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"Prove there does not exist a finite simple non-abelian group of order of a Fibonacci number"

I would like to answer the above question, but I currently have few ideas of where to begin.

I understand we will likely only be using results about the prime factorisation of fibonacci numbers. I considered basic results about simple groups, e.g. from the sylow theorems, if a prime $p | |G| $ and $ kp+1$ does not divide $|G|$ for all integers $k$, then the sylow-p subgroup is normal.

However I fail to see how exactly to use this, and other standard techniques.

I have heard before that no Fibonacci number is a perfect number, but again, I cannot see how to use this exactly.

Would someone be able to provide me with hints/ideas?

In particular, is there a specific property of simple groups or Fibonacci numbers that I need consider?

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    Very naive heuristic: Fibonacci numbers are very rare (asymptotically $c \cdot \log n$ of them up to $n$), and orders of nonabelian simple groups are pretty rare (it should follow from the classification that most of them are $\mathrm{PSL}_2(\mathbb F_p)$, with order $\sim p^3/2$, so asymptotically $\frac{(2n)^{1/3}}{\log((2n)^{1/3})}$ of them up to order $n$). So coincidences should be unlikely, at least with large numbers. However, many Fibonacci numbers are highly divisible (https://en.wikipedia.org/wiki/Fibonacci_number#Divisibility_properties), which makes them more likely to coincide. – 2017-01-15
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    If you are allowed to use Feit-Thompson, then you know that a non-abelian simple group has order a multiple of four. This immediately rules out $F_n$ as the order unless $6\mid n$. Doesn't really help, I know. – 2017-01-15
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    I found this: http://mathoverflow.net/a/11970/35478 Apparently the result is due to Florian Luca (who also proved that no Fibonacci number is a perfect number). – 2017-01-16
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    @user1729 yes, thanks. I did just come across this too earlier today, however I was unable to find any details of the proof online. Don't suppose you know where the best place is to look? – 2017-01-16
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    @EvgenyT No, sorry. I had a quick look around before I posted the link. According to Wikipedia, Florian Luca apparently has over 500 research articles. I could not find a personal web page, so there was no simple list of all their papers. If you need this result for research (as opposed to Prof. Luca being your prof. and they set you this question...) I would suggest posting this question on MathOverflow and linking to the question I gave above. – 2017-01-18
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    Or you could e-mail the user who made the comment on MO (they really should have given a citation!). The user gives their e-mail address on the profile: http://mathoverflow.net/users/1593/jos%c3%a9-hdz-stgo You could also e-mail Florian Luca himself (but with 500 papers he may never reply!). – 2017-01-18

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