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# Number Theory

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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

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