{
  "id": "2301.06970",
  "version": "v1",
  "published": "2023-01-17T15:44:20.000Z",
  "updated": "2023-01-17T15:44:20.000Z",
  "title": "Infinitely many primes in each of the residue classes $1$ and $8$ modulo $9$ are sums of two rational cubes",
  "authors": [
    "Somnath Jha",
    "Dipramit Majumdar",
    "B. Sury"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given an integer $n$, it is a classical Diophantine problem to determine whether $n$ can be written as a sum of two rational cubes. The study of this problem has a copious history that can be traced back to the works of Sylvester, Satg{\\'e}, Selmer etc. and up to the recent works of Alp{\\\"o}ge-Bhargava-Shnidman. In this short note, we prove that infinitely many primes congruent to $1$ modulo $9$ are sums of two rational cubes and that infinitely many primes congruent to $8$ modulo $9$ are sums of two rational cubes. Among other results, we prove also that every residue class $a \\pmod {q}$ (for any prime $q$ and any integer $a$ co-prime to $q$) contains infinitely many primes which are sums of two rational cubes. Further, we provide many infinite families of composite numbers which are rational cube sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-17T15:44:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "residue class",
      "primes congruent",
      "rational cube sums",
      "classical diophantine problem",
      "infinite families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}