Bulletin of the
Korean Mathematical Society
BKMS

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Bull. Korean Math. Soc. 2013; 50(4): 1243-1259

Printed July 1, 2013

https://doi.org/10.4134/BKMS.2013.50.4.1243

Copyright © The Korean Mathematical Society.

On some behavior of integral points on a hyperbola

Yeonok Kim

SoongSil University

Abstract

In this paper, we study the root system of rank 2hyperbolic Kac-Moody algebras. We give some sufficient conditions for the existence of imaginary roots of square length $-2k \, ( k\in {\mathbb Z}_{>0})$. We also give several relations between the integral points on the hyperbola ${\mathfrak h}$ to show that the value of the symmetric bilinear form of any two integral points depends only on the number of integral points between them. We also give some generalizations of Binet formula and Catalan's identity.

Keywords: Kac-Moody algebra, hyperbolic type, integral point, Binet formula, Catalan's identity

MSC numbers: 17B67, 17B65, 11B39

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