & \color{blue}{12} & -\dfrac{56}{165} & -\dfrac{383021}{380940}\\
\hline
5 & -\dfrac{136}{133} & +\dfrac{201}{4} & & \color{blue}{13} & -\dfrac{125}{92} & -\dfrac{936}{5281}\\
\hline
\color{red}6 & +\dfrac{201}{4} & -\dfrac{136}{133} & & 14 & -\dfrac{361}{540} & +\dfrac{1861}{240}\\
\hline
7 & -\dfrac{5}{8} & -\dfrac{477}{692} & & 15 & -\dfrac{817}{660} & -\dfrac{1581}{1520}\\
\hline
\color{red}8 & -\dfrac{477}{692} & -\dfrac{5}{8} & & 16 & -\dfrac{865}{592} & -\dfrac{14177}{20156}\\
\hline
\end{array}
\]
根据这个链接,https://math.stackexchange.com/q ... 4z4-1-come-in-pairs, 又补充了第17组解$(u,v)=(\frac{553}{80},-\frac{33400}{19537})$

验证代码

1. func=Function[{x,y,z},((x-y)^2-z^2-1)/(x^2-x y+y^2+(x-y))];
   
2. Factor[2(func[x,y,z]+func[y,z,x]+func[z,x,y])-func[x,y,z]func[y,z,x]func[z,x,y]-4]
   
3. Block[{u=553/80,v=-33400/19537},Solve[d^2==4 (-6-2 u+u^2) (2-2 u+u^2)-8 (-2-4 u+u^2) (2-2 u+u^2) v-16 u (4-3 u+u^2) v^2-4 (4-12 u+4 u^2-2 u^3+u^4) v^3+(4-8 u-4 u^3+u^4) v^4,d]]

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wayne

从目前整理的信息来看, 所有人都不确定是否还有更小的解. 没准咱们基于这个思路 能够 捡到漏, 调整一下这93个解的排位
https://math.stackexchange.com/q ... 4b4/4857107#4857107

1. 422481; 414560, 217519, 95800 (Frye, 1988); C1, u = -9/20.
   
2. 2813001; 2767624, 1390400, 673865 (MacLeod 1997); C1, u = -9/20.
   
3. 8707481; 8332208, 5507880, 1705575 (Bernstein, 2001); u = -29/12.
   
4. 12197457; 11289040, 8282543, 5870000 (Bernstein, 2001); u = -93/80, -400/37.
   
5. 16003017; 14173720, 12552200, 4479031 (Bernstein, 2001); u = -136/133, 201/4.
   
6. 16430513; 16281009, 7028600, 3642840 (Bernstein, 2001); u = 12185/432.
   
7. 20615673; 18796760, 15365639, 2682440 (Elkies, 1986); C0, u = -5/8.
   
8. 44310257; 41084175, 31669120, 2164632 (Gerbicz, 2006); u = -817/660.
   
9. 68711097; 65932985, 42878560, 10409096 (Gerbicz, 2006); u = -21021/9788.
   
10. 117112081; 106161120, 87865617, 34918520 (Gerbicz, 2006); u = (big).
    
11. 145087793; 122055375, 121952168, 1841160 (Rathmann, 2007); u = -361/540.
    
12. 156646737; 146627384, 108644015, 27450160 (Rathmann, 2007); u = -136/133.
    
13. 589845921; 582665296, 260052385, 186668000 (Tomita, 2006); C0, u = -5/8.
    
14. 638523249; 630662624, 275156240, 219076465 (MacLeod, 1998); C0, u = -5/8.
    
15. 873822121; 769321280, 606710871, 558424440 (GDRZ, 2007); u = -12285/4112.
    
16. 1259768473; 1166705840, 859396455, 588903336 (GDRZ, 2008); C5, u = -41/36.
    
17. 1679142729; 1670617271, 632671960, 50237800 (Tomita, 2006); C1, u = -9/20.
    
18. 1787882337; 1662997663, 1237796960, 686398000 (GDRZ, 2007); u = -93/80.
    
19. 1871713857; 1593513080, 1553556440, 92622401 (GDRZ, 2007); u = -865/592.
    
20. 3393603777; 3134081336, 2448718655, 664793200 (Tomita, 2007); C0, u = -5/8.
    
21. 5179020201; 24743080, 3971389576, 4657804375 (Fulea, Feb 2024); u = 553/80.
    
22. 12558554489; 11988496761, 7813353720, 4707813440 (Bremner, 2015); u = 233/60.
    
23. 15434547801; 15355831360, 5821981400, 140976551 (Tomita, 2007); C1
    
24. 39871595729; 36295982895, 29676864960, 11262039896 (Tomita, Feb 2024)
    
25. 46055390617; 18125123544, 41714673255, 34169217200 (Tomita, 2024); C5
    
26. 64244765937; 52289667920, 17111129720, 55479193841 (Piezas, 2024); C9, u = -125/92
    
27. 76973733409; 39110088360, 49796687200, 71826977313 (Tomita, Feb 2024)
    
28. 521084370137; 372623278887, 435210480720, 369168502640 (Tomita, Feb 2024)
    
29. 597385645737; 443873167360, 142485966505, 544848079888 (Tomita, Feb 2024)
    
30. 820234293081; 78558599440, 814295112544, 337210257575 (Tomita, Feb 2024)
    
31. 1059621884297; 535914713672, 1041572957760, 187577183625 (Piezas, 2024)
    
32. 1367141947873; 1226022682752, 1047978087905, 408600530760 (Bremner, 2015)
    
33. 1682315502153; 468405247415, 1657554153472, 801719896720 (Tomita, Feb 2024)
    
34. 2051764828361; 125777308440, 894416022327, 2032977944240 (Tomita, 2024)
    
35. 5062297699257; 4987588419655, 2480452675600, 502038853976 (Tomita, 2008); C0
    
36. 6014017311081; 66822832760, 1313903832425, 6010589044544 (Fulea, Feb 2024)
    
37. 6382441853233; 2927198165920, 613935345969, 6310500741600 (Tomita, Feb 2024)
    
38. 7082388012473; 4408757988560, 5819035124295, 5611660306848 (Tomita, Feb 2024)
    
39. 25866132798297; 23449050222680, 18776929334105, 12035933588696 (Piezas, 2024); C9
    
40. 26969608212297; 487814048600, 8528631804200, 26901926181047 (Fulea, Feb 2024)
    
41. 27497822498977, 19031674138785, 25762744660064, 2054845288320 (Tomita, Feb 2024)
    
42. 29999857938609; 27239791692640, 22495595284040, 7592431981391 (Tomita, 2006); C1
    
43. 45556888578449; 27546142170735, 7908038161032, 43940127884360 (Tomita, 2024)
    
44. 58844817090201; 34511786481280, 56329979520665, 26636493544576 (Tomita, Feb 2024)
    
45. 230791363907489, 148739531603136, 32467583677535, 220093974949320 (Tomita, Feb 2024)
    
46. 573646321871961; 514818101299289, 440804942580160, 130064300991400 (Tomita, 2008); C1
    
47. 5380742305932201; 1554532675059625, 1841841620201576, 5352683902805120 (Fulea, Feb 2024)
    
48. 20249506709579721; 18565945114216720, 14890026433468471, 3579087147375440 (Tomita, 2008); C0
    
49. 62940516903410601; 56827813308111785, 47886740272114976, 8813425670440240 (Tomita, 2008); C0
    
50. 87486470529871881; 16306696482461560, 21794572772239369, 87375622888246360 (Fulea, Feb 2024)
    
51. 103117303193818953; 4092004076331400, 24975412054750025, 103028409596553328; (Fulea, Feb 2024)
    
