Carmichael's Theorem

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Theorem

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

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

The exceptions for $n \le 12$ are:

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


Proof


Also see


Source of Name

This entry was named for Robert Daniel Carmichael.


Sources