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.

中文翻译：

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具有立方和和立方和的丢番图方程

我们求解 $a(x^3+y^3+z^3)=(x+y+z)^3$ 类型的丢番图方程，其中 $x$, $y$, $z$ 是整数变量，并且系数 $a \neq 0$ 是有理数。我们展示了这些方程的无限族，包括那些 $a$ 是任何立方或某些有理分数的方程，它们具有非平凡解。也有无限的方程族没有任何非平凡解，包括那些 $1/a=1-24/m$ 对整数 $m$ 有限制的方程。方程可以用椭圆曲线表示，除非 $a=9$ 或 $1$，并且任何非零 $j$-不变量和挠群 $\mathbb{Z}/3k\mathbb{Z}$ 的椭圆曲线对于 $k= 2,3,4$ 或 $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ 对应于特定的 $a$。我们证明，对于任何 $a$，非平凡解的数量最多为 $3$ 或者是无限的，对于整数 $a$，它是 $0$ 或 $\infty$。对于$a=9$，我们找到了通解，它取决于两个整数参数。这些三次方程在粒子物理学中很重要，因为它们确定了 $U(1)$ 规范组下的费米子电荷。