52. 481334894209428521; 343651286746207896, 438980913824794665, 225712385669145920 (Piezas, 2024)
    
53. 1592672455342770513; 208032601069058735, 851144034922098880, 1559028675188874616 (Piezas, 2024)
    
54. 2778996090487120353; 2556827383749699103, 2024155336530384440, 585715960903147640 (Bremner, 2015); C7
    
55. 10816708329115215113; 9585769407872803575, 8510180374729994520, 985735303963754488 (Piezas, 2024)
    
56. 20234461127553384633; 19399184483902029008, 2329747842666412840, 12696186158476139705 (Tomita, 2024); C7
    
57. 77107030404994920297; 69320669852667799672, 38320435200564613600, 56375727168307546985 (Tomita, 2024); C9
    
58. 101783028910511968041; 99569174129827461335, 21710111037730547416, 54488888702794271560 (Piezas, 2024); C1
    
59. 108593344076382641697; -46196947347028916440, -107238802094189542120, 38751631463616255521 (Bremner, 2024); C0
    
60. 202540855134365138633; 201236910265023650505, 705558147137161920, 80940380256877627544 (Piezas, 2024)
    
61. 228746036963039501833; 163180699054891578792, 84616109521023161865, 210878774189729581880 (Piezas, 2024)
    
62. 375075545025537358721; 335981923744570504065, 188195571677171463096, 275897431444390465240 (Piezas, 2024)
    
63. 1671674986261410994097; 1199828498161126807800, 1534990269771364822095, 655960628418767673472 (Piezas, 2024)
    
64. 2711100675240842912689; 1977857900813232827064, 1617105485720597938520, 2376217986337223238735 (Piezas, 2024)
    
65. 3037467718844497770129; 2877363855098380947880, 444897078221606141840, 2016612085130087009647 (Tomita, 2024); C7
    
66. 4069249774250960557713; 3819055879832290430609, 842141328509923524200, 2794258267049888616280 (Piezas, 2024)
    
67. 9649219915259253551497; 7599957410902753037705, 8407785501400674212160, 4280294741983707700872 (Tomita, Feb 2024)
    
68. 10739931407728904606857; 772654695228940017240, 10320518856970101984393, 6653143628547990852040 (Piezas, 2024)
    
69. 11305555143522867817873; 10539980352556633840239, 7799922278924748599160, 4141571237269338150920 (Tomita, 2024); C5
    
70. 12214291847502204701241; 7745659501403353894384, 2120589250533219579335, 11684173258429439467360 (Tomita, Feb 2024)
    
71. 17503689286309573964097; 15876595946759369395903, 7188470920864810763360, 12896301483090810351440 (Tomita, 2024)
    
72. 18276027741543869996617; 13226266181198583365544, 16841033682021117865520, 4780632380106855105975 (Tomita, 2024); C5
    
73. 24504057146788194291849; 1519814187310380835480, 23896480714429616100215, 13623248018235893097232 (Tomita, 2024); C7
    
74. 29998124444432653523113; 12036780855644297767488, 23415987016826083521705, 26432693245716083746520 (Tomita, Feb 2024)
    
75. 120175486227071990769561; 30248376090268690676600, 118508989446504950664160, 56915898438422390129561 (Tomita, 2024); C1
    
76. 409840652625395469143913; 381461080909525552802665, 168213921178037201816584, 281048473715879152495040 (Tomita, 2024)
    
77. 587020625514136613276553; 179164925544119666072000, 222787467202130880567415, 582653975191641098286104 (Piezas, 2024)
    
78. 864745895259187110399737; 57810716855047169409080, 591519768111748983750888, 812937165464036006213895 (Piezas, 2024)
    
79. 1088768337585323892067521; 76399836994875695614145, 974071910293355929264000, 842960029363955380661896 (Piezas, Feb 2024)
    
80. 1171867103503245199920081; 1165970778032514255823760, 440517744543240750721000, 59421842165791512201169 (Tomita, 2024); C1
    
81. 1779979592349189232414713; 1724845107301282006322000, 574585584668340612894713, 1018986340666195845813760 (Tomita, 2024)
    
82. 1796867575393608033006561; 21904850878998429166561, 1771894249938641198780200, 867970652747799735398360 (Piezas, 2024)
    
83. 3122849928997768901912409; 1891988836723177605880960, 985329200220584284726784, 3003225858017812695181145 (Tomita, Feb 2024)
    
84. 6714012701109174954871521; 1758067984180618846616200, 6632467268281371571709360, 3057432874236989781768479 (Tomita, 2024); C1
    
85. 8997319881974346759473697; 8281143989708209432415360, 1749772249172099623115896, 6550300128305909879699935 (Tomita, 2024)
    
86. 19874054816411213708481009; -5967420362778572362681840, -19270755733101284410120384, 11389900458885552539102735 (Bremner, 2024); C0
    
87. 21291952935426564624339201; 5328636655728999148343576, 20991236668646283695879935, 10137374115207940432133560 (Tomita, 2024); C1
    
88. 31293260543726494476580617; 27386104940472276169105720, 25024939958628554701755145, 8089277164034877786318544 (Bremner, 2015); C9
    
89. 34497456764264994703368889; 12209879806944320496330055, 26621272474250391413865480, 30730370351168229154149048 (Tomita, Feb 2024)
    
90. 96242977191578497031965033; 47172089378698523207965335, 26409847035187091768472744, 94680315476024517009462320 (Tomita, 2024); C5
    
91. 133140691304639620846181457; 129410861225043592041256520, 41328162329293632574512440, 74522041242387759937530799 (Bremner, 2024); C0
    
92. 227529118288906398066378489, 85818832944459457142858489, 226369052354324181334408840, 2650718685573298353948640 (Piezas, 2024); C1
    
93. 452835938257547709708389177, 451622371501239854889723705, 10453265194894185904695360, 145561707582919191804801464 (Piezas, 2024)

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wayne

不过, 我自己也有一个思路, 对于$a^4+b^4+c^4=d^4$, 只考虑$gcd(a,b,c,d)=1$的情况,那么一定是一对偶数,一对奇数. 于是,不妨设$p\leftarrow\frac{a+b}{2},q\leftarrow\frac{a-b}{2},u\leftarrow\frac{c+d}{2},v\leftarrow\frac{d-c}{2}$,
得到 $4 u v (u^2+v^2)=(p^2+q^2)^2+(2 p q)^2$ ,而这个方程 就是涉及 高斯整数分解了. $u,v$都是形如$x^2+y^2$的形式.

wayne

问题的症结还是回到了 之前绕不过去的地方, 就是 对于一个 无法转化成Weierstrass形式的四次的 椭圆曲线,该如何求解.

wayne

通过待定系数反复调整,降低系数的复杂性,发现 可以设$z=D,u v = 4 x + 2,  u + v = b y + 2 $,得到$-512 x^3-256 x^4+256 b x^2 y+256 b x^3 y+32 b^2 y^2+64 b^2 x y^2+32 b^3 x y^3-4 b^4 y^4+z^2=0$
然后给定有理数$x$,只要满足$x^2+2x =p^2$关系, 就能将该四次曲线 转化成椭圆曲线 的Weierstrass形式:
\[
\begin{align}
Y^3 + \frac{64}{3} b^4 (12 x^4-4 x^2-4 x-1) Y - \frac{1024}{27} b^6 (216 x^6+576 x^5+396 x^4+100 x^3-12 x^2-6 x-1) + Z^2
\end{align}\]

