## 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?

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

math grad - Interest: Number theory
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### 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.

• benvitalis says:

Yes, 9^21 is the limit. There are other smaller ones, though.