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Proof of Beal's Conjecture

Manouchehr Shaghaghi*

Department of Engineering, István Széchenyi University, Győr, Hungary

*Corresponding Author:
Manouchehr Shaghaghi
Department of Engineering, István Széchenyi University, Győr, Hungary
E-mail:
[email protected]

Received: 11-Jul-2022, Manuscript No. JSMS-22-68995; Editor assigned: 14-Jul-2022, Pre QC No. JSMS-22-68995 (PQ); Reviewed: 28-Jul-2022, QC No. JSMS-22-68995; Revised: 04-Aug-2022, Manuscript No. JSMS-22-68995 (A); Published: 11-Aug-2022, DOI: 10.4172/JStatsMath Sci.8.6.001.

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Abstract

Beal's hypothesis states if Ax + By = Cz where A, B, C, x, y, and z are positive integers and x, y, and z are greater than 2, then A, B, and C must share a prime component. The most crucial justification for conjecture proof is that the conjecture itself contains the answer to Beal's conjecture. The largest common integer factor between A, B, C attendants us in the simplest forms of Equation Ax + By = Cz

Keywords

Beal's conjecture, Fermat's last theorem, Simplest form, Common integer factor

Introduction

The most crucial justification for conjecture proof is that the conjecture itself contains the answer to Beal's conjecture. The largest common integer factor between A, B, C attendants us in the simplest forms of Equation (1).

Ax + By = Cz.....Equation(1)

The powers of positive integer numbers are displayed in Table 1.

First of all, we gain the possible simplest form of three - elements equations with the numbers in Table 1. These equations have simplest forms as Equations (2) to (5)

10 11 12 13 14 15 16 17 18 19 110
20 21 22 23 24 25 26 27 28 29 210  
30 31 32 33 34 35 36 37 38 39 310  
40 41 42 43 44 45 46 47 48 49 410  
50 51 52 53 54 55 56 57 58 59 510  
60 61 62 63 64 65 66 67 68 69 610  
70 71 72 73 74 75 76 77 78 79 710  
80 81 82 83 84 85 86 87 88 89 810  
90 91 92 93 94 95 96 97 98 99 910  
100 101 102 103 104 105 106 107 108 109 1010  
                     

Table 1. The powers of positive integer numbers.

The following equations have the simplest forms for Equation (1):

equation

K is a positive integer number, and Equations (2) to (5) have not common factor between any of the equations. By proving these four cases and proving the absence of the fifth case, Beal's conjecture is also proved.

In other words, Beal's conjecture is true if and only if the simplest form of the Equation (1) is one of the Equations (2) to (5).

The substantiation should look like this:

•Let the Equation (2) to (5) are the simplest form of the Equation (1) therefore have a largest common factor and it will be explained in Chapter 2

•Let Ax + By = Cz have a greatest common factor; therefore, Equations (2) to (5) are the simplest form of the Equation (1), and it will be explained in Chapter 3.

EQUATION (2) TO EQUATION (5) IS THE SIMPLEST FORM OF EQUATION (1)

Equations (2) and (3)

Equation (1) can be formed by multiplying Km in the sides of Equations (2) and (3). Therefore, Equations (2) and (3) are the most basic of Equations (1).

equation

equation

Equations (4) and (5)

Equation (1) can be formed by multiplying Km in the sides of Equations (4) and (5). We come to the conclusion that Equation (4) and (5) are the most basic version of Equation (1).

equation

THE SIMPLEST FORM OF THE EQUATION Ax +By = Cz IS REPRESENTED BY EQUATIONS (2) TO (5).

P is assumed to be the most frequent component of the equation Ax +By = Cz .

equation

a, b, and c are numbers from Table 1. and they must possess some prerequisite in order to be expressed in Equation (1) in their simplest form.

These are the prerequisites:

If a=1, then the numbers b and c have different exponents.

Proof:

We suppose that b = i1 and c = j1 with power 1>2 in Equation (6).

Equation (6)⇒ 1 + i1 = j1 , because j is always greater than i this equation has no solution for any i and j.

The power inequality for b and c in Equation (6) shows that Equation (4) and (5) are the simplest forms for Equation (1), in this case K = Gn  in Equation (4) and Fn in Equation (5) for n ≥1  and n ≠ m

equation

b, c are to numbers from Table 1. With different exponents 3 and 2.

equation

b, c are to numbers from Table 1. With different exponents 1 and 3.

If c = 1

In Equation (6) the lowest value of a and b is 1 and c can therefore never be assigned one.

1 + 1 ≠ 1

If a = K > 1, then b and c are two numbers with the same exponents.

If a = 1, b and c are two numbers with different exponents therefore if

a = K > 1, then b and c are two numbers with the same exponents. The only thing that needs to be proven is that K does not have an exponent greater than one as shown in Equation (7).

K + i1 = j1…..Equation (7)

This is demonstrated by the following contraposition: Equation (7) is a simplest form of Equation (1) and if K is a number with exponent greater than one, no common factor leads us to Equation (1) therefore K is not a number with an exponent greater than one.

equation

b, c are to numbers from Table 1. With the same exponents 3.

If c = K > 1, then a and b are two numbers with the same exponents, if c = 1, b and a are two numbers with different exponents therefor if c=K > 1, b and a are two numbers with the same exponents.

The only thing that needs to be proven is that K does not have an exponent greater than one, as shown in Equation (8).

i1 + j1 = K…..Equation (8)

This is demonstrated by the following contraposition:

Equation (8) is the simplest form of Equation (1), and if K is a number with an exponent greater than one, no common factor leads us to Equation (1), therefore K is not a number with an exponent greater than one.

equation

a, b are to numbers in Table 1. with the same exponents 3.

Finally, we conclude that Beal's conjecture is correct if and only if Equation (1)'s simplest form is one of the following equations:

equation

Conclusion

Fermat's Last Theorem is generalized in Beal's conjecture. It states that if AX +BY = CZ , where A,B,C,x, y, and z are all positive integers and all greater than 2, then A , B , and C must have a common prime factor.

The solution to Beal's conjecture lies in the conjecture itself, and it is the most important reason for conjecture proof.

The simplest forms of Equation (1) are guided by the largest common integer factor between A ,B , and C .

Ax + By = Cz.....Equation(1)

equation

We can prove Fermat's conjecture much more easily with proof of Beal's conjecture. To reach the Beal's conjecture, the sides of Equations (2) to (5) must be multiplied by Km.

This leads us to an important conclusion that follows the proof of Fermat's theorem in a more direct way. Multiplying Km by the sides of Equations (2) to (5) never yields the same exponent in the equations.

Appreciation

A sincere thank you to Tom Hedalen for his support in writing the article.

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