求所有x,y满足(x+7^(1/3)y)^3=(a+7^(1/3)b)(c+7^(1/3)d)

yigo

如题\(x,y,a,b,c,d\)均为整数，

求满足\((x+\sqrt[3]{7}y)^3=(a+\sqrt[3]{7}b)(c+\sqrt[3]{7}d)\)的所有\(x,y\)。

wayne

需要满足 $x^3 + 7 y^3 = a c, 3  x^2 y = a d + b c, 3  x y^2 = b d$,  消元得到 条件需要满足 $b^3 c^3 + a^3 d^3 + 7 b^3 d^3 = 0$

yigo

显然\((x,y)=(0,\ZZ),(\ZZ,0)\)是方程的解，下面考虑\(x,y\)为非零整数的解。

\((x+\sqrt[3]{7}y)^3=3xy^2(\sqrt[3]{7})^2+3x^2y\sqrt[3]{7}+x^3+7y^3\)，

把\(\sqrt[3]{7}\)看做未知元，题目关系式表明这个二次多项式可以因式分解，

则判别式\(\Delta=9x^4y^2-12xy^2(x^3+7y^3)=-3xy^2(x^3+28y^3)\)为完全平方数。

即\(-3x(x^3+28y^3)\)需为完全平方数，令\(z^2=-3x(x^3+28y^3)\)。

以下是问豆包给的解答，说要用椭圆曲线。

等式\(z^2=-3x(x^3+28y^3)\)两边除以\(x^4\)。

令\(u=\frac{z}{x^2},v=\frac{y}{x}\)，上式化为:\(u^2=-84v^3-3\)。

豆包让我去软件上求这个椭圆曲线的秩，求得秩为1，然后豆包告诉我这个椭圆曲线有无限多组有理数解。

可仅考虑\(x,y\)互质的解，\((x,y)=(1,-1),(3,-1),(28,-19)\)是满足要求的3组解。

lsr314

x/y的值不同的解有无穷多个,通过两边同时乘以适当的$k^6$可以求得整数解，此时$x,y$放大为$k^2$倍，$a,b,c,d$放大为$k^3$倍：
{x,y,a,b,c,d}=
{3,-3,18,-9,-9,-9}
{-5,5/3,100/9,-(25/3),-(25/3),5}
{-(219/38),219/56,-(815337/80864),-(47961/2128),-(47961/2128),657/56}
{-(1265/61),1265/3,-(2009882600/33489),-(1600225/183),-(1600225/183),1265}
{-(270813/40049),270813/24389,-(4831764861599658/39118063437989),-(73339680969/976755061),-(73339680969/976755061),812439/24389}
{-(9226981/1663260),9226981/4536000,372987596064464653859/37645631525980800000,-(85137178374361/7544547360000),-(85137178374361/7544547360000),9226981/1512000}
{-(573343926/2252725111),573343926/4500297451,1416528610056008649301254084/22837976411322734247522343771,-(328723257481093476/10137933074836992061),-(328723257481093476/10137933074836992061),1720031778/4500297451}
{2452184545855/244345860301,-2452184545855/48066200283),15612594627057314629877157182772091600/8609362963977438386011930189800249,-6013209046930092597681025/11744777059549804665183),-6013209046930092597681025/11744777059549804665183),-(2452184545855/16022066761)}
{18146281582545429/2349209147442082,-(18146281582545429/734115129078184),2233516407023034315927623154235643397976317964407/4051422548395042757838006904558987162171501216,-(329287535273027439158274848794041/1724589976506094615442189739088),-(329287535273027439158274848794041/1724589976506094615442189739088),-(54438844747636287/734115129078184)}