Solving Pell's Equation Using Suborbital Graphs

Abstract

The purpose of this paper is to establish a relationship between the suborbital graphs and integer solutions of Pell's equation of the form $x^2-Ny^2=1$, where $N$ is a non-square positive integer. This involves exploring novel suborbital graphs generated by the action of some specific modular subgroups on extended rational numbers. We developed an innovative combinatorial notation for integer solutions of Pell's equation.  Additionally, we present convincing results  concerning the vertices of the suborbital graphs that we construct.