{
  "id": "2312.09985",
  "version": "v1",
  "published": "2023-12-15T17:57:14.000Z",
  "updated": "2023-12-15T17:57:14.000Z",
  "title": "On differences of perfect powers and prime powers",
  "authors": [
    "Pedro-José Cazorla García"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a prime number $q$ and a squarefree integer $C_1$, we develop a method to explicitly determine the tuples $(y, n, \\alpha)$ for which the difference $y^n-q^\\alpha$ has squarefree part equal to $C_1$. Our techniques include the combination of the local information provided by Galois representations of Frey-Hellegouarch curves with the effective resolution of Thue-Mahler equations, as well as the use of improved lower bounds for $q-$adic and complex logarithms. As an application of this methodology, we will completely resolve the case when $1 \\le C_1 \\le 20$ and $ 2 \\le q < 25$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-15T17:57:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41",
      "11D59",
      "11F80",
      "11F11"
    ],
    "keywords": [
      "prime powers",
      "perfect powers",
      "difference",
      "squarefree part equal",
      "squarefree integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}