Beal conjecture

The Beal conjecture is the following conjecture in number theory:

Equivalently,

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 venues, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture.

To illustrate, the solution ${\displaystyle 3^{3}+6^{3}=3^{5}}$ has bases with a common factor of 3, the solution ${\displaystyle 7^{3}+7^{4}=14^{3}}$ has bases with a common factor of 7, and ${\displaystyle 2^{n}+2^{n}=2^{n+1}}$ has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively

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