{
  "id": "1604.06586",
  "version": "v1",
  "published": "2016-04-22T09:34:27.000Z",
  "updated": "2016-04-22T09:34:27.000Z",
  "title": "On the Representation of Primes by Binary Quadratic Forms, and Elliptic Curves",
  "authors": [
    "Michele Elia",
    "Federico Pintore"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\\mathbb F_q$-points of an elliptic curve over a finite field $\\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic forms of given discriminant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-22T09:34:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D09",
      "11Y40",
      "11E12"
    ],
    "keywords": [
      "binary quadratic forms",
      "elliptic curve",
      "representation",
      "mild technical conditions",
      "exploiting schoofs algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160406586E"
    }
  }
}