The well known Fibonacci sequence $F_{0} = 0, F_{1} = 1$ and, by recurrence law, $F_{n+1}:=F_{n} +F_{n-1}$ for all $n\geq 1$, has the following property (proved by Carmichael in 1913):
With the exception of $F_{1} = F_{2} = 1, F_{6} = 8$ and $F_{12} = 144$ every Fibonacci number $F_{n}$ has a prime factor that is not a factor of any smaller Fibonacci number.
Such a prime factor is called a Primitive Divisor of $F_{n}$.
Anybody knows if in the decomposition in prime factors of $F_{n}$, with $n\not\in \{1,2,6,12\}$ can appear a power greater than $1$ of some primitive divisor of $F_{n}$? That is, if $p$ is some primitive divisor of $F_{n}$, then $p^k$ does not divide $F_{n}$ for $k > 1$?
Many thanks in advanced for your comments.
I don't have an answer, but here are some consolatory news: nobody has.
The question is hinged on the existence of the so-called Wall–Sun–Sun primes, which itself is an open question. See, it says right there, under "Equivalent definitions":
(They use some savage terminology in which the relation "$p$ is a primitive divisor of $F_n$" must be restated as "$n$ is the rank of apparition modulo $p$", or "$n=\alpha(p)$". With that in mind, the message is as clear as the day.)
The search of Wall–Sun–Sun primes up to $2^{64}$ brought no results.
So it goes.