The length of cycles in Fibonacci-sequences modulo $M$. (English)

Fritzlar, Torsten (ed.), Problem solving in mathematics education. Proceedings of the 9th ProMath conference, Lüneburg, Germany, August 30‒ September 2, 2007. Hildesheim: Franzbecker (ISBN 978-3-88120-473-6/pbk). 153-168 (2008).

Summary: A simple logical conclusion, using Dirichlet’s box principle, proves that the Fibonacci-sequence module $M$ is cyclic for every $M$. But how long are the cycles? Material collected for the purpose of pattern-identification rapidly leads to different hypotheses. For some, we will outline proofs; others are intended as challenges for further work in incentive groups and seminars. The same applies to the second problem area in which a cyclic fill-in process (model of Phyllotaxis) is examined. In this process, $n$ regularly arranged positions on the circumference of a circle (places for leaves or kernels) are occupied, advancing invariably by the same place number $k$, that is to say: with an unchanging rotation angle. If $n$ and $k$ are Fibonacci-numbers, amazing patterns appear in the sequence of the respective minimal distance between the positions occupied up to that time.