Fibonacci[1,000,000,000] contains 208,987,640 decimal digits

From: Michael Taktikos (michael.taktikos_at_hanse.net)
Date: 08/09/04 

Date: Mon, 9 Aug 2004 16:21:17 +0200

"Ed Pegg Jr" <edpegg@gmail.com> wrote in comp.soft-sys.math.mathematica:
> For finding the number of digits in any sufficently large Fibonacci
> number in base 10, let k = (ArcCsch[2])/Log[10]. The number of
> digits in Fibonacci[n] is Round[n k].
>
> k ~ 0.20898764024997873376927208923755541682245923991821
>
> Fibonacci[10^9] has 10^9 k ~ 208987640 digits.

Thank you for confirming this result. It means, that the number mentioned in
Sloane's sequence A068070
(number of digits of the 10^n_th Fibonacci number), like some other numbers
in this sequence,
are wrong.
N. Sloane get now a comment to A068070 with the corrected sequence
{0,2,21,209,2090,20899,208988,2089877,20898764,208987640,2089876402,20898764
0
25,208987640250,2089876402500,20898764024998,208987640249979,
2089876402499787,20898764024997873,208987640249978734,...}.

In addition, here is a fast way to find the n_th Tribonacci number:

rho=1/3*((19+3*Sqrt[33])^(1/3)+(19-3*Sqrt[33])^(1/3)+1);
triboappr[n_]:=N[(rho-1)/(4rho-6)*rho^(n-3),3000];

For getting the digits of the 10^n_th Tribonacci number, we could try a
constant k_2 = 0.264649...
(similar to the k above), or quick and dirty

Table[MantissaExponent[triboappr[10^i]][[2]],{i,1,7}]

and get

{2, 26, 264, 2646, 26464, 264649, 2646494,...}

Until now, this sequence was not in Sloane's database and is sent there.

Greetings from Hamburg,

Michael Taktikos

P.S. Because the last mail I sent to comp.soft-sys.mathematica somehow got
lost, this mail will appear in sci.math.symbolic, too.

Relevant Pages

  • Re: user defined function that converts string to float
    ... > I need user defined function that converts string to float in c. ... initial, possibly empty, sequence of white-space characters (as ... point character, then an optional exponent part as defined in ... then a nonempty sequence of hexadecimal digits ...
    (comp.lang.c)
  • Re: determinism, freewill, chaos, and circular causality
    ... classes, including sequence prediction, strategic games, function ... the squares each with its decimal digits ... Do the algorithms for generating Pi all require infinite memory? ... numbers" and the qualifcation is that there is no shorter input string ...
    (comp.ai.philosophy)
  • Re: Randomness
    ... I started with the first however many digits of pi (a number whose ... the first sequence gives a 1 for every digit that is a 0 or 1 ... designating a coin or die as fair in order to establish the normality ... weighting, determine whether this sequence is random" you ...
    (sci.math)
  • Re: Randomness
    ... I started with the first however many digits of pi (a number whose ... the first sequence gives a 1 for every digit that is a 0 or 1 ... designating a coin or die as fair in order to establish the normality ... weighting, determine whether this sequence is random" you ...
    (sci.math)
  • Re: New countable infiniity logic
    ... >infinite number of digits, and therefore 0.333... ... you are confusing the number of digits in the sequence with the ... it is an infinite sequence that *BY YOUR DEFINITION* does not ... A list of naturals is not a list of digits used to express those ...
    (sci.math)