{
  "id": "2406.00840",
  "version": "v1",
  "published": "2024-06-02T19:22:48.000Z",
  "updated": "2024-06-02T19:22:48.000Z",
  "title": "Improved upper bounds on Diophantine tuples with the property $D(n)$",
  "authors": [
    "Chi Hoi Yip"
  ],
  "comment": "short notes, 4 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $n$ be a non-zero integer. A set $S$ of positive integers is a Diophantine tuple with the property $D(n)$ if $ab+n$ is a perfect square for each $a,b \\in S$ with $a \\neq b$. It is of special interest to estimate the quantity $M_n$, the maximum size of a Diophantine tuple with the property $D(n)$. In this notes, we show the contribution of intermediate elements is $O(\\log \\log |n|)$, improving a result by Dujella. As a consequence, we deduce that $M_n\\leq (2+o(1))\\log |n|$, improving the best-known upper bound on $M_n$ by Becker and Murty.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-02T19:22:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D09",
      "11D45"
    ],
    "keywords": [
      "diophantine tuple",
      "best-known upper bound",
      "perfect square",
      "intermediate elements",
      "non-zero integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}