Beal's conjecture is a conjecture in number theory proposed by the Texas billionaire and amateur mathematician Andrew Beal.
While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:
If
where A, B, C, x, y, and z are positive integers with x, y, z > 2 then A, B, and C must have a common prime factor.
By computerized searching, greatly accelerated by aid of modular arithmetic, this conjecture has been verified for all values of all six variables up to 1000.[1] So in any counterexample, at least one of the variables must be greater than 1000.
To illustrate, the solution 33 + 63 = 35 has bases with a common factor of 3, and the solution 76 + 77 = 983 has bases with a common factor of 7. Indeed the equation has infinitely many solutions, including for example
![\left[a \left(a^m + b^m\right)\right]^m + \left[b \left(a^m + b^m\right)\right]^m = \left(a^m+b^m\right)^{m+1}](http://wpcontent.answers.com/math/a/0/4/a04866e6b07e69158801f5f43762dc36.png)
for any a, b, m > 3. But no such solution of the equation is a counterexample to the conjecture, since the bases all have the factor am + bm in common.
It can happen that the exponents are pairwise coprime, as for example in 274 + 1623 = 97.
Beal's conjecture is a generalization of Fermat's last theorem, which corresponds to the case x = y = z. If ax + bx = cx with 
, then either the bases are coprime or share a common factor. If they share a common factor, it can be divided out of each to yield an equation with smaller, coprime bases.
The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided (−2 + i)3 + (−2 − i)3 = (1 + i)4.[2]
Beal has offered a prize of US$100,000 for a proof of his conjecture or a counterexample[3].
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