变换公式是
\[Y=\frac{8 (2 b^2 x y^2+b^2 y^2+24 b x^3 y+24 b x^2 y-3 p x z-48 x^4-96 x^3)}{3 y^2}\]
\[Z=\frac{32 x (2 b^3 p x y^3+8 b^2 p x y^2+4 b^2 p y^2+48 b p x^3 y+48 b p x^2 y+b x^2 y z+b x y z-64 p x^4-128 p x^3-4 x^3 z-8 x^2 z)}{y^3}\]
逆变换公式是
\[y=\frac{48 x \left(192 b^3 x^4+448 b^3 x^3+96 b^3 x^2+32 b^3 x-12 b x^2 Y-12 b x Y+3 p Z\right)}{2304 b^4 x^4+4608 b^4 x^3-256 b^4 x^2-256 b^4 x-64 b^4+96 b^2 x Y+48 b^2 Y-9 Y^2}\]

\[z=\frac{16 x \left(5308416 b^8 p x^8+24772608 b^8 p x^7+42467328 b^8 p x^6+38928384 b^8 p x^5+15597568 b^8 p x^4+1998848 b^8 p x^3-491520 b^8 p x^2-163840 b^8 p x-20480 b^8 p-2654208 b^6 p x^6 Y-7299072 b^6 p x^5 Y-4976640 b^6 p x^4 Y-1155072 b^6 p x^3 Y+258048 b^6 p x^2 Y+129024 b^6 p x Y+21504 b^6 p Y+165888 b^5 x^6 Z+608256 b^5 x^5 Z+626688 b^5 x^4 Z+147456 b^5 x^3 Z+23040 b^5 x^2 Z+4608 b^5 x Z+124416 b^4 p x^4 Y^2+124416 b^4 p x^3 Y^2-27648 b^4 p x^2 Y^2-27648 b^4 p x Y^2-6912 b^4 p Y^2-20736 b^3 x^4 Y Z-48384 b^3 x^3 Y Z-10368 b^3 x^2 Y Z-3456 b^3 x Y Z+864 b^2 p x Y^3+432 b^2 p Y^3+648 b x^2 Y^2 Z+648 b x Y^2 Z+81 p Y^4\right)}{\left(2304 b^4 x^4+4608 b^4 x^3-256 b^4 x^2-256 b^4 x-64 b^4+96 b^2 x Y+48 b^2 Y-9 Y^2\right)^2}\]

-------
而方程$x^2+2x =p^2$的有理数解是 $x=\frac{(s-1)^2}{2 s}, p =\frac{s^2-1}{2 s}$, 代入$(9)$即可. 为了使得 椭圆曲线求出的解尽可能的小,我们需要选择合适的参数$b,s$
我们取 $s=2$,那么$x=\frac{1}{4},p=\frac{3}{4}$,代入得到方程是$-18 b^6-47 b^4 Y+Y^3+Z^2=0$,所以我们可以继续取 $b=1$,得到一个曲线$Y^3-47 Y+Z^2-18=0$.
该曲线的秩为2, https://beta.lmfdb.org/EllipticCurve/Q/203272/b/1

数学星空

利用Maple软件容易得到：

\((-u^4 + 4u^3 + 8u - 4)v^4 + (4u^4 - 8u^3 + 16u^2 - 48u + 16)v^3 + (16u^3 - 48u^2 + 64u)v^2 + (8u^4 - 48u^3 + 64u^2 - 32u - 32)v - 4u^4 + 16u^3 + D^2 - 32u + 48=0\)

Weierstrass标准式：

V^3 + (1280/3*u^6 - 1280*u^5 + 6784/3*u^4 - 2560*u^3 + 5120/3*u^2 + 16*u^8 - 128*u^7 - 1024*u + 256)*V + 8192 - (114688*u)/3 + (212992*u^2)/3 + (257024*u^9)/27 - (40448*u^8)/3 + (106496*u^7)/9 - (212992*u^5)/9 + (161792*u^4)/3 - (2056192*u^3)/27 - 128*u^12 + (3584*u^11)/3 - (13312*u^10)/3 + Y^2=0,

有理变换：

V=2*(12*u^4*v + 8*u^3*v^2 - 12*u^4 - 72*u^3*v - 24*u^2*v^2 + 48*u^3 + 96*u^2*v + 32*u*v^2 - 3*D*p - 48*u*v - 96*u - 48*v + 144)/(3*v^2),

Y=4*(p*u^4*v^3 - 2*p*u^3*v^3 + 2*D*u^4*v + 6*p*u^4*v + 8*p*u^3*v^2 + 4*p*u^2*v^3 - 4*D*u^4 - 12*D*u^3*v - 4*p*u^4 - 36*p*u^3*v - 24*p*u^2*v^2 - 12*p*u*v^3 + 16*D*u^3 + 16*D*u^2*v + 16*p*u^3 + 48*p*u^2*v + 32*p*u*v^2 + 4*p*v^3 - 8*D*u*v - 24*p*u*v - 32*D*u - 8*D*v - 32*p*u - 24*p*v + 48*D + 48*p)/v^3,

v=(288*u^8 - 1344*u^7 + 5760*u^5 - 13056*u^4 + 26880*u^3 - 43008*u^2 + 44544*u - 13824 + (-72*u^4 + 432*u^3 - 576*u^2 + 288*u + 288)*V + 18*p*Y)/(-9*V^2 + (96*u^3 - 288*u^2 + 384*u)*V + 144*u^8 - 1152*u^7 + 2048*u^6 + 1536*u^5 - 5504*u^4 + 10752*u^3 - 13312*u^2 + 18432*u - 6912),

y=(20736*p*u^16 - 110592*p*u^15 - 552960*p*u^14 + 6524928*p*u^13 - 26206208*p*u^12 + 66748416*p*u^11 - 138838016*p*u^10 + 242270208*p*u^9 - 343105536*p*u^8 + 363921408*p*u^7 - 276758528*p*u^6 + 90636288*p*u^5 + 78905344*u^4*p - 113246208*u^3*p + 49545216*u*p - 15925248*p + (-41472*p*u^12 + 373248*p*u^11 - 1285632*p*u^10 + 2328576*p*u^9 - 1714176*p*u^8 - 2912256*p*u^7 + 12496896*p*u^6 - 23592960*p*u^5 + 31629312*p*u^4 - 33447936*p*u^3 + 27205632*p*u^2 - 13271040*p*u + 2654208*p)*V + (7776*p*u^8 - 62208*p*u^7 + 158976*p*u^6 - 269568*p*u^5 + 400896*p*u^4 - 331776*p*u^3 + 55296*p*u^2 + 248832*p*u - 124416*p)*V^2 + (864*p*u^3 - 2592*p*u^2 + 3456*p*u)*V^3 + 81*p*V^4 + (10368*u^12 - 117504*u^11 + 516096*u^10 - 981504*u^9 + 446976*u^8 + 866304*u^7 - 1253376*u^6 - 2469888*u^5 + 9345024*u^4 - 16330752*u^3 + 15630336*u^2 - 8626176*u + 1990656)*Y + (-5184*u^8 + 24192*u^7 - 103680*u^5 + 235008*u^4 - 483840*u^3 + 774144*u^2 - 801792*u + 248832)*V*Y + (648*u^4 - 3888*u^3 + 5184*u^2 - 2592*u - 2592)*V^2*Y)/(20736*u^16 - 331776*u^15 + 1916928*u^14 - 4276224*u^13 - 929792*u^12 + 22069248*u^11 - 48791552*u^10 + 63111168*u^9 - 35659776*u^8 - 67829760*u^7 + 290455552*u^6 - 510394368*u^5 + 649658368*u^4 - 639369216*u^3 + 523763712*u^2 - 254803968*u + 47775744 + (27648*u^11 - 304128*u^10 + 1167360*u^9 - 1769472*u^8 - 368640*u^7 + 6414336*u^6 - 12976128*u^5 + 19464192*u^4 - 22167552*u^3 + 18137088*u^2 - 5308416*u)*V + (-2592*u^8 + 20736*u^7 - 27648*u^6 - 82944*u^5 + 255744*u^4 - 414720*u^3 + 387072*u^2 - 331776*u + 124416)*V^2 + (-1728*u^3 + 5184*u^2 - 6912*u)*V^3 + 81*V^4)

其中p满足： \(p^2=4u^4-16u^3+32u-48\) (这个有理解如何得到？）

wayne

没有一个趁手的好工具,目前来看,网上最好的工具 还是magma, 光一个椭圆曲线的数据库压缩包就有 1GB,  https://magma.maths.usyd.edu.au/magma/download/db/
根据lmfdb网站, https://beta.lmfdb.org/EllipticCurve/Q/203272/b/1, 可以玩椭圆曲线的工具有四个:
1) Magma ,目前来看,最强大,数据库也大, 但是闭源的.
2) PariGP
3) SageMath
4) Oscar ,  是一个julia包,底层基于 FLINT, GAP, polymake, Singular

以这个曲线$y^2=x^3-47x+18$为例,拿到所有的整数点,只有magma 和 SageMath 提供了函数, 而PariGP 需要借用 ellratpoints(E,100000) ,并且是限高搜索.

wayne

PARI/Gp有个hyperellratpoints 函数挺不错的, 傻瓜式的返回一个超椭圆曲线指定高度的的有理解.
$u=-\frac{5}{8}$试手,解得的v如下

1. f(u,v)=u^4*v^4 - 4*u^4*v^3 - 4*u^3*v^4 + 8*u^3*v^3 - 8*u^4*v - 16*u^3*v^2 - 16*u^2*v^3 - 8*u*v^4 + 4*u^4 + 48*u^3*v + 48*u^2*v^2 + 48*u*v^3 + 4*v^4 - 16*u^3 - 64*u^2*v - 64*u*v^2 - 16*v^3 + 32*u*v + 32*u + 32*v - 48;
   
2. hyperellratpoints(f(-5/8,x),10000000)
   
3. %4 = [[-1617/200, 708019737/2560000], [-1617/200, -708019737/2560000], [-477/692, 64335705/30647296], [-477/692, -64335705/30647296], [20824/2003, 32269755735/128384288], [20824/2003, -32269755735/128384288], [-34272/4885, 164968598721/763623200], [-34272/4885, -164968598721/763623200], [36696/8687, 2141459895/344978144], [36696/8687, -2141459895/344978144], [398113/66200, 16290521488377/280476160000], [398113/66200, -16290521488377/280476160000], [-124529/68084, 7518853086255/296667587584], [-124529/68084, -7518853086255/296667587584], [-176752/157345, 9325236881511/792238368800], [-176752/157345, -9325236881511/792238368800], [4037701/712772, 1548010052740935/32514811134976], [4037701/712772, -1548010052740935/32514811134976], [4718261/816880, 2172780193392063/42706747801600], [4718261/816880, -2172780193392063/42706747801600], [-3589408/3049765, 3795691506791841/297634129767200], [-3589408/3049765, -3795691506791841/297634129767200]]

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代入,得到了8个解，

1. {2682440,18796760,15365639,20615673}
   
2. {260052385,582665296,186668000,589845921}
   
3. {275156240,630662624,219076465,638523249}
   
4. {2448718655,664793200,3134081336,3393603777}
   
5. {2480452675600,4987588419655,502038853976,5062297699257}
   
6. {3579087147375440,14890026433468471,18565945114216720,20249506709579721}
   
7. {8813425670440240,47886740272114976,56827813308111785,62940516903410601}
   
8. {46196947347028916440,107238802094189542120,38751631463616255521,108593344076382641697}

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$u=-\frac{9}{20}$,解得的v如下

1. ? hyperellratpoints(f(-9/20,x),10000000)
   
2. %8 = [[1000/47, 495260031/441800], [1000/47, -495260031/441800], [5728/215, 16723774473/9245000], [5728/215, -16723774473/9245000], [-1041/320, 2126704839/40960000], [-1041/320, -2126704839/40960000], [-1425/412, 3894577617/67897600], [-1425/412, -3894577617/67897600], [-4209/3500, 6507898503/700000000], [-4209/3500, -6507898503/700000000], [30080/6007, 195619772649/7216809800], [30080/6007, -195619772649/7216809800], [34225/6692, 75863359911/2559020800], [34225/6692, -75863359911/2559020800], [-41952/33865, 2280363347913/229367645000], [-41952/33865, -2280363347913/229367645000]]

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得到了6个解，

1. {95800,414560,217519,422481}
   
2. {673865,2767624,1390400,2813001}
   
3. {632671960,1670617271,50237800,1679142729}
   
4. {5821981400,15355831360,140976551,15434547801}
   
5. {7592431981391,22495595284040,27239791692640,29999857938609}
   
6. {130064300991400,440804942580160,514818101299289,573646321871961}

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$u=-\frac{29}{12}$,解得的v如下

1. ? hyperellratpoints(f(-(29/12),x),10000000)
   
2. %9 = [[1865/132, 4566509815/2509056], [1865/132, -4566509815/2509056], [6280/1359, 15222896105/132975432], [6280/1359, -15222896105/132975432], [-30768/57253, 1190554344625/33715604664], [-30768/57253, -1190554344625/33715604664], [-3333/107368, 27781785391625/1660015789056], [-3333/107368, -27781785391625/1660015789056]]

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对应3个解

1. {8332208, 5507880, 1705575, 8707481}
   
2. {125777308440, 894416022327, 2032977944240, 2051764828361}
   
3. {27546142170735, 7908038161032, 43940127884360, 45556888578449}

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wayne

功夫不负有心人，我捡到好几个漏了，排行榜排位发生变化。原来的第91名和第93名被两个更小的答案挤掉了,应该是

1. {123188180833923372056627153, 118194421251475239056505903, 66491673395168374249746120, 62831773759131557571594880}
   

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对应的(u,v,w)是$ (-12065/12396, -84558637/193874100, 21792/5035)$

1. {174088703841632292189275073, 170289556324371670328363560, 92419682545114696981174360, 46402888024739111034420161}

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对应的(u,v,w)是 $(-136/133, -63528125/85096232, 23685689/3885556)$
。
其他新答案如下：

1. {123188180833923372056627153, 118194421251475239056505903, 66491673395168374249746120, 62831773759131557571594880}
   
2. {174088703841632292189275073, 170289556324371670328363560, 92419682545114696981174360, 46402888024739111034420161}
   
3. {504068891841730072306483681, 435117527990435060232042280, 411177854471028470696556192, 106958136069417067994530335}
   
4. {918310142223097801006397529, 889106559932545369165762471, 535607712470322407570378200, 250530677254015598463660440}
   
5. {1447760188128589806063912993, 1312026469210482429462939440, 1087682246112509824947055976, 417484587979578624273415135}
   
6. {3677215554888336428049606273, 3368564311979655025855012600, 2320714811284022366485718240, 2237780791013852505780494977}
   
7. {5627688864176878140758748009, 5585259068218868809780112240, 2333861078054615396464078960, 700306884975436628944196759}
   
8. {10018903766533510889057062601, 9667333776938051160393251960, 6052043593595878314368914743, 168645743122342622620098480}
   
9. {12629217765947417896034599809, 12228221407419874341992355560, 7328739297665501682037024184, 3739210609433203972994481025}
   
