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n-digit ... n-th power (Posted on 2015-10-20) Difficulty: 3 of 5
How many n-digit positive integers exist which are an n-th power?

No Solution Yet Submitted by Ady TZIDON    
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Solution Non-counting solution Comment 2 of 2 |
The solution can easily be found by trying powers.  For example with a base of 6.  6^4 has 4 digits but 6^5 falls short.

What is then sought is for b^n > 10^(n-1)
nlog(b)=n-1
log(b)=1-1/n
1/n = 1-log(b)
n=1/(1-log(b))
for b=9 this gives 21.85 implying 21 is the largest n.

b largest n
9 21
8 10
7 6
6 4
5 3
4 2
3 1
2 1
1 1
The sum of the second column is 49.


  Posted by Jer on 2015-10-20 13:42:48
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