Beal conjecture

The Beal conjecture is the following conjecture in number theory: If A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} , where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor. Equivalently, The equation A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z > 2. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million. In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture.