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One of my favourite easy group theory problems:

Any prime number $p$ divides $f_{2p(p^2-1)}$, where $f_n$ is the Fibonacci sequence.

Proof: Let

$$G:= \{ A \in M_2 (Z_p) | \det(a)= \pm1 }$$\}$$and let$$F=\left( \begin{array}{c c} 1 & 1 \\ 1 & 0 \\ \end{array}\right) $$Then F \in G and G is a group of order 2p(p^2-1). Thus$$\left( \begin{array}{c c} f_{n+1+2p(p^2-1)} & f_{n+2p(p^2-1)} \\ f_{n+2p(p^2-1)} & f_{n+2p(p^2-1)-1} \end{array}\right)= F^{n+2p(p^2-1)}= F^n = \left( \begin{array}{c c} f_{n+1} & f_{n} \\ f_{n} & f_{n-1} \end{array}\right) \mod p \,.$$P.S. Better periods can be obtained by solving the linear reccurence in Z_p if p =\pm1 \mod 5 or in an algebraic extension if p =\pm 2 \mod 5, but that's exactly the same thing as calculating the order of the matrix F by diagonalizing it. The same idea can be used for any linear recurrence, but one has to replace G by GL_2(Z_p), and discuss the cases when p divides or doesn't divide the free term of the polynomial associated to teh recurrence. PPS: Can anyone please fix my matrices. 1 [made Community Wiki] One of my favourite easy group theory problems: Any prime number p divides f_{2p(p^2-1)}, where f_n is the Fibonacci sequence. Proof: Let$$G:= { A \in M_2 (Z_p) | \det(a)= \pm1 }$$and let$$F=\left( \begin{array}{c c} 1 & 1 \ 1 & 0 \ \end{array}\right) $$Then F \in G and G is a group of order 2p(p^2-1). Thus$$\left( \begin{array}{c c} f_{n+1+2p(p^2-1)} & f_{n+2p(p^2-1)} \ f_{n+2p(p^2-1)} & f_{n+2p(p^2-1)-1} \end{array}\right)= F^{n+2p(p^2-1)}= F^n = \left( \begin{array}{c c} f_{n+1} & f_{n} \ f_{n} & f_{n-1} \end{array}\right) \mod p \,.

P.S. Better periods can be obtained by solving the linear reccurence in $Z_p$ if $p =\pm1 \mod 5$ or in an algebraic extension if $p =\pm 2 \mod 5$, but that's exactly the same thing as calculating the order of the matrix $F$ by diagonalizing it.

The same idea can be used for any linear recurrence, but one has to replace $G$ by $GL_2(Z_p)$, and discuss the cases when $p$ divides or doesn't divide the free term of the polynomial associated to teh recurrence.

PPS: Can anyone please fix my matrices.