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A NOTE ON THE INTEGRAL POINTS ON SOME HYPERBOLAS Hansaem Ko a and Yeonok Kim b; ¤ 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 su±cient conditions for the existence of the integral points on the hyperbola ha : x2 ¡ axy + y2 = 1 and hk : x2 ¡ axy + y2 = ¡k (k 2 Z>0). To list all the integral points on that hyperbola, we ¯nd the number of elements of k. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print) 1226-0657 http://dx.doi.org/10.7468/jksmeb.2013.20.3.137 ISSN(Online) 2287-6081 Volume 20, Number 3 (August 2013), Pages 137{148)

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