关于网上【最贱的数学题】帖子的提问

nyy

nyy 发表于 2021-12-31 14:20
x = (-28 (a + b + 2 c))/(6 a + 6 b - c), \quad  y = (364 (a - b))/(
6 a + 6 b - c)  这个二元变换 ...

https://www.math.tamu.edu/~rojas/cubic2weierstrass.pdf

Transforming a general cubic elliptic curve equation to Weierstrass form

95% 的人解不出这道题《史上最贱的数学题》能否用数学软件计算出来？ - Nemesis XX的回答 - 知乎
https://www.zhihu.com/question/267427508/answer/323883323

如何得到那个变换，答案也许在这里面！

nyy

nyy 发表于 2023-3-28 16:01
https://www.math.tamu.edu/~rojas/cubic2weierstrass.pdf

Transforming a general cubic elliptic cu ...

数学中最著名的曲线和曲面分别是什么？ - 刘醉白的回答 - 知乎
https://www.zhihu.com/question/23697239/answer/2961227001

这个带图片的回答，似乎还不错！

nyy

nyy 发表于 2023-4-3 08:08
数学中最著名的曲线和曲面分别是什么？ - 刘醉白的回答 - 知乎
https://www.zhihu.com/question/2369723 ...

1. Clear["Global`*"];(*删除所有变量*)
   
2. (*通分,合并同类项,取得分子*)
   
3. f=a/(b+c)+b/(a+c)+c/(a+b)-4//Together//Numerator
   
4. (*找到适合方程的整数解*)
   
5. Do[If[f==0&&And[a+b!=0,b+c!=0,c+a!=0],Print[{a,b,c}]],{a,-12,12},{b,-12,12},{c,-12,12}]
   

复制代码

先找部分整数解

1. {-11,-9,5}
   
2. {-11,-4,1}
   
3. {-11,1,-4}
   
4. {-11,5,-9}
   
5. {-9,-11,5}
   
6. {-9,5,-11}
   
7. {-5,9,11}
   
8. {-5,11,9}
   
9. {-4,-11,1}
   
10. {-4,1,-11}
    
11. {-1,4,11}
    
12. {-1,11,4}
    
13. {1,-11,-4}
    
14. {1,-4,-11}
    
15. {4,-1,11}
    
16. {4,11,-1}
    
17. {5,-11,-9}
    
18. {5,-9,-11}
    
19. {9,-5,11}
    
20. {9,11,-5}
    
21. {11,-5,9}
    
22. {11,-1,4}
    
23. {11,4,-1}
    
24. {11,9,-5}

复制代码

先搞部分解再说，这只是一小步！

nyy

https://link.zhihu.com/?target=h ... I_43_from29to41.pdf

看不懂

An unusual cubic representation problem

但是转发过来，希望有人能看懂

补充内容 (2023-4-4 09:27):
https://ami.uni-eszterhazy.hu/up ... I_43_from29to41.pdf

nyy

nyy 发表于 2023-4-3 15:10
https://link.zhihu.com/?target=http%3A//ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf

...

The general procedure to transform a genus one cubic projective curve into Weierstrass form is outlined in the book by Silverman/Tate "Rational points on elliptic curves", Chapter I, Section 3, pages 22-23.

Some software packages can do this for you, for instance, Magma. There is an online Magma calculator http://magma.maths.usyd.edu.au/calc/ which is free to use as long as the calculation does not go over 120 seconds. Here is some code that verifies the equations in the paper.

F<N>:=FunctionField(Rationals());
PS<a,b,c>:=ProjectiveSpace(F,2);
C:=Curve(PS, N*(a+b)*(b+c)*(c+a)-a*(a+b)*(a+c)-b*(b+c)*(b+a)-c*(c+a)*(c+b));
P0:=C![1,-1,0];
F, CtoF:=EllipticCurve(C,P0);
E<X,Y,Z>:=EllipticCurve([0, 4*N^2+12*N-3, 0, 32*(N+3), 0]);
test,FtoE:=IsIsomorphic(F,E);
assert test eq true;
CtoE:=Expand(CtoF*FtoE);
CtoE;

Good wishes,
Andrew Bremner

这个作者给我的回信！