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? 



Squarefree Fibonacci Numbers
I assume the traditional definition with $F\_0=0$ and $F\_1=1$. 


Most likely there are infinitely many squarefree Fibonacci numbers. A simple way to construct them is to consider a subsequence $F\_p$ for prime $p$. Notice that if $q^2\mid F\_p$ for some prime $q$, then $q$ must be a [Wall-Sun-Sun prime](https://en.wikipedia.org/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime), whose existence is a big open question (and even if they exist, they would be very rare).


