# Thread: Leftmost digit of a randomly chosen Fibonacci number

1. ## Leftmost digit of a randomly chosen Fibonacci number

.

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}?

2. ## Re: Leftmost digit of a randomly chosen Fibonacci number

Originally Posted by greg1313
.

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}?
IIRC, doesn't the leftmost digit have repetition? Something like it repeats every 67 numbers?

3. ## Re: Leftmost digit of a randomly chosen Fibonacci number

Originally Posted by SlipEternal
IIRC, doesn't the leftmost digit have repetition? Something like it repeats every 67 numbers?
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.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Possible open hint: Consider the distribution in Benford's Law.

4. ## Re: Leftmost digit of a randomly chosen Fibonacci number

Originally Posted by greg1313
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

These have leftmost digits of 1.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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

5. ## Re: Leftmost digit of a randomly chosen Fibonacci number

$\phi^{67}\approx 10^{14}$ is why the digits are so close.

Anyway, the split is almost 50/50 based on Bedford's law.

6. ## Re: Leftmost digit of a randomly chosen Fibonacci number

Originally Posted by SlipEternal

Anyway, the split is almost 50/50 based on Bedford's [sic] law.
Yes, one can look at the distribution of Benford's Law at this site: https://en.wikipedia.org/wiki/Benford%27s_law