{
  "id": "1305.3285",
  "version": "v3",
  "published": "2013-05-14T20:12:12.000Z",
  "updated": "2014-01-16T17:39:42.000Z",
  "title": "On the Hermite problem for cubic irrationalities",
  "authors": [
    "Nadir Murru"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional continued fraction (with pre--period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences. This multidimensional continued fraction is derived from a modification of the Jacobi algorithm, which is proved periodic if and only if the inputs are cubic irrationals. Moreover, this representation provides simultaneous rational approximations for cubic irrationals.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-01-16T17:39:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "11J68"
    ],
    "keywords": [
      "cubic irrationality",
      "hermite problem",
      "cubic irrationals",
      "linear recurrent sequences",
      "periodic multidimensional continued fraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.3285M"
    }
  }
}