v-palindromes: an analogy to the palindromes

TL;DR

This paper introduces the concept of v-palindromes, numbers related to their reversals through prime factorizations, proves their infinite existence across bases, and discusses related conjectures.

Contribution

It formally defines v-palindromes in various bases, proves their infinite occurrence, and explores conjectures, expanding the understanding of number properties related to reversals and factorizations.

Findings

v-palindromes exist in infinitely many bases

Prime factorization sums of numbers and their reversals are equal for v-palindromes

The paper proposes several conjectures on v-palindromes

Abstract

In the year 2007, the author discovered an intriguing property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, prime factorize each number, and sum the numbers appearing in each factorization, both sums are $18$. Such numbers are formally introduced in a short published note in 2018. These numbers are later named $v$-palindromes because they can be viewed as an analogy to the usual palindromes. In this article, we introduce the concept of a $v$-palindrome in base $b$, and prove their existence for infinitely many bases. Finally, we collect some conjectures on $v$-palindromes.