The Fibonacci sequence (with F_0=0, F_1=1) has F_1=1, F_5=5, and F_{12}=144. I was wondering whether F_n = n^e for any other n>12 and 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 n>12 such that F_n=n^e for any 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 F_n are the prime factors of n, so if I could show that every one of n's prime factors was a prime factor of some F_i for 1\leq i\leq n-1, then I would have my contradiction, but I can't get it.)