Actually, I'm solving some exercises from the book "Finite Field" by Rudolf Lidl et al.
There is an exercise for which the idea is missing to solve it:
Let $r$ be the least period of the Fibonacci sequence in the finite field $F_q$ i.e. the sequence with $s_0= 0, s_1= 1$, and $s_{n+2}=s_{n+1}+s_n$, for $n \geq 0$. Let $p$ be the characteristic of $F_q$. Prove that $r=20$ if $p = 5$, that $r$ divides $p-1$ if $p =$ +/- 1 mod 5 and that $r$ divides $p^2-1$ in all other cases.
Could anyone help me with a good idea / a good proposition?
My favorite Fibonacci technique is the matrix formulation, which is well worth knowing and easily proved: $$ A^n= \begin{pmatrix}1&1\\1&0\end{pmatrix}^n= \begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix} $$
The first part of the question follows easily. Indeed, $A^2=A+I$ and so $A^5=5A+3I \equiv 3I \bmod 5$. Therefore, $A^{20} \equiv 3^{4} I \equiv I \bmod 5$. Since $A^4 = 3A + 2I \not\equiv I \bmod 5$, the period mod $5$ is $20$.