The fundamental period of a periodic phenomenon pertaining to $v$-palindromes

Abstract

Natural numbers satisfying a certain unusual property are defined by the author in a previous note. Later, the author called such numbers $v$-palindromic numbers and proved a periodic phenomenon pertaining to such numbers and repeated concatenations of the digits of a number. It was left as a problem of further investigation to find the smallest period. In this paper, we provide a method to find the smallest period. Some theorems from signal processing are used, but we also supply our own proofs.