Number Theory

   

Beal's Conjecture as Univariate Polynomial Identity Derived from Algebraic Expansion of Powers of Binomials: Analysis and Proof

Authors: Kamal Barghout

The general equation of Beal’s conjecture + = , at points where its variables are numerically equal, is identified as a univariate polynomial identity derived from algebraic expansion of powers of binomials which upon expansion and reduction to two terms produces + ≡ , where the general polynomial equation has integer solution at the intersection with the line − = 0 as a special case and satisfies Beal’s condition of perfect power terms; , , , are positive integers, > 2 and + = . This algebraic identity can be represented by the addition of two vectors in the vector space of the set of all polynomials in the form () = for ∈ ℚ as a subspace of the infinite vector space over ℚ of all polynomials with basis 1, , 2 , … with the ordinary addition of polynomials and multiplication by a scalar from ℚ, where is particular to any solution to the equation. Here we look for elements in the ℚ field where the rational number can be converted to a number in exponential form that successfully combines with the basis-element to produce perfect power terms. Accordingly, it is shown that all three monomials of the identity equation numerically produce terms of perfect powers by following the rules of exponentiation which produces integer solutions to Beal’s general equation and can be obtained by expanding the corresponding binomial identity.

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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
[vI] 2019-04-29 10:30:25

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