{
  "id": "2012.04139",
  "version": "v1",
  "published": "2020-12-08T00:54:23.000Z",
  "updated": "2020-12-08T00:54:23.000Z",
  "title": "Diophantine equations with sum of cubes and cube of sum",
  "authors": [
    "Bogdan A. Dobrescu",
    "Patrick J. Fox"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT",
    "hep-ph",
    "hep-th"
  ],
  "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 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-08T00:54:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D25",
      "11G05",
      "11D45",
      "11D85"
    ],
    "keywords": [
      "diophantine equations",
      "nontrivial solution",
      "infinite families",
      "gauge group",
      "integer variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}