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 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. 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$是有理数。 我们证明存在无限多族这样的方程，包括那些$a$是任意立方数或某些有理分数的情况，它们具有非平凡解。 也存在无限多族没有任何非平凡解的方程，包括那些$1/a = 1- 24/m$在整数$m$的限制下的情形。 这些方程可以用椭圆曲线表示，除非$a = 9$或 1，任何非零$j$-不变量和挠群$\mathbb{Z}/3k\mathbb{Z}$的椭圆曲线，对于$k = 2,3,4$，或$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z} $对应于一个特定的$a$。 我们证明，对于任何 $a$，非平凡解的数量最多为3个或无限个，而对于整数 $a$，其解的数量要么是0，要么是 $\infty$。 对于 $a = 9$，我们找到了通解，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 $U(1)$ 规范群下的费米子电荷。