----- Original Message -----

From: Gershon Bialer

To: math-fun@mailman.xmission.com

Sent: Saturday, January 18, 2003 6:12 PM

Subject: Re: [math-fun] "Beal's Conjecture"

I looked at Liff's article and he discusses the equation, but he doesn't make the conjeture that there is no solution with positive integers x,y,z and a,b,c>2. According to the AMS, at http://www.ams.org/ams/ams-administered-prizes.html, the Beal conjecture was first proposed in 1997. There is a link to an article on the page with references to lots of similar problems.

 

Gershon Bialer

----- Original Message -----

From: Steve Gray

To: math-fun@mailman.xmission.com

Sent: Saturday, January 18, 2003 6:05 PM

Subject: [math-fun] "Beal's Conjecture"

    I thought this comjecture was relatively new, following Wiles' proof of FLT. But it is discussed, not in connection with Beal's name, in 1968. See "On Solutions of the Equation x^a + y^b = z^c" in Mathematics Magazine, 1968, p. 174, by Allan I. Liff, who presents some facts and conditions on its solution. I think the usual notation now is a^x + b^y = c^z. So far as I know, in all solutions found so far, one of x,y,or z is 2.

 

Steve Gray

 

    I forgot to say that a,b,c must be co-prime for a valid counterexample. At the site http://www.norvig.com/beal.html , which has more information, the statement reads

 

There are no positive integers x,m,y,n,z,r satisfying the equation

x^m + yⁿ = z^r

where m,n,r > 2 and x,y,z are co-prime (that is, gcd(x,y) = gcd(y,z) = gcd(x,z) = 1).