From: steiner@bgnet0.bgsu.edu (Ray Steiner)
Newsgroups: sci.math.research
Subject: Re: Perfect Powers
Date: Thu, 27 Aug 1998 13:00:32 -0500

In article <35E44750.95516589@SouthWind.Net>, rgwv@southwind.net wrote:

> Et al,
> 
>         It is well known that only the perfect powers that differ by 1
> are the integers 8 & 9.  Perfect powers are integers which are the
> result of raising the natural numbers to any power greater than one.
> See Neil J.A. Sloane, M3326 in Encyc. of Integer Sequences by Academic
> Press, 1995.  For a difference of tow, we have 25 & 27, 3 has 1 & 4 and
> 125 & 128, 4 has 4 & 8 and 32 & 36, and a difference of 5 has 4 & 9 and
> 27 & 32.  Question, what is the first pair (not necessarily consecutive)
> perfect power numbers which differ by 6, 14, 34, 42 and 50.  Thank you
> in advance. I've checked all perfect power numbers up to 2*10^10.
> 
> Mathematically yours,
> Robert G. Wilson v,
> PhD ATP / CF&GI
Eh, so sorry, but it is NOT "well-known" that the only powers that differ 
by 1 are 8 and 9.
This is Catalan's conjecture. The problem has been shown to be equivalent 
to solving
 x^p- y^q=1, where p and q are odd primes satisfying

10^5 < p < 3.31*10^12,
10^6 < q < 4.13*10^17,
assuming p < q.
Question: How do we resolve the problem for the ranges of p and q listed here??
For more on Catalan's conjecture, see the book by Paulo Ribenboim
CATALAN'S CONJECTURE
Are 8 and 9 the only consecutive powers?
Academic Press, New York, 1994.

Incidentally, a famous mathematician once told me that the solution of 
x^p- y^q = 6
would lead to the solution of another famous problem. Could anyone tell me what
he was referring to?
Regards,
Ray Steiner

==============================================================================

From: gerry@mpce.mq.edu.au (Gerry Myerson)
Newsgroups: sci.math
Subject: Re: x^2-1=y^m
Date: Thu, 11 Mar 1999 09:58:13 +1100

In article <abms2gq6g1wk@forum.swarthmore.edu>, blang@club-internet.fr
(Bruno Langlois) wrote:

> Who can solve the diophantine equation :
> x^2-1=y^m  (x,y,m > 1)

Chao Ko, On the diophantine equation x^2 = y^n + 1, xy \ne 0, Scientia 
Sinica (Notes) 14 (1965) 457--460, MR 32 #1164. 

See also Mordell, Diopantine Equations, pp. 302--304. 

Gerry Myerson (gerry@mpce.mq.edu.au)