Fibonacci Matrix

Let

Then

for the Fibonacci Sequence.

Theorem

If prime and then

Proof

Follows from Wall’s Theorem.

Theorem

If prime and then

So the period but .

Proof

Notice if and with

so if has period
then has period dividing .
Conversely if is invertible
then the period of must divide that of .
It is invertible so long as

which is impossible for .
Then all starting points must have the same period.
Let be minimal such that .
Then

is a Group which acts on

by left multiplication sending

But as all elements of are invertible,
no has any fixpoints except for the identity which has .
All equivalence classes under multiplication by have size
Therefore

so and thus
We know that:

so

and we finally get thus: