# Carmichael's Theorem

## Theorem

Let $n \in \Z$ such that $n > 12$.

Then the $n$th Fibonacci number $F_n$ has at least one prime factor which does not divide any smaller Fibonacci number.

The exceptions for $n \le 12$ are:

$F_1 = 1, F_2 = 1$: neither have any prime factors
$F_6 = 8$ whose only prime factor is $2$ which is $F_3$
$F_{12} = 144$ whose only prime factors are $2$ (which is $F_3$) and $3$ (which is $F_4$).

## Proof

We have that:

$1$ has no prime factors.

Hence, vacuously, $1$ has no primitive prime factors.

$8 = 2^3$

and $2 \divides 2 = F_3$

$144 = 2^4 3^2$

and:

$2 \divides 8 = F_6$
$3 \divides 21 = F_8$

for example.

## Source of Name

This entry was named for Robert Daniel Carmichael.