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
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): $144$