# Carmichael's Theorem

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## Contents

## 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.

## Also see

## Source of Name

This entry was named for Robert Daniel Carmichael.

## Sources

- 1913: R.D. Carmichael:
*On the numerical factors of the arithmetic forms $\alpha^n + \beta^n$*(*Ann. Math.***Vol. 15**,*no. 1/4*: pp. 30 – 70) www.jstor.org/stable/1967797 - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $144$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $144$