Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 04 Jan 1995 17:17:31 GMT
Subject: Re: Number theory question

geoff...@oracorp.com (Geoffrey Hird) asked:

> Do there exist powers 2^n and 3^m such that |2^n - 3^m| = 1,
> other than the two cases 2,3 and 8,9 ?

Yes, the cases 2,1 and 4,3.  Other than those,

kevin2...@delphi.com (Kevin Brown) notes:

> This problem is not open, and is not difficult to prove [proof].

which goes to show I should have thought a bit rather than looking up
Guy.  Kevin only proves half the statement, though, and somewhat
awkwardly.  An easy proof is the following.  Inspect to 3^4, and
Consider the smallest larger counterexample.

  Case A: 3^m-1=2^n, m even => 3^(m/2)-1 divides 2^n.
  Case B: 3^m-1=2^n, m odd => 3^m-1 is 2 mod 4.
  Case C: 3^m=2^n-1, n even => 2^(n/2)-1 divides 3^m.
  Case D: 3^m=2^n-1, n odd => 2^n-1 is 1 mod 3.

In each case we find a contradiction or a smaller counterexample.

Dan
Hoey@AIC.NRL.Navy.Mil