{
  "id": "2307.02749",
  "version": "v1",
  "published": "2023-07-06T03:18:40.000Z",
  "updated": "2023-07-06T03:18:40.000Z",
  "title": "The Local-Global Conjecture for Apollonian circle packings is false",
  "authors": [
    "Summer Haag",
    "Clyde Kertzer",
    "James Rickards",
    "Katherine E. Stange"
  ],
  "comment": "25 pages, 4 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a primitive integral Apollonian circle packing, curvatures that appear must fall into one of six or eight residue classes modulo 24. The Local-Global Conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. We then formulate a new conjecture, and give computational evidence in support of it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-06T03:18:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C26",
      "11D99",
      "11-04",
      "20H10",
      "11E12",
      "11A15",
      "11B99"
    ],
    "keywords": [
      "residue classes modulo",
      "local-global conjecture states",
      "sufficiently large integer",
      "computational evidence",
      "primitive integral apollonian circle packing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}