10. {16013927795894026959663168953, 13664102621078861306004840840, 13258851160243699422011467335, 552797283800610952652413328}
    
11. {31117794612098148260469657593, 27393584701634528092690803705, 24738316179435439698109667120, 808590818355102638853673176}
    
12. {133490320060360914535848328209, 119298860162529095599542795640, 103549547575488612225699187040, 10521526640620348981446408209}
    
13. {175968975906723433958996724209, 163272917896204873786050444815, 123781164812377477678355882712, 60534442433151314096924049480}
    
14. {245878569117573833167406061393, 221725734480997509909988175768, 182603232896217072374572339025, 105992175134428412918623707160}
    
15. {444997840250072084560273090857, 412381504908379528321286079785, 315781014866997985099251407224, 136742510896974060556436491360}
    
16. {601546336586676758217069747353, 507376500694215143603840904000, 504285647197281669856795386728, 6571990083263349384871692135}
    
17. {780564938636045965056301701249, 779297054673311472994724152360, 211846693725917814575333200001, 140703840741824401472415321160}
    
18. {860433206740544424461178566577, 845916519446670855349916469440, 426695309330591314699902838264, 232356172746879400195244503985}
    
19. {2677724555233904033024321940201, 2650599803386595785432318503160, 1187505796585909023692004388960, 501284627341500145592546539799}
    
20. {8229646208430186915167706621881, 7918174790330642885254291363640, 5060779866902849712056608508592, 425728077239657769125737790535}
    
21. {8653092317071774719310202489073, 8099532884728944729940462847464, 6001671979737012735726619523855, 1516716120293429599551302835760}
    
22. {10830828827804801622456578003121, 10707756967155140530274037773135, 4979640258082663479364236259240, 246940186636356727683332717504}
    
23. {12137566662146665711886365252737, 10460832656100181686547108582112, 9872001147462059766910335739775, 3898022937032042181537566499160}
    
24. {77214586804364169308050947074361, 76668894313793210460603803478440, 31176096882978693586981687856264, 14921528807524384363774241268025}
    
25. {1098951104967340650758908677898497, 938322348395206180783921371381503, 837343681998415354786117892702000, 661717340341387736183451188650240}
    
26. {1166771203657741610801659018209393, 1145400860459371360702253709703768, 602860664444767185991355248844720, 43015381264158179520049525808015}
    
27. {2985772999602784901636123486486649, 2693763654464217358310238734024120, 2269321665966462462485146483298632, 739419769536770231012635302250375}
    
28. {3762098874072146335431610024521161, 3544724218928041607923371624299240, 2511945393847796136912099490227552, 1272617115110814812810988604290615}
    
29. {3991096389697638324039142815248313, 3619759640991069943317745685641720, 2798227399699632035038808311974841, 2133999097873523205616379862269920}
    
30. {17726595196734905515392721211749761, 17007029137278431161651951093629136, 9645408167029758797295194170291585, 8953806277800028928928574895292560}
    
31. {27234039469010571287750749472806593, 25097616096502142214946285532508824, 19788748762317394956388167436667840, 685489169050827570208239564001985}
    
32. {140537630371183160245850568603894617, 123349662390916854001862509235330905, 112214442819394909948440677165601840, 13593048082866832685243215228608456}
    
33. {575814109241966259438071103769180457, 568495181674030634997227526296268240, 268742894833497872412755452524709832, 127884971884543172835628915182179625}
    
34. {830656188985023870948584779728249417, 716161782031675610165431528427020400, 679368538882959228476067055088860745, 58032618743000072263285011580324856}
    
35. {1701149366116763201315972094656489529, 1505798350865774387042674647101996096, 1340939407906109772768541246995561415, 125408329526832811338657001571718760}
    
36. {6247103423292859699056869181299776761, 6186967282345145998802943536561096200, 2757331153546626033198927459840538873, 28450385633108239635917565180981520}
    
37. {10299384611211248340376023286842835433, 10031362083685034676043313839954861545, 5251921176120295799149380679888209944, 4372583605557734074973725524282199920}
    
38. {10373358865070540657105617744371110617, 10036370941197170791465088824362503544, 5916310526039507031961165533723980720, 3796307295487803837472619636479447335}
    
39. {67434082137309994555247453811022197057, 58825777559993370245474213094921674240, 54256586296486494216437458509656437057, 13938951315456976978513685926934640560}
    
40. {242538180175270610853290853090287229769, 236937233673417726067275845959333925705, 122707666757292152245213732016236276920, 95165392799313961110399476887579379208}
    
41. {806138838545758367937672064817957800737, 726447315918756597554941566956138194616, 604700759189277502518379225699911213440, 317124731692923422812335044519683520735}
    
42. {1871719083282687906202560764473598495521, 1747706727327181776430166707334029969185, 1102486892305607002168440071204344088928, 1100377611684590854488735787299681397720}
    
43. {3395929891485334592370746651336421162321, 3360388888883721642901019519945403343304, 1475840549437269376782395216972838074000, 926430612195290681941470814843189475665}
    
44. {15972961180516771719532723631898149809961, 15451300536692003301752989081291359395640, 8057625913255747468244742036175871587760, 7892758983116882563694252553878845426473}
    
45. {19934350711896960649365583332561489559721, 19537629354156506960751018658172933774505, 10240767198398148770624283487986683810904, 5887827370046553813670911877907196495680}
    
46. {34363901684184698539080985638973937196873, 31297237528652492191374671448716327803760, 25637787666341642063737209582545890147145, 7388644063119524070799636844138879027864}
    
47. {45844192201888470314405213353949226832897, 42785823428040041234161501258166782422015, 32113076851593729899259358463785814766496, 7021521131014292996997517970292353757120}
    
48. {86005230383956921385792579690168474216577, 82860902927671527147578479599418011711736, 52456251116927883513093749646292242340735, 6240825928464181630956269883794822302960}
    
49. {370178787342908519545804626629109524196313, 342926094771396607922211737347247063070216, 261289014778029757306114612499727621175335, 130207738461126188904986843727436181309360}
    
50. {459300900508807447879376165305627118309057, 452679652603063022497164751569715401046335, 223855569708056236637936238204314784983432, 17926415451798202836937173484154386635240}
    
51. {500796050001427880070678938556915333014849, 474852175855673033084619074909651063938040, 330354100428652215769025528607019680881985, 109825562277571613049068495579199043286232}
    
52. {977460797198232920093887556716932292060753, 973433274958851473729632905204552594628320, 334229339844264464206041083686350217717329, 223010558671964311114051804654219491905440}
    
53. {1470695115188634600420193135982485445470881, 1467627323920247878783602785437560274771295, 443184584373500545468832476446952876225256, 135298029890536514615749627509396118115320}
    
54. {5082559754777633036582163979494292876893337, 4492209740965145731782789443715553202358584, 4013477702776553653893950470981211910469785, 885745318409953247534849392851560923632720}
    
55. {6143539032661373070735769580379563143044881, 5852230257834881029367629460392322577618705, 3593221165723286365451189689649125338450744, 3035234250371793173002501048943483041012760}
    
56. {6275137477254099552690315260094419636025849, 6006100538619513212890558195061841980522200, 3771284895110544283874170485413841936716840, 2618535932270839713747743806066980040134151}
    
57. {9114172774086281066413676070790614410779473, 8944672405792029461839096353940380640553135, 4726479082843848686237115074773144463194664, 579120214506563635174419598136966085554560}
    
58. {11827000417138776237238567751611797835894529, 11703497880395831612988069202805744430635528, 5304881704326318326137500214801627488843720, 1883580279065188183577494265944805898270975}
    
