{
  "id": "2404.14098",
  "version": "v1",
  "published": "2024-04-22T11:35:09.000Z",
  "updated": "2024-04-22T11:35:09.000Z",
  "title": "Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$",
  "authors": [
    "Pedro-José Cazorla García"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-22T11:35:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41",
      "11F80",
      "11F11"
    ],
    "keywords": [
      "asymptotic fermats",
      "galois theory",
      "integer solutions",
      "modular method",
      "diophantine equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}