论文标题
用立方体和总和立方体总和的双方方程
Diophantine equations with sum of cubes and cube of sum
论文作者
论文摘要
我们求解了类型$ a \，（x^3 \！ + \！y^3 \！ + \！z^3）的diophantine方程）=（x \！ + \！y \！ + \！ + \！z）^3 $，其中$ x，y，z $是整数变量，并且是整数变量，并且该系数$ a \ a \ neq 0 $ neq neq neq neq neqitations rational rationals rational。我们表明，有这些方程式的无限家庭，包括$ a $是任何立方体或某些理性分数，具有非平凡的解决方案。也有无限的方程式家庭没有任何非平凡的解决方案，其中包括$ 1/a = 1-24/m $的方程式，并且对整数$ m $限制了。等式可以用椭圆曲线表示，除非$ a = 9 $或1，否则任何椭圆曲线nonzero $ j $ -invariant and Torsion组$ \ mathbb {z}/3K \ Mathbb {z} $ for $ k = 2,3,4 $，或$ k = 2,3,4 $，或\ Mathbb {z}/6 \ Mathbb {z} $对应于特定的$ a $。我们证明，对于任何$ a $，非平凡解决方案的数量最多是3或无限，对于整数$ a $，它是0或$ \ iffty $。对于$ a = 9 $，我们找到了一般解决方案，该解决方案取决于两个整数参数。这些立方方程在粒子物理学中很重要，因为它们确定了$ u（1）$量规组下的费米电荷。
We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a = 1- 24/m$ with restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1, and any elliptic curve of nonzero $j$-invariant and torsion group $\mathbb{Z}/3k\mathbb{Z}$ for $k = 2,3,4$, or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} $ corresponds to a particular $a$. We prove that for any $a$ the number of nontrivial solutions is at most 3 or is infinite, and for integer $a$ it is either 0 or $\infty$. For $a = 9$, we find the general solution, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group.
