Diophantine equations with sum of cubes and cube of sum
Bogdan A. Dobrescu, Patrick J. Fox
TL;DR
This work analyzes the Diophantine problem $a(x^3+y^3+z^3)=(x+y+z)^3$ for rational $a$, using elliptic curves to classify when primitive solutions exist and how many occur. It identifies infinite families of $a$ with primitive solutions, and infinite families with no primitive solutions, via explicit parametrizations and Dofs-type results; it then translates the problem to an elliptic curve whose rank and torsion controls the number of primitive triples $(x,y,z)$. For integer $a$, the number of primitive solutions is dichotomous (0 or $\infty$), with $a=9$ singled out as a case admitting a complete two-parameter general solution; a broader set of square-$a$ cases yields further infinite families and direct applications to particle-physics anomaly constraints. The work thus provides a comprehensive elliptic-curve framework linking Diophantine cubic equations to $U(1)$ charge assignments, torsion structures, and arithmetic geometry, while delivering explicit families and a detailed taxonomy of solvability. These results illuminate the interplay between Diophantine analysis and physics-inspired constraints, offering concrete parametrizations and a practical method to gauge the existence and abundance of primitive solutions across $a$.
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.