{
  "id": "2404.09171",
  "version": "v1",
  "published": "2024-04-14T07:31:10.000Z",
  "updated": "2024-04-14T07:31:10.000Z",
  "title": "On the solutions of the generalized Fermat equation over totally real number fields",
  "authors": [
    "Satyabrat Sahoo"
  ],
  "comment": "14 pages. arXiv admin note: text overlap with arXiv:2403.14640, arXiv:2301.09263",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be a totally real number field, and $ \\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation, i.e., $Ax^p+By^p+Cz^p=0$ over $K$ of prime exponent $p$, where $A,B,C \\in \\mathcal{O}_K \\setminus \\{0\\}$ with $ABC$ is even (in the sense that $\\mathfrak{P}| ABC$, for some prime ideal $\\mathfrak{P}$ of $ \\mathcal{O}_K$ with $\\mathfrak{P} |2$). For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$ (resp., of certain type in $K^3$), under some assumptions on $A,B,C$ (resp., for all $A,B,C \\in \\mathcal{O}_K \\setminus \\{0\\}$ with $ABC$ is even). We also present several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-14T07:31:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11R80",
      "11F80",
      "11G05",
      "11Y40"
    ],
    "keywords": [
      "totally real number field",
      "generalized fermat equation",
      "asymptotic solution",
      "prime exponent",
      "prime ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}