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From: ian@sdcsma.UUCP (Ian Ferris)
Newsgroups: net.math
Subject: Re: Need proof for density problem
Message-ID: <127@sdcsma.UUCP>
Date: Fri, 27-Sep-85 13:17:41 EDT
Article-I.D.: sdcsma.127
Posted: Fri Sep 27 13:17:41 1985
Date-Received: Wed, 2-Oct-85 08:52:01 EDT
References: <58@unc.unc.UUCP>
Reply-To: ian@sdcsma.UUCP (Ian Ferris)
Organization: System Development Corp. R+D, Santa Monica
Lines: 26

In article <58@unc.unc.UUCP> southard@unc.UUCP (Scott Southard) writes:
>
>Is the set of numbers of the form 2^m * 3^n (that's 2 to the m power times
>3 to the n power) where m and n are integers, dense in the positive
>rational numbers?
>

If I'm not mistaken, if p and q are relatively prime integers,
then the set S of numbers of the form p^m * q^n is dense in the
positive reals.

I suspect, since school has just started in most places, that
this is a homework problem in either a junior level analysis
course or an honors freshman calculus course.  (If my suspicion is
unfounded, I apologize).  Since I think it's a bad idea to
do other people's homework, I will only remark that one possible
proof of this fact (the only one that I came up with) can be
gotten by considering the logarithms of the numbers in S
and proving that they are dense in the reals.  This approach
also requires you to know the following theorem:

if r is any irrational number and e is any positive real number,
then there exist integers m and n such that  0 < abs( m + n * r ) < e.

However, thanks for the problem!  I had a very pleasant noontime
walk yesterday while working on it.


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