{
  "id": "2307.00770",
  "version": "v1",
  "published": "2023-07-03T06:20:16.000Z",
  "updated": "2023-07-03T06:20:16.000Z",
  "title": "A characterization of prime $v$-palindromes",
  "authors": [
    "Muhammet Boran",
    "Garam Choi",
    "Steven J. Miller",
    "Jesse Purice",
    "Daniel Tsai"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "An integer $n\\geq 1$ is a $v$-palindrome if it is not a multiple of $10$, nor a decimal palindrome, and such that the sum of the prime factors and corresponding exponents larger than $1$ in the prime factorization of $n$ is equal to that of the integer formed by reversing the decimal digits of $n$. For example, if we take 198 and its reversal 891, their prime factorizations are $198 = 2\\cdot 3^2\\cdot 11$ and $891 = 3^4\\cdot 11$ respectively, and summing the numbers appearing in each factorization both give 18. This means that $198$ and $891$ are $v$-palindromes. We establish a characterization of prime $v$-palindromes: they are precisely the larger of twin prime pairs of the form $(5 \\cdot 10^m - 3, 5 \\cdot 10^m - 1)$, and thus standard conjectures on the distribution of twin primes imply that there are only finitely many prime $v$-palindromes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-03T06:20:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A45",
      "11A63"
    ],
    "keywords": [
      "characterization",
      "prime factorization",
      "twin prime pairs",
      "corresponding exponents larger",
      "standard conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}