$v$-Palindromes: An Analogy to the Palindromes

TL;DR

This paper introduces the concept of v-palindromes, explores their existence across infinitely many bases, and provides infinite families of such numbers in specific bases, along with conjectures and open problems.

Contribution

It defines v-palindromes in base b, proves their infinite existence, and constructs infinite families in bases p+1 and p^2+1 for odd primes p.

Findings

Existence of v-palindromes in infinitely many bases

Construction of infinite families in bases p+1 and p^2+1

Collection of conjectures and open problems

Abstract

Around the year 2007, one of the authors, Tsai, accidentally discovered a property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, which have prime factorizations $198 = 2\cdot 3^2\cdot 11$ and $891 = 3^4\cdot 11$ respectively, and sum the numbers appearing in each factorization getting $2+3+2+11 = 18$ and $3+4+11 = 18$, both sums are $18$. Such numbers were 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. We also exhibit infinite families of $v$-palindromes in bases $p+1$ and $p^2+1$, for each odd prime $p$. Finally, we collect some conjectures and problems involving $v$-palindromes.