Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 17, No. 1, pp. 19-24 (2001)

More on the distribution of the Fibonacci numbers modulo $\mathbf{5c}$

Gyula Darvasi

Department of Mathematics and Computer Science College of Nyíregyháza Po Box 166, Nyíregyháza, Hungary, H-4401

Abstract: The Fibonacci sequence $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\ge 2$ is purely periodic modulo $m$ with $2\le m\in\mathbb{N}$. Take any shortest full period and form a frequency block $B_m\in \mathbb{N}^m$ to consists of the residue frequencies within any full period. The purpose of this paper is to show that such frequency blocks can nearly always be produced by repetition of some multiple of their first few elements a certain number of times. The four theorems contain our main results where we show when this repetition does occur, what elements will be repeated, what is the repetition number and how to calculate the value of the multiple.

Keywords: Second-order linear recurrences, uniform distribution.

Classification (MSC2000): 11B39

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