59. {12559995101780464868351574860663895593652089, 11564020060970766874354364810293518612242880, 8550192913944418222929749083159204355797880, 6381982797350999465695854651454845891400839}
    
60. {16930431869968555250050230333369291823325721, 16722151415210738551512069214227404055910240, 7556939299858548120883368285232293844669904, 5158024399779229722215011965963364160970215}
    
61. {23292027092250100127874940152491663685896001, 23156921512324918814164893743769567325575376, 9059330016193842060221876334617571924657985, 2415739591667285135424221066398517686453040}
    
62. {33373562681668093445861955364547455104962057, 32649064673153450829436315208092610082433545, 17640967117458259135734133923460933978454640, 9280222253003957569623319645389600144384376}
    
63. {34238025288322679405354616563477029005724489, 30065183349599749353652252170416471089881752, 27019982440646405002224149701865168556376905, 12456165604980448961708437936308199693721880}
    
64. {1134284386346724151363651000320391991301733417, 1032609803380513184909795531534698477102396856, 848327426869312864774603097398961498872787760, 147487711831232703392729490663904776597182505}
    
65. {1485319978088610874772587015727151055020211473, 1286574740535738707167440773242497789045153680, 1201664975215670323433872762392001485297819152, 453089166851196484578476500609393843402652945}
    
66. {1545876932044961499994504420017547380040919673, 1497715832818468945880745540824932809695125760, 867479830643708710233050296693323999236618960, 579548962635019950126280259669775577793320327}
    
67. {2111194167978422288220921255365867051196794393, 2089717145764499016944579167741933418625835545, 794746331186817432094227002787045298634324480, 793869142619513622281841596664558491280370776}
    
68. {2374902405268544156003955493525772232090590553, 2364179084161218516599144160529002560832423920, 838951429158856278829815610575389260158180697, 523803017372010966009313491168271066116491920}
    
69. {3811844410730149410687121517968558056866543897, 3762736287177600681424722023412611629632252647, 1785749307339601641041230194085043539067310960, 841825955601636887278486971756260420166660840}
    
70. {4359453760016731480341559048760092353445158393, 3755677900085395394517147290349149195700019240, 3549239935664878385299887730614787797397708760, 1371914780945160657480571387727362688956590599}
    
71. {15812220228973978029640545092760827730729586761, 15796268259750535399121357323071506402161885760, 3712162287258766514894362261184279475656584265, 2805937385907916698694532219596648902968866136}
    
72. {47495446719420628851882981590552211302229878697, 40201850014156946145960402821235256903275744640, 39491889176636414193004404950719346759851424855, 14504924792674282059881740229589583428350105896}
    
73. {50850387717526082374854170580381669126913882753, 47994627294635220655319628420215891231758648520, 34275156842649042025698564180116719276001302128, 1196165497178991078511743748654292211747408255}
    
74. {61773983416363337070401598227333909934876779377, 56541966500458363000561559181800928586654852239, 45640359472895282187354073055003880273698544760, 6857076456214041801253822623630409875859862280}
    
75. {69751243118562330823889798476000284455547179409, 68153172183689262267448592863699209952421875040, 35066778012681271091786396663864666683979135240, 27640993066460423875775176412252797556736340591}
    
76. {165863826757829334016288350445306547111729869953, 156960145284219259407960968722441510786198304720, 110647433399478938169758487559272230733321962696, 4999255307566191240836242234455714209054519425}
    
77. {235330445700501159858787905821316726845325706593, 201250975205387708645126290259718166098130747745, 191722654928290095680347390901095351758768308824, 93203774317958905875123208651612222143956320960}
    
78. {429743683696497862908946921973872134110241764097, 387601083793902125736364769834148333834225686528, 327681172232494969258587163020444460111450229360, 50975076867131678259567605351981276102818401535}
    
79. {763660316535847336100572602521466005340135157281, 752932548477756505439823728841382425781846753040, 353206228650528968498081979906555913216071735760, 236864725299048270364761586920409130549561496031}
    
80. {867080795573415261846041568022401507120424349873, 738646873410735401580081582729428786424654440625, 718786885148112455723179058063160994689574385752, 158839500159704960415730696880379641913325282440}
    
81. ...省略100个...
    

复制代码

更多数据放在这里，https://nestwhile.com/res/a4b4c4d4/ 里的abcd-new.txt 文件
持续更新中，。。。敬请关注。

数学星空

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，\([u,v,D,x,y,z]\)]

[1, [-9/20, 1000/47, 495260031/441800, 414560/422481, 95800/422481, 217519/422481]],
[2, [-9/20, 1000/47, -495260031/441800, -632671960/1679142729, -1670617271/1679142729, 50237800/1679142729]],
[3, [-9/20, 5728/215, 16723774473/9245000, 2767624/2813001, 673865/2813001, 1390400/2813001]],
[4, [-9/20, 5728/215, -16723774473/9245000, -5821981400/15434547801, -15355831360/15434547801, 140976551/15434547801]],
[5, [-9/20, -1041/320, 2126704839/40960000, 1670617271/1679142729, 632671960/1679142729, -50237800/1679142729]],
[6, [-9/20, -1041/320, -2126704839/40960000, -95800/422481, -414560/422481, -217519/422481]],
[7, [-9/20, -1425/412, 3894577617/67897600, 15355831360/15434547801, 5821981400/15434547801, -140976551/15434547801]],
[8, [-9/20, -1425/412, -3894577617/67897600, -673865/2813001, -2767624/2813001, -1390400/2813001]],
[9, [-9/20, -4209/3500, 6507898503/700000000, 414560/422481, 95800/422481, -217519/422481]],
[10, [-9/20, -4209/3500, -6507898503/700000000, 7592431981391/29999857938609, -22495595284040/29999857938609, -27239791692640/29999857938609]],
[11, [-9/20, 30080/6007, 195619772649/7216809800, 22495595284040/29999857938609, -7592431981391/29999857938609, 27239791692640/29999857938609]],
[12, [-9/20, 30080/6007, -195619772649/7216809800, -95800/422481, -414560/422481, 217519/422481]],
[13, [-9/20, 34225/6692, 75863359911/2559020800, 440804942580160/573646321871961, -130064300991400/573646321871961, 514818101299289/573646321871961]],
[14, [-9/20, 34225/6692, -75863359911/2559020800, -673865/2813001, -2767624/2813001, 1390400/2813001]],
[15, [-9/20, -41952/33865, 2280363347913/229367645000, 2767624/2813001, 673865/2813001, -1390400/2813001]],
[16, [-9/20, -41952/33865, -2280363347913/229367645000, 130064300991400/573646321871961, -440804942580160/573646321871961, -514818101299289/573646321871961]],

[17, [-29/12, 1865/132, 4566509815/2509056, 894416022327/2051764828361, -125777308440/2051764828361, 2032977944240/2051764828361]],
[18, [-29/12, 1865/132, -4566509815/2509056, -8332208/8707481, -5507880/8707481, -1705575/8707481]],
[19, [-29/12, 6280/1359, 15222896105/132975432, -7908038161032/45556888578449, -27546142170735/45556888578449, 43940127884360/45556888578449]],
[20, [-29/12, 6280/1359, -15222896105/132975432, -8332208/8707481, -5507880/8707481, 1705575/8707481]],
[21, [-29/12, -30768/57253, 1190554344625/33715604664, 5507880/8707481, 8332208/8707481, 1705575/8707481]],
[22, [-29/12, -30768/57253, -1190554344625/33715604664, 125777308440/2051764828361, -894416022327/2051764828361, -2032977944240/2051764828361]],
[23, [-29/12, -3333/107368, 27781785391625/1660015789056, 5507880/8707481, 8332208/8707481, -1705575/8707481]],
[24, [-29/12, -3333/107368, -27781785391625/1660015789056, 27546142170735/45556888578449, 7908038161032/45556888578449, -43940127884360/45556888578449]],

