Problem #63 Difficulty: 5%

Powerful Digit Counts

Solution Language: Python
View Solution View on Project Euler

Problem Statement

The 5-digit number, 16807 = 7⁵, is also a fifth power. Similarly, the 9-digit number, 134217728 = 8⁹, is a ninth power.

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

Approach

The solution involves:

  1. Recognizing that for n-digit numbers to be nth powers, the base must be less than 10
  2. For each base b (1-9), computing b^n for increasing n
  3. Checking if the number of digits equals n
  4. Stopping when b^n has fewer than n digits (this happens eventually for all bases)
  5. Counting all valid combinations