TITLE: Squarefree Fibonacci Numbers
QUESTION [8 upvotes]: Let $F_n$, $n\geq 0$, be the sequence of Fibonacci numbers, where $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 1$. A number is squarefree if it is is not divisible by the square of a prime number. 
Question: Are there infinitely many squarefree Fibonacci numbers?

REPLY [2 votes]: For prime $p$, let $M(p)$ be the least positive $n$ such that $p^2 \mid F_n$.  Then $p^2 \mid F_n$ iff $M(p) \mid n$.  Thus at most $N/M(p)$ of the first $N$ Fibonacci numbers
are divisible by $p^2$.  If we could prove that $\sum_p 1/M(p) < 1$, then  at least a positive fraction of all Fibonacci numbers will be squarefree.
Well, it seems to be true numerically; I don't know if it's provable.