🚀 go-pugleaf

Date: 21 Aug 1987 09:24:06-EDT (Friday)
From: "Victor S. Miller" <VICTOR@yktvmx.bitnet>
Subject: Counting perfect powers

In the interval $I_x = [x-\sqrt{x},x+\sqrt{x}] I would like to estimate
the number of perfect powers: i.e. the number of integers of the form
$n^r$ with r>1.  Now it is easy to see that for each r>1 there is at most
1 integer of the form $n^r$ in the interval.  That crude estimate
gives an upper bound of $\log x / \log 2$.  If you work a little harder
you can get that down to $(1+\epsilon)\log x / \log \log \log x$. Namely,
count all the exponents up to $\log x / \log \log \log x$.  If r is
bigger than that, then you must have $n<=\log \log x$.  What is the
true asymptotic?
                      Victor Miller -- IBM, Research