Some equations with features of digit reversal and powers

TL;DR

This paper investigates specific digit-based equations involving reversals and powers in base 10, analyzing their solutions and asymptotic densities, and compares these results across different numerical bases.

Contribution

It introduces new equations involving digit reversal and powers, studies their solutions, and explores their asymptotic densities across various bases.

Findings

Identifies solutions to equations involving digit reversal and powers in base 10.

Analyzes the asymptotic density of these solutions.

Compares results across different numerical bases.

Abstract

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