# Carmichael's Theorem

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## Contents

## Theorem

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

Then the $n$th Fibonacci number $F \left({n}\right)$ has at least one prime factor which does not divide any smaller Fibonacci number.

The exceptions for $n \le 12$ are:

- $F \left({1}\right) = 1, F \left({2}\right) = 1$: neither have any prime factors
- $F \left({6}\right) = 8$ whose only prime factor is $2$ which is $F \left({3}\right)$
- $F \left({12}\right) = 144$ whose only prime factors are $2$ (which is $F \left({3}\right)$) and $3$ (which is $F \left({4}\right)$).

## Proof

## 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*: 30 – 70) www.jstor.org/stable/1967797