**Authors:** A. A. Frempong

Using a direct construction approach, the author proved the original Beal conjecture that if A^x + B^y = C^z , where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. Two main types of equations were involved, namely, the equation A^x + B^y = C^z and an equation which was called a tester equation. A tester equation has similar properties as A^x + B^y = C^z and was used to determine the properties of A^x + B^y = C^z . Also, two types of tester equations, namely, a literal tester equation and a numerical tester equation were applied. Each side of A^x + B^y = C^z and a tester equation was reduced to unity by division. The non-unity sides were justifiably equated to each other to produce a new equation which was called the master equation. The side of the master equation involving the terms of the tester equation was called the tester side of the master equation. Three versions of the proof were presented. In Version 1 proof, the tester equation was the literal equation G^m + H^n = I^p, but in Versions 2 and 3 proofs, the tester equations were the numerical tester equations, 2^9 + 8^3 = 4^5 and 3^3 + 6^3 = 3^5, respectively. By a comparative analysis, in which the corresponding "terms" on the right and left sides of the master equation were equated to each other, it was determined that if A^x + B^y = C^z , then A, B and C have a common prime factor. The proof is very simple, and occupies a single page, and even, high school students can learn it.

**Comments:** 5 Pages. Copyright © by A. A. Frempong

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[v1] 2017-02-28 01:30:08

[v2] 2017-03-04 01:18:52

[v3] 2017-03-07 17:04:09

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