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What are the first 10 digits of 2^(1000!). Or 2 raised to 1000 factorial? Any help or hint would be appreciated. Thank you.

 Sep 15, 2019
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2^(1000!):
1 - Calculate 1000! to its full accuracy or to 2,568 digits. There are "arbitrary precision" calculators on the Internet that can calculate such big numbers. Or, if you know a programming language, it can easily be calculated as well. 


2 - The above calculation comes to :
4.02387260077093773543702433923....... x  10^2567.


3 - Calculate log(2) base 10, to  at least the same accuracy as in (1) above, or even to 3000 decimal places, which would give: 0.30102999566398119521373889472449....etc.


4 - Multiply (2) above x (3) above also to a precision of about 3000 digits. It should start with:
1.2113063515624881214916220007003..... x  10^2567


5 - The fractional part of (4) above should start with:
0.8024233892872257177919623......etc.


6 - Raise the frational part in (5) above to the power of 10, or 10^ 0.8024233892872257177919623.....etc.


7 - The above should give you the first digits of 2^(1000!), which are:


8 - 63448796572785746651.......etc.


9 - Count the first 10 digits from the left:
6,344,879,657.........etc.


10 - And that is the END!!.

 Sep 15, 2019
edited by Guest  Sep 15, 2019

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