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ISSN : 1226-0657

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Vol.20 No.3

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A NOTE ON THE INTEGRAL POINTS ON SOME HYPERBOLAS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2013, v.20 no.3, pp.137-148
https://doi.org/10.7468/jksmeb.2013.20.3.137
Ko, Hansaem
Kim, Yeonok
Ko,, H. , & Kim,, Y. (2013). A NOTE ON THE INTEGRAL POINTS ON SOME HYPERBOLAS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 20(3), 137-148, https://doi.org/10.7468/jksmeb.2013.20.3.137

Abstract

In this paper, we study the Lie-generalized Fibonacci sequence and the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We derive several interesting properties of the Lie-Fibonacci sequence and relationship between them. We also give a couple of sufficient conditions for the existence of the integral points on the hyperbola <TEX>$\mathfrak{h}^a:x^2-axy+y^2=1$</TEX> and <TEX>$\mathfrak{h}_k:x^2-axy+y^2=-k$</TEX> (<TEX>$k{\in}\mathbb{Z}_{</TEX><TEX>></TEX><TEX>0}$</TEX>). To list all the integral points on that hyperbola, we find the number of elements of <TEX>${\Omega}_k$</TEX>.

keywords
Lie-Fibonacci sequence, Lie-Fibonacci number, Kac-Moody algebra, hyperbolic type

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Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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