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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.

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 I guess that you should look at the following article ams.org/notices/199711/beal.pdf It mentions some related conjectures made prior to Beal. – Adrián Barquero Jun 19 2010 at 17:32
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 I think you should edit this. Right now the only actual question in your post is about a controversial subject. – Steve Huntsman Jun 19 2010 at 17:42
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 In this interesting overview paper, thehcmr.org/issue1_1/elkies.pdf, the conjecture is referred to as the Tijdeman-Zagier conjecture. There is no explicit reference, though. – Halfdan Faber Jun 19 2010 at 18:13
Nils Bruin covers the conjecture briefly in Chabauty methods and covering techniques applied to generalised Fermat equations, PhD-thesis, University of Leiden, 1999. See also: springerlink.com/content/a177k83512kp6301. – Halfdan Faber Jun 26 2010 at 16:26
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 Perhaps it is time to close the question. – S. Carnahan♦ Oct 3 2010 at 5:34
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closed as no longer relevant by Felipe Voloch, Yemon Choi, Andy Putman, Daniel Moskovich, Andres Caicedo Jun 11 at 23:29

3 Answers

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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.

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Thanks, Charles. See: math.univ-lyon1.fr/~roblot/ihp/Fermatlectures.pdf (later version, it appears). It would be nice to see some informal documentation predating 1993, in the absence of any concise reference. – Halfdan Faber Jul 13 2010 at 3:38
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 In one of those sci.math discussions, I wrote, "The December 1992 Western Number Theory meeting was held in Corvallis. The problem list was edited by Richard Guy and is dated 9 June 93. The relevant part of Problem 92:12 reads as follows. 92:12 (Andrew Granville) Find examples of $x^p + y^q = z^r$ with $1/p + 1/q + 1/r \lt 1$ other than $2^3 +1^7 = 3^2$ and $7^3 + 13^2 = 2^9$." I consider that to be a concise reference to formal documentation preceding 1993. – Gerry Myerson Oct 3 2010 at 0:11
Fantastic! I'm going to accept the answer, with the note from Gerry. – Halfdan Faber Oct 3 2010 at 6:31
The second part of this question was also answered indirectly. There appears to have been very little, if any, work related to the BGTZ Conjecture in recent years. – Halfdan Faber Oct 3 2010 at 6:42
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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.

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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.

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 "As with most sci.math discussions, they generated more heat than light." You can say that again. Wow... let's hope MO steers clear of that sort of discussions. – José Figueroa-O'Farrill Jun 19 2010 at 21:45
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 I've been very impressed with the MO moderator's ability to prevent cranks from taking over the site. When I first heard about it, I expected that it would quickly degenerate into something akin to sci.math... – Andy Putman Jun 19 2010 at 23:41
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 sci.math.research never degenerated into that sort of thing, but that's because it's moderated. There's the advantage of moderation, but the disadvantage is that is slows the site down (e.g. I'm supposed to be the moderator today, but I've just been asleep for 8 hours). At the time of setting sci.math.research up there wasn't really the machinery available to have the whole community moderating. That's the breakthrough this site offers. – Kevin Buzzard Jun 20 2010 at 6:36

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