2178 And All That

TL;DR

This paper studies numbers that are reverse multiples in various bases, introduces Young graphs to analyze their structure, and uses transfer-matrix methods to enumerate such numbers for bases 2 to 100.

Contribution

It extends Young's tree approach to classify (g,k)-reverse multiples and applies transfer-matrix techniques for enumeration across multiple bases.

Findings

Determined possible k values for bases 2 to 100.

Developed Young graphs to analyze reverse multiples.

Applied transfer-matrix method for enumeration.

Abstract

For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood.