Question

the relationship between the Fibonacci sequence and the periodic table​

Answer 1

Answer:

The periodic nature of the fibonacci sequence modulo m

linear-algebra group-theory matrices finite-groups

Let xn denote the n-th element of the fibonacci sequence and

A:=(0111)

It's easy to show, that it holds:

An=(Fn−1FnFnFn+1)

However, I want to show that

(Fn mod m)n(m∈N)

is a periodic sequence. Therefor, it's sufficient to show, that

(An mod m)n

is periodic. In other words: We need to show, that A is an element of finite order in GL(2,Z/mZ). What's the most elegant way to do that?

PS: I know that it might be better to choose A and thereby An in an other way, but I'm asked to show the statement for the given choice of A.

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