How Many Perfect Powers Are There Below 1000? Step-by-Step Count
What Are Perfect Powers and Why Count Them?
First of all, thank you to everyone who thought about the problem, worked through the numbers, and shared their answers — you are all great!
As promised, here is the answer: 41 out of 1000. There are exactly 41 perfect powers below 1000. How do we count them? The straightforward approach is brute-force enumeration by exponent.
Counting Perfect Powers: Squares and Cubes
Step 1. Exponent 2 (perfect squares): 31 numbers — from 1² through 31², since 32² = 1024 > 1000.
Step 2. Exponent 3 (perfect cubes): Among the cubes below 1000 — that is, 1³, 2³, 3³, 4³, 5³, 6³, 7³, 8³, 9³, 10³ — three values were already counted in Step 1: 1³ = 1², 4³ = 8², and 9³= 27². So we add 7 new numbers.
Step 3. Exponent 4: The largest fourth power below 1000 is 5⁴ = 25². But 5⁴, 4⁴, 3⁴, and 2⁴ were all already counted in Step 1 as perfect squares. 0 new numbers.
Higher Exponents Among Perfect Powers Below 1000
Step 4. Exponent 5: The largest fifth power below 1000 is 3⁵, since 4⁵ = 2¹⁰ = 1024 > 1000. The number 3⁵ has not appeared before. The other fifth power, 2⁵ = 32, also has not been seen yet. 2 new numbers.
Step 5. Exponent 6: The largest sixth power below 1000 is 3⁶, but 3⁶ = 9² was already counted in Step 1, and 2⁶ = 4³ was counted in Step 2. 0 new numbers.
Step 6. Exponent 7: Only 2⁷ = 128 is below 1000, and it has not appeared before. 1 new number.
Step 7. Exponents 8 and 9: 2⁸ = 16² was already counted in Step 1. 2⁹ = 512 = 8³ was already counted in Step 2. 0 new numbers from either.
The Final Count of Perfect Powers Below 1000
Adding everything up:
Exponent 2: 31
Exponent 3: +7
Exponent 5: +2
Exponent 7: +1
Total: 41 perfect powers below 1000, including the number 1.
The key to getting the correct count is carefully tracking overlaps — many perfect powers belong to multiple categories simultaneously (for example, 64 = 8² = 4³ = 2⁶), so each must be counted only once.
