>>12506336I conjecture, but haven’t yet proven that Fibonacci numbers are never a non-trivial power of 7. I did manage to use the fact the Fibonacci numbers are a strong divisibility sequence (
https://en.m.wikipedia.org/wiki/Divisibility_sequence) to show that if x is the smallest positive integer such that Fibonacci(x)=7^k for some positive integer k, then x is either prime or even. The “or even” is a consequence of the second Fibonacci number being 1.
You can prove that every prime power divides a Fibonacci number because every positive integer, n, divides some Fibonacci number. The proof of this follows from looking at a mod n analog of the Fibonacci numbers and seeing that it has to be periodic. (Look up Pisano period)
I’m giving up for now to spend time with family.
