The Beal Conjecture:

A Proof and Counterexamples

Charles William Johnson

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 and the Limitations of Algebraic Notation (View in html file, 30kb) The Beal Conjecture: A Proof and Counterexamples ( View full version in pdf file, 194 kb) The Beal Conjecture: A summary of an extensive essay on the theme. (Version in html file, 29 kb) The Beal Conjecture Submission Number Two. (Version in html file, 17 kb) A Submission to the Beal Conjecture Competition Last-Digit Terminations and the Beal Conjecture: An Explanation (Version in html file, 16 kb)

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The Beal Conjecture Essay