A characterization of prime $v$-palindromes

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.