Coincidences in numbers of graph vertices corresponding to regular planar hyperbolic mosaics

TL;DR

This paper investigates the relationships between sequences associated with regular hyperbolic mosaics, solving specific Diophantine equations to identify coincident elements in these sequences.

Contribution

It introduces a method to determine common elements in sequences linked to hyperbolic mosaics by solving related Diophantine equations.

Findings

Identified elements common to sequences of mosaics {4,5} and {p,q}.

Developed a framework for solving Diophantine equations arising from hyperbolic mosaic sequences.

Enhanced understanding of the numerical coincidences in hyperbolic tessellations.

Abstract

The aim of this paper is to determine the elements which are in two pairs of sequences linked to the regular mosaics $\{4,5\}$ and $\{p,q\}$ on the hyperbolic plane. The problem leads to the solution of diophantine equations of certain types.