Asymptotic formulae for the square root of the n-th perfect power

Abstract

In this note we obtain asymptotic formulae for the square root of the n-th perfect power.

Sums of perfect powers

“…Equation (1) implies, as is well-known, that the series ∞ i=1 1 P i converges. Equation (1) gives P n ∼ n.…”

A conjecture on the difference between consecutive perfect powers

“…Finally, substituting equation (4) and equation (5) into (3) we obtain equation (1). The theorem is proved.…”

Some results on the difference between consecutive perfect powers

“…Proof. We have P 1 = 1 2 = 1, P 2 = 2 2 = 4, P 3 = 8 and P 4 = 3 2 . Consequently if n ≥ 4 then P n ≤ (n − 1) 2 , since the squares are perfect powers.…”