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# 20 digits....

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How do I find the first 20 digits and the last 20 digits of this very large number: 2027^2029^2039? Any hint or help would be greatly appreciated. Thank you.

Mar 25, 2019

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To find the first and the last 20 digits of this very large number: 2027^2029^2039 proceed as follows:
1 - Take the log of 2029 and multiply by 2039 to get:6743.548 093 900 992 469 000............etc.

2 - To (1) above add the log(log(2027)) to get:6744.067 845 424 099 918 773 355.............etc.

3 - The above calculations should be carried to an accuracy of a minimum of 7000 digits in (2) above, and preferably, to 8000 digits to get the fractional part of the log accurate enough.
4 - Raise the number in (2) above to the power of 10 and again must carry the caculation to an accuracy of about 8,000 digits.
5 - Raise the fractional part in (4) above to the power of 10 and that should give you the first few hundred ACCURATE digits of the beginning of the number in (1) above, which begins with: 1,203, 717, 813, 292, 743, 410, 727.....

An alternative approach to the above is as follows:
1 - Take the log of 2027 to an accuracy of 8,000 digits and multiply by:  2029^2039 and that should give the requisite  accuracy required for the fractional part.
2 - Take the fractional part in (1) above and raise it to the power of 10 and that should give a few hundred accurate digits of the beginning of 2027^2029^2039.

To get the last 20 digits of the same number follow this procedure:
Take (2027^2029^2039) mod 10^20 and should get the last 20 digits:  16, 412, 296, 620, 638, 392, 187.
However, few calculators can tackle such gigantic numbers such as Wolfram/Alpha, or a programming language such as C++, which is what I used to calculate these numbers.

Mar 25, 2019