Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site mmintl.UUCP Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!genrad!panda!talcott!harvard!cmcl2!philabs!pwa-b!mmintl!franka 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 Brought to you by Super Global Mega Corp .com