From: steiner@math.bgsu.edu (steiner)
Subject: Re: Catalan's Conjecture:
Date: Thu, 07 Jan 1999 18:21:53 -0500
Newsgroups: sci.math
Keywords: Current status: consecutive proper powers => {8,9}

In article <3691746A.2A67BFD7@idt.net>, xxyzz <xxyzz@idt.net> wrote:

>What is the current state of Catalan's Conjecture?  I know that people
>have proven that the equation has only finitely many solutions, and
>there are no solutions outside of the obvious one up to a very large
>number.  Can anyone tell me if there have been any recent developments?
Here are the latest results as I know them:
Consider the equation
x^p-y^q=1.
It is enough to assume p and q are primes. We may also assume p<q.
It is known, through work of Mignotte, Baker, et al.
That
10^5 < p < 3.31*10^12
and
10^6 < q < 4.13*10^17
A very important subcase of this conjecture asks whether there are any 
double Wieferich
pairs of primes in this range. If so, what are they?
Double Wieferich primes satisfy both
p^(q-1)= 1(mod q^2)
and
q^(p-1)= 1(mod p^2).
At present, only 6 pairs of dw primes are known:
(2,1093), (3,1006003), (5,1645333507), (83,4871), (911, 318917) and (2903, 
18787).
Answering the above question would go a long way toward resolving 
Catalan's conjecture.
BTW: See my article in the July, 1998 issue of Math. Comp. for more info.
Regards,
Ray Steiner
==============================================================================

From: ray steiner <steiner@math.bgsu.edu>
Subject: Possible breakthrough in Catalan's conjecture
Date: Wed, 15 Dec 1999 18:06:38 GMT
Newsgroups: sci.math

 It seems that there has been a recent breakthrough in solving the Catalan
conjecture, which is to prove that the equation |x^p - y^q| =1	(*) has no
nontrivial integer solutions if p and q are odd primes. (This eventually
shows that the only integer solutions of (*) are (x,y,p,q)= (3,2,2,3) and
(2,3,3,2) ). The results are due to Mignotte, Bugeaud, Hanrot and Mihailescu.
1). It's no loss of generality to assume that p < q. Then Mignotte has
deduced that p < 7.15*10^11, q < 7.78*10^16 by using recent results in linear
forms in logarithms of algebraic numbers. He has also deduced by brute-force
search that p > 10^7.

2). (Bugeaud and Hanrot) If x and y are positive integers satisfying (*) then
q | h_p^-, where h_p^- is the relative class number of the cyclotomic field
Q(zeta_p).

3). (Mihailescu) If any nontrivial solution to (*) exists then p and q must
form a double Wieferich pair, i.e., p^{q-1} = 1(mod q^2) and q^{p-1} = 1(mod
p^2}.

Of course, these results are now being carefully checked.
Would anyone care to join me in a search for relevant double Wieferich primes?

It seems that if we could get 1000 fast machines working on this at once,
perhaps we could narrow the 10^20 or so possible pairs quite rapidly!
Regards, Ray Steiner -- steiner@math.bgsu.edu


Sent via Deja.com http://www.deja.com/
Before you buy.