TITLE: Fibonacci numbers with largest prime factor appearing more than once
QUESTION [7 upvotes]: $F_6=2^3$ and $F_{12}=2^43^2$. Is there an $n>12$ such that $F_n=p^2k$ with $p$ prime and $k$ is $p$-smooth?

REPLY [2 votes]: There are $x\exp\left(-(1+o(1))\sqrt{\log x\log\log x}\right)$ numbers up to x which are divisible by the square of their largest prime factor. Interpreting this as a probability, there are heuristically about
$$
\int^\infty_{1000}e^{-\sqrt{\log\varphi\cdot x\log x}}dx\approx2.8\cdot10^{-24}
$$
such n. Insofar as there is no "conspiracy" among Fibonacci numbers, it seems likely that $F_{12}$ is the last example.