{
  "id": "2307.01815",
  "version": "v1",
  "published": "2023-07-04T16:41:12.000Z",
  "updated": "2023-07-04T16:41:12.000Z",
  "title": "On perfect powers that are sums of cubes of a nine term arithmetic progression",
  "authors": [
    "Nirvana Coppola",
    "Mar Curcó-Iranzo",
    "Maleeha Khawaja",
    "Vandita Patel",
    "Özge Ülkem"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that the only integral solutions to the equation $(x-4r)^3 + (x-3r)^3 + (x-2r)^3+(x-r)^3 + x^3 + (x+r)^3+(x+2r)^3 + (x+3r)^3 + (x+4r)^3 = y^p$ satisfy the condition $xy=0$ if $p\\geq 5$ is a prime. We also show that there are infinitely many solutions for $p=2$ and $p=3$. This is a natural continuation of previous work carried out by A. Arg\\'{a}ez-Garc\\'{i}a and the fourth author. We use an amalgamation of existing methods to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier's Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-04T16:41:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41",
      "11D59",
      "11J86",
      "14H52"
    ],
    "keywords": [
      "term arithmetic progression",
      "perfect powers",
      "thue equation solver earlier",
      "voutiers primitive divisor theorem",
      "significant computational efficiency"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}