Beal's conjecture

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Beal's conjecture is a conjecture in number theory proposed by the Texas billionaire and mathematical amateur Andrew Beal.

While investigating generalizations of Fermat's last theorem in 1993, Andrew Beal formulated the following conjecture:

$\left. A^x +B^y = C^z \right.$ ,

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]^[2]. 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

$\left[a \left(a^m + b^m\right)\right]^m + \left[b \left(a^m + b^m\right)\right]^m = \left(a^m+b^m\right)^{m+1}$

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 $a m + b m$ in common.

It can happen that the exponents are pairwise coprime, as for example in $274 + 162 3 = 9 7$ .

Beal's conjecture is a generalization of Fermat's last theorem, which corresponds to the case $x = y = z$ . If $a x + b x = c x$ with $x \ge 3$ , 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. In either case, a counterexample to Fermat's Last Theorem yields a counterexample to Beal's conjecture.

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.^[3] In this respect Beal's statement differs from Fermat's; given a specific small integer exponent, Fermat's Last Theorem tends to be true, and indeed easier to prove, in a suitable extension of the ring of integers than in the ring of integers alone.

Beal has anted up a prize of US$100,000 for a proof of his conjecture or a counterexample^[4].