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From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.math
Subject: Re: Need proof for density problem
Message-ID: <701@mmintl.UUCP>
Date: Tue, 1-Oct-85 06:42:29 EDT
Article-I.D.: mmintl.701
Posted: Tue Oct  1 06:42:29 1985
Date-Received: Sat, 5-Oct-85 02:20:20 EDT
References: <58@unc.unc.UUCP>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 17

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?

Yes.  Here is an outline of a proof.

Take the logarithm of the numbers, getting m*log(2) + n*log(3).
It is fairly easy to show that the numbers of the form m*x + n*y,
where y/x is not a rational number, are dense in the reals.  Since
the logarithm is order preserving, 2^m * 3^n must be dense in the
positive reals (equivalent to dense in the positive rationals).
If log(3)/log(2) were rational, there would be numbers n and m such
that 2^m = 3^n, which violates unique factorization.

Frank Adams                           ihpn4!philabs!pwa-b!mmintl!franka
Multimate International    52 Oakland Ave North    E. Hartford, CT 06108