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.

This theorem requires a proof. In particular: It remains to be shown that these are the only examples. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Zsigmondy's Theorem

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): $144$