The Beal Conjecture: A Proof and Counterexamples Charles William Johnson The conjecture A^{x} + B^{y} = C^{z} made by Mr. Andrew Beal is concerned with the common prime factor for positive integers and their exponents. "If , a^{x} + b^{y} = c^{z}, 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.
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