{
  "id": "2211.11012",
  "version": "v1",
  "published": "2022-11-20T16:18:57.000Z",
  "updated": "2022-11-20T16:18:57.000Z",
  "title": "Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials",
  "authors": [
    "Matteo Bordignon",
    "Ethan Simpson Lee"
  ],
  "comment": "18 pages, one table, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\\leq x$ such that a non-empty set of irreducible polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this set can contain as many polynomials as desired. To demonstrate, we present computations for some irreducible polynomials and obtain an explicit upper bound for the number of Sophie Germain primes up to $x$, which have practical applications in cryptography.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-20T16:18:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N32",
      "11N35",
      "11N36"
    ],
    "keywords": [
      "explicit upper bound",
      "irreducible polynomials",
      "primes simultaneously representable",
      "sophie germain primes",
      "selbergs upper sieve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}