A complete comparison for the number of palindromes in different bases

Abstract

Let $ b $ and $ b_1 $ be distinct positive integers larger than $ 1 $, and let $ A_{b} (n) $ and $ A_{b_1} (n) $ be the number of palindromes in bases $ b $ and $ b_1 $ that are less than or equal to $ n $, respectively. In this article, we finish the comparative study of the functions $ A_b (n) $ and $ A_{b_1} (n) $. As a result, we present the full picture of the asymptotic behavior of their difference.

Keywords

• palindrome
• palindromic number
• b-adic expansion
• extremal order
• sign change