Diophantine equations with sum of cubes and cube of sum

Bogdan A. Dobrescu, Patrick J. Fox

Published 2020-12-08Version 1

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.