Status of Beal, Granville, Tijdeman-Zagier Conjecture 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.
 A: 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.
A: 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.

A: 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.  
