{
  "id": "2212.06127",
  "version": "v1",
  "published": "2022-12-12T18:55:55.000Z",
  "updated": "2022-12-12T18:55:55.000Z",
  "title": "On the index of appearance of a Lucas sequence",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbf{u} = (u_n)_{n \\geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \\geq 2$, where $a_1$ and $a_2$ are fixed nonzero integers. For each prime number $p$ with $p \\nmid 2a_2D_{\\mathbf{u}}$, where $D_{\\mathbf{u}} := a_1^2 + 4a_2$, let $\\rho_{\\mathbf{u}}(p)$ be the rank of appearance of $p$ in $\\mathbf{u}$, that is, the smallest positive integer $k$ such that $p \\mid u_k$. It is well known that $\\rho_{\\mathbf{u}}(p)$ exists and that $p \\equiv \\big(D_{\\mathbf{u}} \\mid p \\big) \\pmod {\\rho_{\\mathbf{u}}(p)}$, where $\\big(D_{\\mathbf{u}} \\mid p \\big)$ is the Legendre symbol. Define the index of appearance of $p$ in $\\mathbf{u}$ as $\\iota_{\\mathbf{u}}(p) := \\left(p - \\big(D_{\\mathbf{u}} \\mid p \\big)\\right) / \\rho_{\\mathbf{u}}(p)$. For each positive integer $t$ and for every $x > 0$, let $\\mathcal{P}_{\\mathbf{u}}(t, x)$ be the set of prime numbers $p$ such that $p \\leq x$, $p \\nmid 2a_2 D_{\\mathbf{u}}$, and $\\iota_{\\mathbf{u}}(p) = t$. Under the Generalized Riemann Hypothesis, and under some mild assumptions on $\\mathbf{u}$, we prove that \\begin{equation*} \\#\\mathcal{P}_{\\mathbf{u}}(t, x) = A\\, F_{\\mathbf{u}}(t) \\, G_{\\mathbf{u}}(t) \\, \\frac{x}{\\log x} + O_{\\mathbf{u}}\\!\\left(\\frac{x}{(\\log x)^2} + \\frac{x \\log (2\\log x)}{\\varphi(t) (\\log x)^2}\\right) , \\end{equation*} for all positive integers $t$ and for all $x > t^3$, where $A$ is the Artin constant, $F_{\\mathbf{u}}(\\cdot)$ is a multiplicative function, and $G_{\\mathbf{u}}(\\cdot)$ is a periodic function (both these functions are effectively computable in terms of $\\mathbf{u}$). Furthermore, we provide some explicit examples and numerical data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T18:55:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11N05",
      "11N37"
    ],
    "keywords": [
      "lucas sequence",
      "appearance",
      "prime number",
      "fixed nonzero integers",
      "legendre symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}