
05602466
a
2009e.00389
Rehlich, Hartmut
The length of cycles in Fibonaccisequences modulo $M$.
Fritzlar, Torsten (ed.), Problem solving in mathematics education. Proceedings of the 9th ProMath conference, L\"uneburg, Germany, August 30 September 2, 2007. Hildesheim: Franzbecker (ISBN 9783881204736/pbk). 153168 (2008).
2008
Hildesheim: Franzbecker
EN
F65
D55
problem solving
Fibonacci sequences
Summary: A simple logical conclusion, using Dirichlet's box principle, proves that the Fibonaccisequence module $M$ is cyclic for every $M$. But how long are the cycles? Material collected for the purpose of patternidentification 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 fillin 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 Fibonaccinumbers, amazing patterns appear in the sequence of the respective minimal distance between the positions occupied up to that time.