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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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    $\begingroup$ 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. $\endgroup$
    – Adrian Barquero-Sanchez
    Commented Jun 19, 2010 at 17:32
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    $\begingroup$ I think you should edit this. Right now the only actual question in your post is about a controversial subject. $\endgroup$
    – Steve Huntsman
    Commented Jun 19, 2010 at 17:42
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    $\begingroup$ 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. $\endgroup$
    – Halfdan Faber
    Commented Jun 19, 2010 at 18:13
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    $\begingroup$ This question remains entirely unanswered. $\endgroup$
    – Halfdan Faber
    Commented Oct 2, 2010 at 19:51
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    $\begingroup$ Perhaps it is time to close the question. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 3, 2010 at 5:34

3 Answers 3

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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.

Edit to expand on a comment:

On sci.math, Gerry Myerson wrote on Aug 22 2000:

Since Andrew Granville's contribution to the Western Number Theory problem list has come up in this discussion, I want to put it on record here. 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 < 1 other than 2^3 +1^7 = 3^2 and 7^3 + 13^2 = 2^9. [Blair Kelly III gave 2^5 + 7^2 = 3^4 and Reese Scott 17^3 + 2^7 = 71^2.]

In Guy's write-up of the 1993 problems, dated 3 March 94, there is a comment about 92:12, wherein Granville agrees with the suggestion that it was intended that x, y and z be relatively prime, and gives 3^5 + 11^4 = 122^2 as another example. Peter Montgomery gave 5 larger examples found by Beukers & Zagier.

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  • $\begingroup$ 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. $\endgroup$
    – Halfdan Faber
    Commented Jul 13, 2010 at 3:38
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    $\begingroup$ 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. $\endgroup$
    – Gerry Myerson
    Commented Oct 3, 2010 at 0:11
  • $\begingroup$ 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. $\endgroup$
    – Halfdan Faber
    Commented 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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    $\begingroup$ "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. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jun 19, 2010 at 21:45
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    $\begingroup$ 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... $\endgroup$
    – Andy Putman
    Commented Jun 19, 2010 at 23:41
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    $\begingroup$ 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. $\endgroup$
    – Kevin Buzzard
    Commented Jun 20, 2010 at 6:36

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