The Fibonacci sequence (with , ) has , , and . I was wondering whether for any other and . 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 such that for any ?

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 are the prime factors of , so if I could show that every one of 's prime factors was a prime factor of some for , then I would have my contradiction, but I can't get it.)