.
Randomly choose one Fibonacci number.
Will its leftmost digit more likely belong to the set {1, 4, 8, 9} or to the set {2, 3, 5, 6, 7}?
I didn't know about that alleged information. I went to this list of the first 300 Fibonacci numbers just
to quickly look at the leading 1's starting at n = 2 for F(2).
The first 300 Fibonacci numbers, factored
n
-------
2
2 + 67 = 69
69 + 67 = 136
136 + 67 = 203
203 + 67 = 270
F(n) corresponding to these have leftmost digits of 1.
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Possible open hint: Consider the distribution in Benford's Law.
If you look at the leftmost digit of $F(n+67)$, apparently, it is extremely similar to the leftmost digit of $F(n)$. It is the same something like 97% of the time. Although, IIRC, that was a heuristic approach that discovered that, and I am not sure if it has been proven yet. It was true for the first thousand or so terms.
This graph does a good job of showing how similar they are:
http://www.wolframalpha.com/input/?i...g%5B10%5D))%5D
Yes, one can look at the distribution of Benford's Law at this site: https://en.wikipedia.org/wiki/Benford%27s_law