Solving Pell's equation using suborbital graphs

KÖROĞLU T.

FILOMAT, vol.38, no.19, pp.6863-6870, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 19
  • Publication Date: 2024
  • Doi Number: 10.2298/fil2419863k
  • Journal Name: FILOMAT
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.6863-6870
  • Karadeniz Technical University Affiliated: Yes

Abstract

We establish a connection between 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. We derive new suborbital graphs generated by the action of some specific modular subgroups on extended rational numbers. By using these graphs, we obtain a new combinatorial notation for the integer solutions of Pell's equation and some results on the vertices of the graphs studied here.