{
  "id": "2410.14133",
  "version": "v1",
  "published": "2024-10-18T03:01:28.000Z",
  "updated": "2024-10-18T03:01:28.000Z",
  "title": "On Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Divisors",
  "authors": [
    "Likun Xie"
  ],
  "comment": "41 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "We establish quantitative lower bounds for the number of primes $p$ such that, for an odd integer $b$, $p - b \\leq N$ and $p - b = 2^{k(N) + m} P_k \\quad \\text{for some } m \\geq 0$, where $P_k$ is a product of at most $k$ primes other than 2, for $k \\geq 2$. These results hold under various upper bounds for $2^{k(N)}$ depending on $k $, specifically when $2^{k(N)} \\leq N^a$ for different values of $a$. Our approach combines techniques from Chen's theorem, weighted sieves, and Selberg's lower bound sieve, as well as results on primes in arithmetic progressions with large power factor moduli.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-18T03:01:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "selbergs lower bound sieve",
      "large power factor moduli",
      "chens theorem",
      "quantitative lower bounds",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}