[25, [-93/80, -400/37, 2960618799/4380800, 11289040/12197457, 5870000/12197457, 8282543/12197457]],
[26, [-93/80, -400/37, -2960618799/4380800, -1237796960/1787882337, -1662997663/1787882337, -686398000/1787882337]],
[27, [-93/80, -2433/920, 372167198439/5416960000, 1662997663/1787882337, 1237796960/1787882337, 686398000/1787882337]],
[28, [-93/80, -2433/920, -372167198439/5416960000, -5870000/12197457, -11289040/12197457, -8282543/12197457]],
[29, [-93/80, -84237/359800, 66222454729143/118359808000000, 11289040/12197457, 5870000/12197457, -8282543/12197457]],
[30, [-93/80, -84237/359800, -66222454729143/118359808000000, 15876595946759369395903/17503689286309573964097, 7188470920864810763360/17503689286309573964097, -12896301483090810351440/17503689286309573964097]],

[31, [-400/37, -93/80, 2960618799/4380800, 686398000/1787882337, 1662997663/1787882337, 1237796960/1787882337]],
[32, [-400/37, -93/80, -2960618799/4380800, -8282543/12197457, -5870000/12197457, -11289040/12197457]],
[33, [-400/37, -2433/920, 985189443879/579360800, 5870000/12197457, 8282543/12197457, 11289040/12197457]],
[34, [-400/37, -2433/920, -985189443879/579360800, -1662997663/1787882337, -686398000/1787882337, -1237796960/1787882337]],

[35, [-136/133, 201/4, 1416600375/141512, 14173720/16003017, 4479031/16003017, 12552200/16003017]],
[36, [-136/133, 201/4, -1416600375/141512, -108644015/156646737, -146627384/156646737, -27450160/156646737]],
[37, [-136/133, -1005/568, 102051891015/2853447968, 146627384/156646737, 108644015/156646737, 27450160/156646737]],
[38, [-136/133, -1005/568, -102051891015/2853447968, -4479031/16003017, -14173720/16003017, -12552200/16003017]],

[39, [201/4, -136/133, 1416600375/141512, 27450160/156646737, 146627384/156646737, 108644015/156646737]],
[40, [201/4, -136/133, -1416600375/141512, -12552200/16003017, -4479031/16003017, -14173720/16003017]],
[41, [201/4, -1005/568, 87999215295/5161984, 4479031/16003017, 12552200/16003017, 14173720/16003017]],
[42, [201/4, -1005/568, -87999215295/5161984, -146627384/156646737, -27450160/156646737, -108644015/156646737]],

[43, [-5/8, -1617/200, 708019737/2560000, 630662624/638523249, 275156240/638523249, 219076465/638523249]],
[44, [-5/8, -1617/200, -708019737/2560000, -260052385/589845921, -582665296/589845921, -186668000/589845921]],
[45, [-5/8, -477/692, 64335705/30647296, 18796760/20615673, 2682440/20615673, -15365639/20615673]],
[46, [-5/8, -477/692, -64335705/30647296, 2448718655/3393603777, -664793200/3393603777, -3134081336/3393603777]],
[47, [-5/8, 20824/2003, 32269755735/128384288, 18796760/20615673, 2682440/20615673, 15365639/20615673]],
[48, [-5/8, 20824/2003, -32269755735/128384288, -2480452675600/5062297699257, -4987588419655/5062297699257, 502038853976/5062297699257]],
[49, [-5/8, -34272/4885, 164968598721/763623200, 582665296/589845921, 260052385/589845921, 186668000/589845921]],
[50, [-5/8, -34272/4885, -164968598721/763623200, -275156240/638523249, -630662624/638523249, -219076465/638523249]],
[51, [-5/8, 36696/8687, 2141459895/344978144, 664793200/3393603777, -2448718655/3393603777, 3134081336/3393603777]],
[52, [-5/8, 36696/8687, -2141459895/344978144, -2682440/20615673, -18796760/20615673, 15365639/20615673]],
[53, [-5/8, 398113/66200, 16290521488377/280476160000, 47886740272114976/62940516903410601, -8813425670440240/62940516903410601, 56827813308111785/62940516903410601]],
[54, [-5/8, 398113/66200, -16290521488377/280476160000, -260052385/589845921, -582665296/589845921, 186668000/589845921]],
[55, [-5/8, -124529/68084, 7518853086255/296667587584, 4987588419655/5062297699257, 2480452675600/5062297699257, -502038853976/5062297699257]],
[56, [-5/8, -124529/68084, -7518853086255/296667587584, -2682440/20615673, -18796760/20615673, -15365639/20615673]],
[57, [-5/8, -176752/157345, 9325236881511/792238368800, 630662624/638523249, 275156240/638523249, -219076465/638523249]],
[58, [-5/8, -176752/157345, -9325236881511/792238368800, 3579087147375440/20249506709579721, -14890026433468471/20249506709579721, -18565945114216720/20249506709579721]],

[59, [233/60, -216285/23504, 675018780820577/1988776857600, -7813353720/12558554489, 4707813440/12558554489, 11988496761/12558554489]],
[60, [233/60, -216285/23504, -675018780820577/1988776857600, -335981923744570504065/375075545025537358721, 188195571677171463096/375075545025537358721, 275897431444390465240/375075545025537358721]],
[61, [233/60, 7584/54605, 42947654622593/5367070845000, -343651286746207896/481334894209428521, 438980913824794665/481334894209428521, -225712385669145920/481334894209428521]],
[62, [233/60, 7584/54605, -42947654622593/5367070845000, -7813353720/12558554489, 4707813440/12558554489, -11988496761/12558554489]],

[63, [-56/165, -383021/380940, 997084931509393/1975381798005000, 1047978087905/1367141947873, -408600530760/1367141947873, -1226022682752/1367141947873]],
[64, [-56/165, -383021/380940, -997084931509393/1975381798005000, 163180699054891578792/228746036963039501833, -84616109521023161865/228746036963039501833, -210878774189729581880/228746036963039501833]],

[65, [-125/92, -936/5281, 514986989385/118026082952, 23449050222680/25866132798297, 18776929334105/25866132798297, -12035933588696/25866132798297]],
[66, [-125/92, -936/5281, -514986989385/118026082952, 52289667920/64244765937, 17111129720/64244765937, -55479193841/64244765937]],

[67, [-361/540, 1861/240, 2085707016991/16796160000, 122055375/145087793, 1841160/145087793, 121952168/145087793]],
[68, [-361/540, 1861/240, -2085707016991/16796160000, -535914713672/1059621884297, -1041572957760/1059621884297, 187577183625/1059621884297]],
[69, [-361/540, -7800/5509, 80577403041031/4424896009800, 1041572957760/1059621884297, 535914713672/1059621884297, -187577183625/1059621884297]],
[70, [-361/540, -7800/5509, -80577403041031/4424896009800, -1841160/145087793, -122055375/145087793, -121952168/145087793]],

[71, [-817/660, -1581/1520, 23439166132879/1006410240000, 9585769407872803575/10816708329115215113, 8510180374729994520/10816708329115215113, 985735303963754488/10816708329115215113]],
[72, [-817/660, -1581/1520, -23439166132879/1006410240000, -2164632/44310257, -31669120/44310257, -41084175/44310257]],
[73, [-817/660, 21792/5035, 148223121024127/5521496805000, -2164632/44310257, -31669120/44310257, 41084175/44310257]],
[74, [-817/660, 21792/5035, -148223121024127/5521496805000, -1199828498161126807800/1671674986261410994097, -1534990269771364822095/1671674986261410994097, 38585919318751039616/98333822721259470241]],

