Interseção de números geométricos via equação de Pell

Abstract

Our work had as main objective to study the intersection of integer sequences, denominated polygonal numbers, through Pell's equation. In this context, the solution of two equations will be treated: x2 􀀀 Dy2 = 1 and x2 􀀀 Dy2 = N, jNj > 1. For the rst one we have used results from the theory of continued fractions. For the last one, we have used the method of solution delineated in literature. Besides, propositions referring to the intersection of polygonal numbers for some particular cases are presented and demonstrated. Also, the proposition of the general case is presented and demonstrated. Finally, we have performed the solution of some of Pell's equations in order to determine the intersection of some polygonal numbers.