# Beal conjecture

## Mathematical conjecture / From Wikipedia, the free encyclopedia

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The **Beal conjecture** is the following conjecture in number theory:

If $A^{x}+B^{y}=C^{z}$ where *A*, *B*, *C*, *x*, *y*, *z* are positive integers and *x*, *y*, *z* are ≥ 3, do *A*, *B*, and *C* have a common prime factor?

- If

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

- 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 $A^{x}+B^{y}=C^{z}$ 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]}