[75, [-865/592, -14177/20156, 2838307044130425/142381024251904, 1553556440/1871713857, 1593513080/1871713857, 92622401/1871713857]],
[76, [-865/592, -14177/20156, -2838307044130425/142381024251904, 208032601069058735/1592672455342770513, -851144034922098880/1592672455342770513, -1559028675188874616/1592672455342770513]],
[77, [-865/592, -230472/438737, 527721585216584535/33730434870574208, 1553556440/1871713857, 1593513080/1871713857, -92622401/1871713857]],
[78, [-865/592, -230472/438737, -527721585216584535/33730434870574208, 179164925544119666072000/587020625514136613276553, -222787467202130880567415/587020625514136613276553, -582653975191641098286104/587020625514136613276553]],

[79, [553/80, -33400/19537, 260117908400151/1221421980800, 24743080/5179020201, 3971389576/5179020201, 4657804375/5179020201]],
[80, [553/80, -33400/19537, -260117908400151/1221421980800, -5352683902805120/5380742305932201, 1554532675059625/5380742305932201, -1841841620201576/5380742305932201]],
[81, [553/80, -294473/635180, 207130360353696711/2582103247360000, -1554532675059625/5380742305932201, 5352683902805120/5380742305932201, 1841841620201576/5380742305932201]],
[82, [553/80, -294473/635180, -207130360353696711/2582103247360000, -3971389576/5179020201, -24743080/5179020201, -4657804375/5179020201]],

[83, [-12065/12396, -1581/1520, 6215830249009663/355017949286400, 812937165464036006213895/864745895259187110399737, 591519768111748983750888/864745895259187110399737, -57810716855047169409080/864745895259187110399737]],
[84, [-12065/12396, -1581/1520, -6215830249009663/355017949286400, 2164632/44310257, -31669120/44310257, -41084175/44310257]],
[85, [-12065/12396, 21792/5035, 33451262757025519/1947744960049800, 2164632/44310257, -31669120/44310257, 41084175/44310257]],
[86, [-12065/12396, 21792/5035, -33451262757025519/1947744960049800, -66491673395168374249746120/123188180833923372056627153, -118194421251475239056505903/123188180833923372056627153, 62831773759131557571594880/123188180833923372056627153]],

[87, [1744/495, 135/1208, 180347147759/178778080800, -435210480720/521084370137, 372623278887/521084370137, -369168502640/521084370137]],
[88, [1744/495, 135/1208, -180347147759/178778080800, -5819035124295/7082388012473, 4408757988560/7082388012473, -5611660306848/7082388012473]],
[89, [1744/495, -977657/480240, 162765645365418367/28255113936720000, -4408757988560/7082388012473, 5819035124295/7082388012473, 5611660306848/7082388012473]],
[90, [1744/495, -977657/480240, -162765645365418367/28255113936720000, -372623278887/521084370137, 435210480720/521084370137, 369168502640/521084370137]],

[91, [-3168/1553, -857/3696, 41906266886375/2353308160896, 19031674138785/27497822498977, 25762744660064/27497822498977, -2054845288320/27497822498977]],
[92, [-3168/1553, -857/3696, -41906266886375/2353308160896, 2927198165920/6382441853233, -613935345969/6382441853233, -6310500741600/6382441853233]],
[93, [-3168/1553, 980785/175296, 5796392699440292705/37055862675228672, 613935345969/6382441853233, -2927198165920/6382441853233, 6310500741600/6382441853233]],
[94, [-3168/1553, 980785/175296, -5796392699440292705/37055862675228672, -25762744660064/27497822498977, -19031674138785/27497822498977, 2054845288320/27497822498977]],

[95, [-1376/705, 14337/340, 22737959090039/1689885000, 148739531603136/230791363907489, 32467583677535/230791363907489, 220093974949320/230791363907489]],
[96, [-1376/705, 14337/340, -22737959090039/1689885000, -36295982895/39871595729, -29676864960/39871595729, -11262039896/39871595729]],
[97, [-1376/705, -81065/89412, 73070745575924711/1986734608705800, 29676864960/39871595729, 36295982895/39871595729, 11262039896/39871595729]],
[98, [-1376/705, -81065/89412, -73070745575924711/1986734608705800, -32467583677535/230791363907489, -148739531603136/230791363907489, -220093974949320/230791363907489]],

[99, [-1152/2345, 19005/3688, 1161796506978039/37397065344800, 443873167360/597385645737, -142485966505/597385645737, 544848079888/597385645737]],
[100, [-1152/2345, 19005/3688, -1161796506978039/37397065344800, -468405247415/1682315502153, -1657554153472/1682315502153, 801719896720/1682315502153]],
[101, [-1152/2345, -15461/13160, 4644845269335303/476175972020000, 1657554153472/1682315502153, 468405247415/1682315502153, -801719896720/1682315502153]],
[102, [-1152/2345, -15461/13160, -4644845269335303/476175972020000, 142485966505/597385645737, -443873167360/597385645737, -544848079888/597385645737]],

[103, [2265/184, -68256/135125, 106975611923719719/309084384500000, -78558599440/820234293081, 814295112544/820234293081, 337210257575/820234293081]],
[104, [2265/184, -68256/135125, -106975611923719719/309084384500000, -7745659501403353894384/12214291847502204701241, -2120589250533219579335/12214291847502204701241, -11684173258429439467360/12214291847502204701241]],

[105, [-1245/5012, 248521/62784, 208445266940805505/99019353702334464, 39110088360/76973733409, -49796687200/76973733409, 71826977313/76973733409]],
[106, [-1245/5012, 248521/62784, -208445266940805505/99019353702334464, 12209879806944320496330055/34497456764264994703368889, -26621272474250391413865480/34497456764264994703368889, 30730370351168229154149048/34497456764264994703368889]],
[107, [-1245/5012, -267904/221337, 2252317127132864545/615318775951504968, 435117527990435060232042280/504068891841730072306483681, -106958136069417067994530335/504068891841730072306483681, -411177854471028470696556192/504068891841730072306483681]],
[108, [-1245/5012, -267904/221337, -2252317127132864545/615318775951504968, 39110088360/76973733409, -49796687200/76973733409, -71826977313/76973733409]],

[109, [1873/200, -51416/9425, 4964755565640087/1776612500000, 487814048600/26969608212297, 8528631804200/26969608212297, 26901926181047/26969608212297]],
[110, [1873/200, -51416/9425, -4964755565640087/1776612500000, -103028409596553328/103117303193818953, 24975412054750025/103117303193818953, -4092004076331400/103117303193818953]],

[111, [-97/400, 78065/484, 2352959771684697/37480960000, 87375622888246360/87486470529871881, 16306696482461560/87486470529871881, 21794572772239369/87486470529871881]],
[112, [-97/400, 78065/484, -2352959771684697/37480960000, -1313903832425/6014017311081, -6010589044544/6014017311081, -66822832760/6014017311081]],

[113, [-1425/412, -9/20, 3894577617/67897600, 1390400/2813001, 2767624/2813001, 673865/2813001]],
[114, [-1425/412, -9/20, -3894577617/67897600, 140976551/15434547801, -5821981400/15434547801, -15355831360/15434547801]],
[115, [-1425/412, 5728/215, 47763328987689/3923208200, 5821981400/15434547801, -140976551/15434547801, 15355831360/15434547801]],
[116, [-1425/412, 5728/215, -47763328987689/3923208200, -2767624/2813001, -1390400/2813001, -673865/2813001]]

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？