Status of Beal, Granville, Tijdeman-Zagier Conjecture - MathOverflow [closed]

Status of Beal, Granville, Tijdeman-Zagier Conjecture

2010-06-19T17:06:37Z

The Beal, Granville, Tijdeman-Zagier Conjecture, i.e.

If $A^x+B^y=C^z$ , where $A, B, C, x, y,z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a common prime factor.

... and its associated $1,000,000 prize for proof or disproof seems to have gone largely unnoticed in the mathematics community. Please answer with (A) references to past or ongoing research or (B) references to equivalent forms of this conjecture known prior to Andrew Beal posing it in 1993.

Answer by Robin Chapman for Status of Beal, Granville, Tijdeman-Zagier Conjecture

2010-06-19T17:21:26Z

There was a great deal of discussion in the sci.math newsgroup about a decade ago. See the threads Beal's Conjecture and Against the term "Beal Conjecture". As with most sci.math discussions, they generated more heat than light.

Answer by Charles for Status of Beal, Granville, Tijdeman-Zagier Conjecture

2010-07-12T18:26:15Z

The sci.math discussions linked to above suggest that Andrew Granville suggested the problem in 1992 and that it was discussed as early as 1985.

I have in my notes:

T-Z predates Beal; see Frits Beukers, "The Diophantine equation $Ax^p+By^q=Cz^r$", Duke Math. J. 91:1 (1998), pp. 61-88.

This kind of informal documentation may be the best available, unfortunately.

Answer by JSE for Status of Beal, Granville, Tijdeman-Zagier Conjecture

2010-10-03T19:13:48Z

At present there is no real strategy for the general problem. But progress on individual cases, or families of cases, keeps moving along. For instance, Poonen, Schaeffer, and Stoll handled the case x^2 + y^3 + z^7 in 2005; last year, Mike Bennett, Nathan Ng and I finished off the case x^2 + y^4 = x^p and David Brown did x^2 + y^3 + z^10.