Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

LlPTAI, K., An approximation problem concerning linear recurrences

An approximation problem concerning linear recurrences 15 Using by (5) we obtain I A

2+5 (12) d

XtVt

Z < r-y x\ V 1 + a

2 + D Let the index n be defined by |Ä

n_i| < |x| < \R

n\. For this n, by (1), (2), (6) and (12), we have B r / A

2 + 5 1 + a

2 + D \B\

n 1 I A

2 + 5 (l-(/Va)

nl)

ID

l» I A

2 + 5 a I B\

n n~

l \a\\l + a

2 + D \a\y/D\R

n^\ ' V 1 + a

2 + D , A

2 +

5 1 > I + a

2+ D ' x >d

x'

y'

z if n is sufficiently large, since |J3| > 1. This shows that, for any lattice point (x,y,z) defined as above, there is an n such that d

x,y

f

Z < d

n and |x| < |Ä

n|. This completes the proof. References [1] G. E. BERGUM, Addenda to Geometry of a generalized Simson's Formula, Fibonacci Quart. 22 N^l (1984), 22-28. [2] A. F. HORADAM, Geometry of a Generalized Simson's Formula, Fibonacci Quart. 20 N^2 (1982), 164-68. [3] D. JARDEN, Recurring Sequences, Riveon Lematematika

, Jerusalem (Israel), 1958. [4] J. P. Jones and P. Kiss, On points whose coordinates are terms of a linear recurrence, Fibonacci Quart. 31, (1993), 239-245. [5] P. KlSS, A Diophantine approximative property of second order linear recurrences, Period. Math. Hungar. 11 (1980), 281-287. [6] C. KLMBERLING, Fibonacci Hyperbolas, Fibonacci Quarterly, 28, N

£1 (1990), 22-27. [7] E. LUCAS, Theorie des fonctions numériques simplement periodiques, American J. Math., 1 (1978), 184-240, 289-321.