The conjecture Ax + By = Cz made by Mr. Andrew Beal is concerned with the common prime factor for positive integers and their exponents.
"If ,ax + by = cz , where A,B,C, x, y and z, are positive integers x, y and z are all greater than 2, then A, B and C must have a common prime factor." [Mauldin, 1997] This represents the original wording of the Beal Conjecture.
The Beal Conjecture requires positive integers in the terms [A, B, C] and exponents [x, y, z] of the equation (the latter whose value must be greater than 2). The products of the terms must reflect the selfsame multiplication of the terms in whole numbers or positive integers. Obviously, no fractional expressions are to appear in any of the three terms or three exponents of the equation. And, the most significant part of the conjecture affirms the necessity that the terms share a common prime divisor. Or, to the contrary, present counterexamples.
The Beal Conjecture has been clarified by Professor R. Daniel Mauldin with the following examples of the resolution of the equation: "Here are some examples of solutions to the equation Ax + By = Cz. Note that all values are positive integers, all exponents are greater than or equal to 3, and A, B, and C always share a common factor."
The Beal Conjecture stipulates, then:
1- the positive integer terms may be assigned any numerical value in any given order of distinction or repetition;
2- the positive integer exponents of the terms, may be assigned any numerical value greater than 2 in any given order of distinction or repetition and;
3- the terms must have a common prime factor.
In the absence of a proof, one must present counterexamples. |
