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Some Fibonacci Fun with Primes

I haven't written in way too long and just wanted to post this fun little proof.

Assertion: Let $F_n$ be the $n$th Fibonacci number defined by

$F_n = F_{n-1} + F_{n-2}, \quad F_0 = 0, F_1 = 1.$

Show that for an odd prime $p \neq 5,$ we have $p$ divides $F_{p^2 - 1}$.

Proof: We do this by working inside $\mathbb{F}_p$ instead of working in $\mathbb{R}$. The recurrence is given by

$\left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{c} F_{n-1} \\ F_{n-2} \end{array} \right) = \left(\begin{array}{c} F_{n-1} + F_{n-2} \\ F_{n-1} \end{array} \right) = \left(\begin{array}{c} F_n \\ F_{n-1} \end{array} \right)$

and in general

$\left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)^{n} \left( \begin{array}{c} 1 \\ 0 \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)^{n} \left( \begin{array}{c} F_1 \\ F_0 \end{array} \right) = \left(\begin{array}{c} F_{n + 1} \\ F_n \end{array} \right)$

The matrix

$A = \left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)$

has characteristic polynomial

$\chi_A(t) = (1 - t)(0 - t) - (1)(1) = t^2 - t - 1$

If this polynomial has distinct roots, then $A$ is diagonalizable (this is sufficient, but not necessary). Assuming the converse we have $\chi_A(t) = (t - \alpha)^2$ for some $\alpha \in \mathbb{F}_p$; we can assume $\alpha \in \mathbb{F}_p$ since $-2\alpha = -1$ is the coefficient of $t,$ which means $\alpha = 2^{-1}$ (we are fine with this since $p$ odd means that $2^{-1}$ exists). In order for this to be a root of $\chi_A,$ we must have

\begin{aligned} 0 \equiv 4 \cdot \chi_A\left(2^{-1}\right) &\equiv 4 \cdot \left(2^{-2} - 2^{-1} - 1\right) \\ &\equiv 1 - 2 - 4 \equiv -5 \bmod{p}. \end{aligned}

Since $p \neq 5$ is prime, this is not possible, hence we reached a contradiction and $\chi_A$ does not have a repeated root.

Thus we may write $\chi_A(t) = (t - \alpha)(t - \beta)$ for $\alpha, \beta \in \mathbb{F}_{p^2}$ (it's possible that $\chi_A$ is irreducible over $\mathbb{F}_p,$ but due to degree considerations it must split completely over $\mathbb{F}_{p^2}$. Using this, we may write

$A = P \left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array} \right) P^{-1}$

for some $P \in GL_{2} \left(\mathbb{F}_{p^2}\right)$ and so

$A^{p^2 - 1} = P \left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array} \right)^{p^2 - 1} P^{-1} = P \left(\begin{array}{cc} \alpha^{p^2 - 1} & 0 \\ 0 & \beta^{p^2 - 1} \end{array} \right)P^{-1}$

Since $\chi_A(0) = 0 - 0 - 1 \neq 0$ we know $\alpha$ and $\beta$ are nonzero, hence $\alpha^{p^2 - 1} = \beta^{p^2 - 1} = 1 \in \mathbb{F}_{p^2}$. Thus $A^{p^2 - 1} = P I_2 P^{-1} = I_2$ and so

$\left(\begin{array}{c} F_p \\ F_{p^2 - 1} \end{array} \right) = \left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right)^{p^2 - 1} \left(\begin{array}{c} 1 \\ 0 \end{array} \right) = I_2 \left( \begin{array}{c} 1 \\ 0 \end{array} \right) = \left( \begin{array}{c} 1 \\ 0 \end{array} \right)$

so we have $F_{p^2 - 1} = 0$ in $\mathbb{F}_p$ as desired.