{
  "id": "1705.06087",
  "version": "v1",
  "published": "2017-05-17T11:01:05.000Z",
  "updated": "2017-05-17T11:01:05.000Z",
  "title": "On short products of primes in arithmetic progressions",
  "authors": [
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We give several families of reasonably small integers $k, \\ell \\ge 1$ and real positive $\\alpha, \\beta \\le 1$, such that the products $p_1\\ldots p_k s$, where $p_1, \\ldots, p_k \\le m^\\alpha$ are primes and $s \\le m^\\beta$ is a product of at most $\\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \\ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-17T11:01:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progressions",
      "short products",
      "reduced residue classes modulo",
      "reasonably small integers",
      "open question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}