Some Thoughts on Determining Symmetric Palintiples

TL;DR

This paper explores methods for identifying symmetric palintiples, classifies their types based on digit patterns, and examines the existence of asymmetric palintiples across different numerical bases.

Contribution

It introduces a new strategy for finding symmetric palintiples and analyzes the conditions under which asymmetric palintiples do not exist.

Findings

Classifies palintiples into symmetric, shifted-symmetric, and asymmetric categories.

Provides a method for determining all symmetric palintiples.

Identifies bases where asymmetric palintiples are absent.

Abstract

A palintiple is a natural number which is an integer multiple of its digit reversal. A previous paper partitions all palintiples into three distinct classes according to patterns in the carries and then determines all palintiples belonging to the shifted-symmetric class. In this paper we consider a strategy for finding all symmetric palintiples based upon recent work. We also discuss the last case of asymmetric palintiples and consider bases for which asymmetric palintiples do not exist.