{
  "id": "2311.12044",
  "version": "v1",
  "published": "2023-11-17T21:00:43.000Z",
  "updated": "2023-11-17T21:00:43.000Z",
  "title": "On Modular Approach to Diophantine Equation $x^4-y^4=nz^p$ over Number Fields",
  "authors": [
    "Erman Isik"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2201.13270",
  "categories": [
    "math.NT"
  ],
  "abstract": "Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over various number fields. In this paper, we prove asymptotic results about the solutions of the Diophantine equation $x^4-y^4=nz^p$ over various number fields using the modular method. For instance, we prove that the asymptotic generalised Fermat Theorem for the equation $x^4-y^4=2^\\alpha z^p$ holds for infinitely many quadratic number fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-17T21:00:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11F80",
      "11F03",
      "11F75"
    ],
    "keywords": [
      "diophantine equation",
      "modular approach",
      "asymptotic generalised fermat theorem",
      "quadratic number fields",
      "modular method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}