$F_6=2^3$ 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?
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I think that this is an open question. Note that if $~p~$ is a prime number then $F_p \equiv \left(\frac{p}{5}\right) \pmod p ~~\text {and}~~ F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p$ It is not known whether there exists a prime $p$ such that: $F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod {p^2}$ See Wikipedia article for more information . |
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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. |
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