{
  "id": "2503.06069",
  "version": "v2",
  "published": "2025-03-08T05:36:47.000Z",
  "updated": "2025-06-25T17:18:47.000Z",
  "title": "Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof --",
  "authors": [
    "Hiroki Aoki",
    "Riku Higa",
    "Ryosei Sugawara"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \\geqq L $, there are at least $ N $ primes in the interval $ [kx, (k+1)x] $'' for $ k \\leqq 15 $. Our proof is based on the idea of the proof by Erd\\\"{o}s for $ k=1 $ and its improvement by Hitotsumatsu and by Sainose for $ k=2 $. In the case of $ k=3 $ and $ k=4 $, the method is very similar to the case of $ k=2 $, however, in the case of $ k \\geqq 5 $, we need new idea to complete the proof.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-25T17:18:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "11N05"
    ],
    "keywords": [
      "elementary proof",
      "real number",
      "hitotsumatsu",
      "proposition",
      "improvement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}