{
  "id": "2409.02933",
  "version": "v1",
  "published": "2024-08-21T17:41:08.000Z",
  "updated": "2024-08-21T17:41:08.000Z",
  "title": "A Pair of Diophantine Equations Involving the Fibonacci Numbers",
  "authors": [
    "Xuyuan Chen",
    "Hung Viet Chu",
    "Fadhlannafis K. Kesumajana",
    "Dongho Kim",
    "Liran Li",
    "Steven J. Miller",
    "Junchi Yang",
    "Chris Yao"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a, b\\in \\mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique. Let $F_n$ be the $n$th Fibonacci number. When $(a,b) = (F_n, F_{n+1})$, it is known that there is an explicit formula for the unique solution $(x,y)$. We establish formulas to compute the solution when $(a,b) = (F_n^2, F_{n+1}^2)$ and $(F_n^3, F_{n+1}^3)$, giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T17:41:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11D04"
    ],
    "keywords": [
      "diophantine equations",
      "integral solution",
      "th fibonacci number",
      "unique solution",
      "explicit formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}