{
  "id": "2407.20048",
  "version": "v1",
  "published": "2024-07-29T14:31:47.000Z",
  "updated": "2024-07-29T14:31:47.000Z",
  "title": "Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers",
  "authors": [
    "Brennan Benfield",
    "Oliver Lippard"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This paper proves several conjectures connecting the zeros in the Pisano period to the prime factors of $K$-Fibonacci numbers. The congruence classes of indices for $K$-Fibonacci numbers that are multiples of the prime factors of $m$ completely determine the number of zeroes in the Pisano period modulo $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T14:31:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B50"
    ],
    "keywords": [
      "fibonacci numbers",
      "prime factors",
      "connecting zeros",
      "fibonacci sequence",
      "pisano period modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}