arvind@utcsri.UUCP (09/08/87)
Date: 3 Sep 1987 15:06:10-EDT (Thursday)
From: "Victor S. Miller" <VICTOR@yktvmz.bitnet>
Subject: Perfect powers in intervals
In response to my query about an upper bound for the number of perfect
powers in intervals, both Andrew Odlyzko and Kevin McCurley pointed out
that the exponents could be restricted to be primes. This, along with
the argument that I gave yields an upper bound of
log x / (\log \log x)^2. Odlyzko then said that the number of perfect
powers was most likely bounded. After thinking about it, I tend to agree:
A simple counting argument shows that the number of intervals
[x-\sqrt(x),x+\sqrt(x)] where A/2 <= x <=A which have more than
\pi(k)+1 perfect powers in them is O((1/2)^(1/k)-1) x^(1/2+1/k)
\log x/\log \log x).
Yesterday, I found a paper by Loxton ("Some problems involving powers
of integers", Acta Arith. 46 (1986), pp. 113-123) which treats almost
the same problem. He looks at the interval [n,n+\sqrt(n)] which is
not much different. His result, using theorems about linear forms
on logarithms is an upper bound of
exp(40(\log \log n \log \log \log n)^(1/2))
when n>=20. Interestingly he conjectures that the number of prime
powers in such an interval is bounded. I'm not sure if he implies
by that that he is not willing to conjecture about general perfect
powers. He notes the example of 11^2, 5^3, and 2^7, and refers to
Stroeker and Tijdeman "Diophantine Equations" in Computationl methods
in Number Theory, Math. Centrum Tracts no. 155. in which it is shown
that there are only 3 solution (given above) to
| p^a - q^b | < p^(a/2)
where p and q are prime, and p < q < 20.
Victor S. Miller
IBM Research