Diophantine equations with sum of cubes and cube of sum

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.