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n-digit ... n-th power (Posted on 2015-10-20)

How many n-digit positive integers exist which are an n-th power?

No Solution Yet Submitted by Ady TZIDON

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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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