[seqfan] Re: Any other thoughts, opinions on (in)finiteness of A020995?
israel at math.ubc.ca
israel at math.ubc.ca
Tue Mar 20 20:19:46 CET 2012
I made a slightly different plot for better visibility, as
David's suggested plot looks too close to a diagonal line.
A: y = digitsum(fib(x)) - x for 0 <= x <= 10000
B: y = (c-1) x
I uploaded this as Fibcomet.gif.
Robert Israel
University of British Columbia
On Mar 19 2012, David Wilson wrote:
>When one looks at a graph of "Goldbach's comet", one "sees" the extreme
>unlikelihood that the comet will ever touch the x-axis, providing a
>counterexample to Goldbach's conjecture.
>
>Perhaps a similar plot would give one a feeling for the likelihood that
>A020995 is complete. I suggest plotting
>
> Plot A: y = digitsum(fib(x))
> Plot B: Line y = cx where c = 4.5 log_10(phi) = .9404443811...
> Plot C: y = x
>
>for, say, 0 <= x <= 10000.
>
>I suspect that Plot A will appear as a "Fibonacci digit sum comet"
>hovering around line B, and that line C will eventually wander far
>enough away from the comet's "tail" that one would be extremely
>confident that there will be no more collisions.
>
>On 3/19/2012 8:11 PM, Sven Simon wrote:
>> Hello,
>>
>> 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?
>>
>> Al
>>
>> --
>> Alonso del Arte
>> Author at
>> SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
>> Musician at ReverbNation.com<http://www.reverbnation.com/alonsodelarte>
>>
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