{
  "id": "2107.05525",
  "version": "v1",
  "published": "2021-07-12T15:54:05.000Z",
  "updated": "2021-07-12T15:54:05.000Z",
  "title": "Intersections of binary quadratic forms in primes] Intersections of binary quadratic forms in primes and the paucity phenomenon",
  "authors": [
    "Alisa Sedunova"
  ],
  "comment": "Accepted for a publication in J. Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "The number of solutions to $a^2+b^2=c^2+d^2 \\le x$ in integers is a well-known result, while if one restricts all the variables to primes Erdos showed that only the diagonal solutions, namely, the ones with $\\{a,b\\}=\\{c,d\\}$ contribute to the main term, hence there is a paucity of the off-diagonal solutions. Daniel considered the case of $a,c$ being prime and proved that the main term has both the diagonal and the non-diagonal contributions. Here we investigate the remaining cases, namely when only $c$ is a prime and when both c,d are primes and, finally, when $b,c,d$ are primes by combining techniques of Daniel, Hooley and Plaksin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-12T15:54:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binary quadratic forms",
      "intersections",
      "paucity phenomenon",
      "main term",
      "non-diagonal contributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}