{
  "id": "2411.11103",
  "version": "v1",
  "published": "2024-11-17T15:49:12.000Z",
  "updated": "2024-11-17T15:49:12.000Z",
  "title": "Sums of S-units in X-coordinates of Pell equations",
  "authors": [
    "Parvathi S Nair",
    "Sudhansu Sekhar Rout"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $S$ be a fixed set of primes and let $(X_{l})_{l\\geq 1}$ be the $X$-coordinates of the positive integer solutions $(X, Y)$ of the Pell equation $X^2-dY^2 = 1$ corresponding to a non-square integer $d>1$. We show that there are only a finite number of non-square integers $d>1$ such that there are at least two different elements of the sequence $(X_{l})_{l\\geq 1}$ that can be represented as a sum of $S$-units with a fixed number of terms. Furthermore, we solve explicitly a particular case in which two of the $X$-coordinates are product of power of two and power of three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-17T15:49:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B37",
      "11D45",
      "11J86"
    ],
    "keywords": [
      "pell equation",
      "non-square integer",
      "x-coordinates",
      "positive integer solutions",
      "finite number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}