The Fibonacci sequence (with $\displaystyle F_0=0$, $\displaystyle F_1=1$) has $\displaystyle F_1=1$, $\displaystyle F_5=5$, and $\displaystyle F_{12}=144$. I was wondering whether $\displaystyle F_n = n^e$ for any other $\displaystyle n>12$ and $\displaystyle e>1$. I wrote a short little program to look for them, but came up with nothing below 100,000.

Can it be proven that there does not exist $\displaystyle n>12$ such that $\displaystyle F_n=n^e$ for any $\displaystyle e$?

This seems to be related to Carmichael's Theorem but I can't seem to use it to get a contradiction. (The prime factors of $\displaystyle F_n$ are the prime factors of $\displaystyle n$, so if I could show that every one of $\displaystyle n$'s prime factors was a prime factor of some $\displaystyle F_i$ for $\displaystyle 1\leq i\leq n-1$, then I would have my contradiction, but I can't get it.)