{
  "id": "1806.09034",
  "version": "v1",
  "published": "2018-06-23T20:34:42.000Z",
  "updated": "2018-06-23T20:34:42.000Z",
  "title": "Almost primes in various settings",
  "authors": [
    "Paweł Lewulis"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k \\geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\\mathcal{P}(n) = \\prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $\\Omega (\\mathcal{P} (n) ) \\leq \\rho_k$ infinitely often for some integer $\\rho_k$. We improve the possible values of $\\rho_k$ for $4 \\leq k \\leq 10$ assuming $GEH$. We also show that we can take $\\rho_5=14$ unconditionally. As a by-product of our approach we reprove the $\\rho_3=7$ result which was previously obtained by Maynard who used techniques specifically designed for this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-23T20:34:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear forms",
      "by-product",
      "techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}