Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 04 Jan 1995 01:46:48 GMT
Subject: Re: Number theory question

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

> I have been told that the following is an open problem in number
> theory.
> 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 ?

This is a case of Catalan's conjecture.  R.K.Guy, in _Unsolved
Problems in Number Theory, 2nd Edition_ (Springer-Verlag, 1994, ISBN
0-387-94289-0), p.155, says:

  Except that there remains a finite amount of computation, Tijdeman
  has settled the old conjecture of Catalan, that the only
  consecutive powers, higher than the first, are 2^3 and 3^2.  But
  this finite amount of computation is far beyond computer range and
  will not be achieved without some additional theoretical ideas.
  Langevin has deduced from Tijdeman's proof that if n,n+1 are
  powers, then n < exp exp exp exp 730 ....

There is nothing in the article to suggest that the problem is easier
if restricted to powers of two and three, and some of the results rely
both bases being greater than two.  Guy cites R. Tijdeman, On the
equation of Catalan, _Acta Arith_, 29(1976) 197-209; MR 53 #7941.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil