Beal conjecture

The Beal conjecture is the following conjecture in number theory:

If

${\displaystyle A^{x}+B^{y}=C^{z},}$

where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.

Equivalently,

The equation ${\displaystyle A^{x}+B^{y}=C^{z}}$ has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3.

The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.[1][2] Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.[3] The value of the prize has increased several times and is currently $1 million.[4]

In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,[5] the Mauldin conjecture,[6] and the Tijdeman-Zagier conjecture.[7][8][9]