Characterization and enumeration of palindromic numbers whose squares are also palindromic

Abstract

Palindromic numbers are positive integers that remain unchanged when their decimal digits are reversed. We characterize palindromic numbers whose squares are also palindromic. We use this to determine the number of $n$-digit palindromic numbers whose squares are palindromic, and the number of palindromic numbers whose squares are palindromic and which are not greater than a fixed positive integer.