Beal's conjecture is a conjecture in number theory proposed by Andrew Beal in 1993. While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:
If
 
where A,B,C,x,y,z are positive integers with x,y,z >2
then A,B,C must have a common prime factor.

THE BEAL PRIZE. The conjecture and prize was announced in the December 1997 issue of the Notices of the American Mathematical Society. 
The prize is named after D. Andrew "Andy" Beal, a Dallas banker and number theory enthusiast, who provided the money. Since that time Andy Beal has increased the amount of the prize for his conjecture.
The prize is now this: $1,000,000 for either a proof or a counterexample of his conjecture.
American Mathematical Society officials say raising the Beal Conjecture Prize to $1 million from $100,000 is meant to stimulate young mathematicians' interest.

Previous partial results
The Beal conjecture has been verified for all values of all six variables up to 1000. So in any counterexample, at least one of the variables must be greater than 1000. See:
http://www.norvig.com/beal.html
http://www.danvk.org/wp/beals-conjecture

More in http://en.wikipedia.org/wiki/Beal%27s_conjecture

PART 1:
My modest contribution: Beal's conjecture is true for small numbers.
Due to the computational cost, I checked the conjecture for small numbers.
When I say "small" I mean the following three restrictions:
1) Due to symmetry of the equation, we assume that A ≤ B
2)  A,B ≤ 2100  and  C ≤ 5100
3)

I checked all possible combinations following these three restrictions.
A total of 855 solutions found in the set of constraints.
No counterexample was found.
Total cpu time = 5 hours

The list of 855 solutions.
in excel format.
in graphic format