n-digit integers which are also an n-th power

 
 

Here are some examples:
 

1-digit:     1^1, \; 2^1, \; 3^1, \; 4^1, \; 5^1, \; 6^1, \; 7^1, \; 8^1, \; 9^1
 
2-digit:     4^2 = 16
 
3-digit:     5^3 = 125

4-digit:     6^4 = 1296

5-digit:     7^5 = 16807

6-digit:     7^6 = 117649

7-digit:     8^7 = 2097152

8-digit:     8^8 = 16777216

9-digit:     8^9 = 134217728           9^9 = 387420489

10-digit:     8^{10} = 1073741824           9^{10} = 3486784401

11-digit:     9^{11} = 31381059609

12-digit:     9^{12} = 282429536481

13-digit:     9^{13} = 2541865828329

14-digit:     9^{14} = 22876792454961

15-digit:     9^{15} = 205891132094649

16-digit:     9^{16} = 1853020188851841

17-digit:     9^{17} = 16677181699666569

18-digit:     9^{18} = 150094635296999121

19-digit:     9^{19} = 1350851717672992089

20-digit:     9^{20} = 12157665459056928801

21-digit:     9^{21} = 109418989131512359209

 
Can you find other examples?

 
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
6^3 = 216
7^3 = 343
8^3 = 512
9^3 = 729
7^4 = 2401
8^4 = 4096
9^4 = 6561
8^5 = 32768
9^5 = 59049
8^6 = 262144
9^6 = 531441
9^7 = 4782969
9^8 = 43046721

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

2 Responses to n-digit integers which are also an n-th power

  1. Paul says:

    There are no more, 9^22 is still only length 21 and 10^22 is length 23 and so 9^21 is the limit.

    Paul.

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