Carmichael's Theorem

Theorem
The only Fibonacci numbers with no primitive prime factors are:
 * $F_1 = F_2 = 1$
 * $F_6 = 8$
 * $F_{12} = 144$

Proof
We have that:


 * $1$ has no prime factors.

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


 * $8 = 2^3$

and $2 \mathrel \backslash 2 = F_3$


 * $144 = 2^4 3^2$

and:
 * $2 \mathrel \backslash 8 = F_6$
 * $3 \mathrel \backslash 21 = F_8$

for example.