arvind@utcsri.UUCP (08/21/87)
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