## Beal’s Conjecture as Sum of Two Vectors in a Polynomial Vector Space that Defines an Identity-Proof

**Authors:** Kamal Barghout

Beal’s equation is identified as polynomial identity and therefore the variable x is considered a countable variable. The central theme to prove Beal’s conjecture here is identifying its numerical solution as a particular solution to the general polynomial identity of αx^l+βx^l=〖δx〗^l, where α,β, δ, and l>2 are positive integers. Beal’s equation can be represented by the addition of two vectors in the vector space of the set of all polynomials in the form p(x)=a〖 x〗^l for a∈ ℚ as a subspace of the infinite vector space over ℚ of all polynomials with basis 1,x,x^2,… with the ordinary addition of polynomials and multiplication by a scalar from ℚ, where l is particular to any solution to Beal’s equation. Accordingly, all three monomials of Beal’s equation numerically produce terms of single power by following the rules of exponentiation. Here we look for elements in the ℚ field where the rational number can be converted to a number in exponential form that successfully combine with the basis-element〖 x〗^l.

**Comments:** 10 Pages. The material in this article is copyrighted. Please obtain authorization from the author before the use of any part of the manuscript

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### Submission history

[v1] 2017-12-05 15:21:45 (removed)

[v2] 2017-12-07 09:57:31 (removed)

[v3] 2017-12-09 13:15:45 (removed)

[v4] 2017-12-12 13:11:08 (removed)

[v5] 2017-12-14 14:10:38 (removed)

[v6] 2017-12-18 06:46:20 (removed)

[v7] 2017-12-30 12:15:53

[v8] 2018-02-05 10:31:56

[v9] 2018-02-20 11:06:15

[vA] 2018-03-13 05:47:37

[vB] 2018-03-17 17:04:34

[vC] 2018-03-27 08:29:08

[vD] 2018-04-05 12:04:09

[vE] 2018-04-10 08:51:25

[vF] 2018-06-27 17:36:27

[vG] 2018-07-14 16:35:00

[vH] 2018-08-02 07:26:36

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