Perfect power: n = aᵇ (a≥1, b≥2). Examples: 8=2³, 64=2⁶=4³=8², 729=3⁶=9³=27². Perfect squares (b=2): 1,4,9,16,25... Perfect cubes (b=3): 1,8,27,64,125... A number can be a perfect power in multiple ways.
How to Use
Enter n:
- Check: Is n a perfect power?
- Representations: All (a,b) pairs
- Type: Square, cube, etc.
Properties
- Perfect powers have density 0 in the integers
- Goldbach conjecture variant: every n>4 is sum of ≤3 perfect powers
- Catalan's conjecture (proved): only consecutive perfect powers are 8,9
Algorithm
For each exponent b=2,3,...,⌊log₂n⌋: compute a=n^(1/b) rounded, check if aᵇ=n. If n=p₁^e₁·...·pₖ^eₖ, then n is a perfect bth power iff b|eᵢ for all i.
Step-by-Step Instructions
- 1Enter n.
- 2Check perfect power.
- 3View all representations.
- 4See minimal base.
- 5Scan range.