Project: The Solutions of a Family of Diophantine Equations
In mathematics, a Diophantine equation is a polynomial equation, usually with two or more unknowns, such that only the integer solutions are sought or studied. An integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation that sums two monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns.
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations; defining an algebraic curve, algebraic surface, or more general objects.
In this program, we will focus on a specific family of Diophantine equations, which is formulated in 1993 by Andrew Beal, a banker and amateur mathematician. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counter example. 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.
The Beal conjecture is the following conjecture in number theory: If Ax+By=Cz, 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, there are no solutions to the above equation in positive integers A, B, C, x, y, z with A, B, and C being pairwise coprime and all of x, y, z being greater than 2.
In this program, we will study some specific equations, in particular, the Fermat's equation.