

Title: A Beastly Number Post by ThudanBlunder on Nov 26^{th}, 2008, 9:05am a) Find the first 6 digits of (10^{666})! b) How many trailing zeros does the above number have? 

Title: Re: A Beastly Number Post by Barukh on Dec 2^{nd}, 2008, 10:32am For an easier part b), I get: [hide]2^{664}(5^{667}  1)[/hide] 

Title: Re: A Beastly Number Post by Barukh on Dec 3^{rd}, 2008, 12:31am According to the following article (http://www.pims.math.ca/pi/issue7/page1012.pdf), solving part a) may require calculation of a certain logarithm to a precision of more than 670 decimal digits! :o 

Title: Re: A Beastly Number Post by ThudanBlunder on Dec 4^{th}, 2008, 6:56pm on 12/02/08 at 10:32:03, Barukh wrote:
If we take n/4 as an estimate for the number of trailing zeros, we get 2.5*10^{665} Your number is 5 times this. 

Title: Re: A Beastly Number Post by Barukh on Dec 5^{th}, 2008, 12:15pm on 12/04/08 at 18:56:10, ThudanBlunder wrote:
Yes, I should've written 5^{666} instead. But now I realized the answer is incorrect anyway (doesn't take into account fractions). To write the answer in a "compact form" may be as difficult as part a) then... 

Title: Re: A Beastly Number Post by SMQ on Dec 5^{th}, 2008, 12:32pm http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subinfty.gif 2) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif _{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gif}10^{666}/5^{n}_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif} ^{n=1} Where in practice the upper bound can be reduced to _{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gif}666 log 10 / log 5_{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif} = 952
SMQ 

Title: Re: A Beastly Number Post by Barukh on Dec 6^{th}, 2008, 9:52am SMQ, you are right. My answer is wrong, since it doesn't take into account the second term in your formula  the sum which is challenging to evaluate. After working it out, I get the following answer to b): 2^{664}5^{666}  143. ::) I will supply details later, after I find the answer to the first question. 

Title: Re: A Beastly Number Post by Eigenray on Dec 6^{th}, 2008, 12:25pm on 12/06/08 at 09:52:06, Barukh wrote:
So is there a clever way to compute [hide]the sum of the digits of 2^{666} = 34004...23324_{5}[/hide]? Actually, the approximation [hide]333*log_{5}(2)[/hide] is pretty good here. 

Title: Re: A Beastly Number Post by Barukh on Dec 7^{th}, 2008, 6:27am on 12/06/08 at 12:25:31, Eigenray wrote:
I don't know. I used highprecision software (MPFR) to compute the number (I still want to get a confirmation it's correct). Quote:
Yes, but it may be quite inaccurate for other exponents. In the attached graph, I plotted the discrepancies between your approximation and actual number for all cases 10^{n} with 10 < n < 2000. Again, if my calculations are correct. 

Title: Re: A Beastly Number Post by Barukh on Dec 7^{th}, 2008, 11:21pm I get the following answer to part a): 13407273847... 

Title: Re: A Beastly Number Post by Eigenray on Dec 8^{th}, 2008, 7:57am on 12/07/08 at 23:21:10, Barukh wrote:
Are you sure it's not 134072738469787? But the first 6 digits are correct :) And the 143 is correct, as the following short (but rather inefficient) program shows: Code:


Title: Re: A Beastly Number Post by Barukh on Dec 8^{th}, 2008, 10:19am on 12/08/08 at 07:57:32, Eigenray wrote:
Hmm... I did calculate the logarithm with very high precision, so at least 15 digits should be accurate. Then, I did exponentiation with a double precision. Could it be I lost 5 digits of accuracy there? ??? What's your method? 

Title: Re: A Beastly Number Post by Eigenray on Dec 8^{th}, 2008, 12:44pm Code:
What do you get? Edit: And here it is with [link=http://www.mpfr.org/]MPFR[/link]: Code:
Gives 1.3407273846978712508 (I already had GMP but apparently it doesn't do logs. So I downloaded the MPFR sources and started ./configure; make. Then I realized I could install it through Cygwin, and wrote the above before it finished compiling.) 

Title: Re: A Beastly Number Post by ThudanBlunder on Dec 9^{th}, 2008, 4:03pm My source (http://books.google.co.uk/books?id=52N0JJBspM0C&pg=PA350&dq=leviathan+1735+n!&ei=zAc_SdD1AqaGzgTPnbnADg)for this problem got yet a different answer, 1.340727397... 

Title: Re: A Beastly Number Post by Eigenray on Dec 9^{th}, 2008, 5:32pm The funny thing is that he did compute the fractional part of 10^{666}/log(10) correctly to 14 digits. But he decided to round it to 8 digits before exponentiating, and then claim 10 digits of accuracy in the result: Quote:
which is silly. 

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