{
  "id": "1707.09581",
  "version": "v1",
  "published": "2017-07-30T04:33:27.000Z",
  "updated": "2017-07-30T04:33:27.000Z",
  "title": "Construction of helices from Lucas and Fibonacci sequences",
  "authors": [
    "Mário M. Graça"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map is defined and we show that its three-dimensional representation is also a helix. After proving that the lattice points of these later helix satisfy certain diophantine Pell's equations we call it a Pell's helix. We prove that for P-Fibonacci and Pell's helices the respective ratio is an invariant, contrasting to the P-Lucas helices whose ratio depends on P. It is also shown that suitable linear combinations of certain complex-valued maps lead to new helices related to Lucas/Fibonacci/Pell numbers. Graphical examples are given in order to illustrate the underlying theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-30T04:33:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "construction",
      "pells helix",
      "diophantine pells equations",
      "suitable linear combinations",
      "p-lucas helices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}