{
  "id": "2106.13796",
  "version": "v1",
  "published": "2021-06-25T17:48:18.000Z",
  "updated": "2021-06-25T17:48:18.000Z",
  "title": "Some Bounds for Number of Solutions to $ax + by + cz = n$ and their Applications",
  "authors": [
    "Damanvir Singh Binner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In a recent work, the present author developed an efficient method to find the number of solutions of $ax+by+cz=n$ in non-negative integer triples $(x,y,z)$ where $a,b,c$ and $n$ are given natural numbers. In this note, we use that formula to obtain some simple looking bounds for the number of solutions of $ax+by+cz=n$. Using these bounds, we solve some special cases of a problem related to the generalization of Frobenius coin problem in three variables. Moreover, we use these bounds to disprove a recent conjecture of He, Shiue and Venkat regarding the solution structure of $ax+by+cz=n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-25T17:48:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "applications",
      "frobenius coin problem",
      "simple looking bounds",
      "non-negative integer triples",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}