## Beal Conjecture Original Directly Proved

**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

**Download:** **PDF**

### Submission history

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

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

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

**Unique-IP document downloads:** 27 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*