Some equations with features of digit reversal and powers

Abstract

In this paper we consider integers in base 10 like $abc$, where $a$, $b$, $c$ are digits of the integer, such that $ab{c}^2-(abc\cdot cba)=\pm n^2$, where $n$ is a positive integer, as well as equations $ab{c}^2-(abc\cdot cba)=\pm n^3$, and $ab{c}^3-(abc\cdot cba)=\pm n^2$ We consider asymptotic density of solutions. We also compare the results with ones with bases different from 10.