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 publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture. Related examples. To illustrate, the solution formula_3 has  bases with a common factor of 3, the solution formula_4 has bases with a common factor of 7, and formula_5 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 and Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution we additionally have where Any solutions to the Beal conjecture will necessarily involve three terms all of  which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; however, such sums are rare. The smallest two examples are: What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power. Relation to other conjectures. Fermat's Last Theorem established that formula_16 has no solutions for "n" > 2 for positive integers "A", "B", and "C". If any solutions had existed to Fermat's Last  Theorem, then by dividing out every common factor, there would also exist solutions with "A", "B", and "C" coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to "x" = "y" = "z". The Fermat–Catalan conjecture is that formula_17 has only finitely many solutions with "A", "B", and "C" being positive integers with no common prime factor and "x", "y", and "z" being positive integers satisfying formula_18 Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent". The abc conjecture would imply  that there are at most finitely many counterexamples to Beal's conjecture. Partial results. In the cases below where "n" is an exponent, multiples of "n" are also proven, since a "kn"-th power is also an "n"-th power. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat–Catalan conjecture#Known solutions. All cases of the form (2, 3, "n") or (2, "n", 3) have the solution 2 + 1 = 3 which is referred below as the Catalan solution. Prize. For a published proof or counterexample, banker Andrew Beal initially offered a prize  of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000. The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved. It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president. Variants. The counterexamples formula_20 and formula_21 show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture is an open conjecture dealing with such cases (the condition of this conjecture is  that the sum of the reciprocals is less than 1). If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case formula_21). If "A", "B", "C" can have a common prime factor then the conjecture is not true; a classic counterexample is formula_23. A variation of the conjecture asserting that "x", "y", "z" (instead of "A", "B", "C") must have a common prime factor is not true. A counterexample is formula_24 in which 4, 3, and 7 have no common prime factor. (In fact, the maximum common prime  factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.) 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 formula_25.