{
  "id": "2104.13256",
  "version": "v1",
  "published": "2021-04-27T15:15:34.000Z",
  "updated": "2021-04-27T15:15:34.000Z",
  "title": "An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots",
  "authors": [
    "Steven Jin",
    "Lawrence C. Washington"
  ],
  "comment": "12 pages, 7 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\\mathbb{F}_p)$. We prove unconditionally that $r(E,p)> 0.72\\log\\log p$ for infinitely many $p$, and $r(E,p) > 0.36 \\log p$ under the assumption of the Generalized Riemann Hypothesis. This can be viewed as an elliptic curve analogue of classical lower bounds on the least primitive root of a prime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-27T15:15:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curve analogue",
      "pillais lower bound",
      "primitive root",
      "classical lower bounds",
      "smallest non-negative integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}