{
  "id": "1311.7003",
  "version": "v3",
  "published": "2013-11-27T15:25:15.000Z",
  "updated": "2014-10-19T18:01:45.000Z",
  "title": "Consecutive primes in tuples",
  "authors": [
    "William D. Banks",
    "Tristan Freiberg",
    "Caroline L. Turnage-Butterbaugh"
  ],
  "comment": "Revised version",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\\mathcal{H}(x) = \\{gx + h_j\\}_{j=1}^k$ of linear forms in $\\mathbb{Z}[x]$, the set $\\mathcal{H}(n) = \\{gn + h_j\\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \\in \\mathbb{N}$. In this note, we deduce that $\\mathcal{H}(n) = \\{gn + h_j\\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \\in \\mathbb{N}$. We answer an old question of Erd\\H os and Tur\\'an by producing strings of $m + 1$ consecutive primes whose successive gaps $\\delta_1,\\ldots,\\delta_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $\\delta_{j-1} \\mid \\delta_j$ for $2 \\le j \\le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \\bmod D$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-02T22:29:26.000Z",
      "abstract": "In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for any admissible k-tuple H(x)={g_jx+h_j : j = 1,...,k} of linear forms in Z[x], the set H(n)={g_jn+h_j: j = 1,...,k} contains at least m primes for infinitely many n. In this short note, we show that if k is even larger in terms of m, then the Maynard-Tao theorem can be used to produce m-tuples H(x) for which the m primes that occur in H(n) are consecutive. We answer an old question of Erdos and Turan by producing strings of m+1 consecutive primes whose successive gaps delta_1,...,delta_m form an increasing (resp. decreasing) sequence. We also show that such strings exist for which delta_{j-1} divides delta_j for j = 2,...,m. For any coprime integers a and D we find strings of consecutive primes of arbitrary length in the congruence class a mod D.",
      "comment": "Theorem 1 improved: it holds for any k > exp(8m+5) --- as in the Marnard-Tao theorem. We thank A. Granville for pointing this out and allowing us to incorporate his proof. Acknowledgements added. Other minor corrections",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-19T18:01:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N35",
      "11N36",
      "11A41"
    ],
    "keywords": [
      "consecutive primes",
      "prime k-tuple conjecture",
      "linear forms",
      "maynard-tao theorem",
      "congruence class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7003B"
    }
  }
}