Here's a more straight-forward (but equally hand-wavy) way to calculate the probability that N is a perfect power: P(N) ≈ (number of perfect powers near N) / (

Here's a more straight-forward (but equally hand-wavy) way to calculate the probability that N is a perfect power: P(N) ≈ (number of perfect powers near N) / (size of the neighborhood) ≈ (ⁿ√(N + r) − ⁿ√N) / r, for some smallish r ≈ d/dN (ⁿ√N) = ⁿ√N / nN