[seqfan] Re: Any other thoughts, opinions on (in)finiteness of A020995?

israel at math.ubc.ca israel at math.ubc.ca
Tue Mar 20 23:14:52 CET 2012

The normal distribution (Central Limit Theorem) is not quite the right 
limit to consider. Instead, one can use the theory of Large Deviations (see 
e.g. Frank den Hollander, "Large Deviations", Fields Institute Monographs, 
American Mathematical Society 2000).

Suppose S_m is the sum of n independent uniformly-distributed random base-b 
digits. The moment generating function of a random base-b digit is phi(t) = 
(1 + e^t + ... + e^((b-1)t)/b. According to Cramér's theorem, for any fixed 
a > (b-1)/2,

lim_{m -> infty} 1/m P(S_m >= a m) = -I(a)

where I(z) = sup_t (z t - log phi(t)).  In particular, I(a) > 0.  
In the case for b=10 and a=1/log[10]((1+sqrt(5))/2)
(notice that here m corresponds to the number of digits, which would be
approximately n/a if we use S_m to model the digit sum of F_n),  
and I get I(a) = .004926712 approximately.  

Anyway, the conclusion is still true: in this random model P(S_m >= am) 
decays exponentially with m, so if we take independent S_m for a linearly 
sequence of m's, with probability 1 only finitely many S_m >= am.

Robert Israel
University of British Columbia

On Mar 19 2012, Sven Simon wrote:

> these numbers were checked up to large indices, and no further numbers 
> with the property were found. They were checked by David Terr for his 
> article (Link in A020995). He made a statistical analysis and predicted 
> 684+-26 numbers for the same property in base 11. In fact I found 710 of 
> these numbers in base 11 (A025490). Lucas numbers have the same size, but 
> there were only 284 having the property in base 11 (A025491). So you see, 
> that Terr's analysis was quite accurate. This difference is explained a 
> little - but not in detail - in Terr's article. With Lucas numbers the 
> sequence ends 948539,973261,983101,2528952, so there happened a 
> statistical runaway (right English?). But these numbers were checked up 
> to indices 7500000, so there is nearly no chance for more numbers even in 
> base 11. And the same is true for Fibonacci in base 10, where T.D. Noe 
> checked up to large indices (A004090). There is no prove for it, but you 
> won't find a new one in base 10 - I would bet. No risk no fun. Sven
> -----Ursprüngliche Nachricht----- Von: seqfan-bounces at list.seqfan.eu 
> [mailto:seqfan-bounces at list.seqfan.eu] Im Auftrag von Alonso Del Arte 
> Gesendet: Montag, 19. März 2012 18:26 An: Sequence Fanatics Discussion 
> list Betreff: [seqfan] Any other thoughts, opinions on (in)finiteness of 
> A020995?
>Long ago, Robert G. Wilson v suggested that the sequence of n such that the
>sum of the base 10 digits of Fibonacci(n) is equal to n (A020995) might be
>infinite, though only twenty terms are known and the largest known term is
>quite small. In 2006, Stefan quoted an argument from Robert Dawson that not
>only is the sequence likely finite, we might already know all the terms.
>Most recently, Charles posted the following argument in a comment for the
>Sequence of the Day for April 20, saying that it needs to be checked:
> "The number of digits in the n-th Fibonacci number is 
> <math>n\log_{10}\varphi+o(1)</math>, so the expected digit sum is about 
> 0.94n. Modeling the result as a normal distribution, the variance is 
> about <math>82.5\log_{10}\varphi</math> and so the heuristic probability 
> that the digits of <math>F_n</math> are large enough to be in the 
> sequence is about 
> <math>\operatorname{erfc}\left(n\frac{1-4.5\log_{10}\varphi}{\sqrt{165}}\rig 
> ht)/2.</math> This decays rapidly: by n = 10,000 it is below 
> <math>10^{-935}.</math>"
>Have you any other thoughts, opinions, those of you who have pondered this
>or similar questions?
>Alonso del Arte
>Author at
>Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>Seqfan Mailing list - http://list.seqfan.eu/
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