Title: Diophantine equations with sum of cubes and cube of sum Show Chinese title

Title: 三次方的和与和的立方的不定方程

Abstract: 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 ratio of cubes 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 certain restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1. If $a$ is an integer and two variables are equal and nonzero, there exist nontrivial solutions only for $a=4$ or 9; there are no solutions for $a = 4$ when $xyz \neq 0$. Without imposing constraints on the variables, we find the general solution for $a = 9$, 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. Show Chinese abstract

Abstract: 我们求解类型的不定方程$ \, a \, (x^3 + y^3 + z^3 ) = (x + y + z)^3$，其中$x,y,z$是整数变量，系数$a \neq 0$是有理数。 我们证明存在无限多这样的方程，包括那些其中$a$是任何立方体的比值或某些有理分数的情况，它们具有非平凡解。 也存在无限多没有任何非平凡解的方程，包括那些其中$1/a = 1 - 24/m$在整数$m$的某些限制下的情况。 这些方程可以通过椭圆曲线表示，除非$a = 9$或 1。 如果$a$是一个整数且两个变量相等且非零，则仅当$a=4$或 9 时存在非平凡解；当$xyz \neq 0$时，$a = 4$没有解。 在不对变量施加约束的情况下，我们找到了$a = 9$的通解，该解依赖于两个整数参数。 这些三次方程在粒子物理中很重要，因为它们决定了在$U(1)$规范群下费米子的电荷。