On the Characterization of Two Canonical Equations Generating Triples Terms Belonging to Beal's Conjecture

7/29/2019 On the Characterization of Two Canonical Equations Generating Triples Terms Belonging to Beal's Conjecture

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On the characterization of two canonical equations generating triples terms

belonging to Beal's Conjecture.

Rodolfo A. Nieves Rivas

[email protected]

Abstract.

We present in this brief article two equations for the canonical representation of the

terms corresponding to particular cases and obtaining the equation triples of the Beal's

conjecture. Then, we establish its characterization and conclude with some examples

that allow us to visualize the behavior and to lay the foundations that guarantee theultimate resolution of this conjecture.

Keywords: Canonical representation, characterization, Beal's triples.
mailto:[email protected]:[email protected]:[email protected]


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Introduction

Since we know the Pythagorean Theorem with its characteristic equation and the

different methods for obtaining their primitive Pythagorean triples and then with the rise

of the study of Diophantine equations and attempt generalization of the Pythagorean

equation to exponents greater than two by Fermat and the results presented by

Matiyasevich with regard to Hilbert's tenth problem. More recently with the solution

presented by Andrew Wiles with modular elliptic curves applied to Fermat's theorem.

Now another problem must be faced and this problem is known as the Beal's conjecture

because it was Andrew Beal who formulated it.

In this short paper we present the characterization and canonical representation of two

equations of general application and partial resolution of this conjecture laying the

foundations that guarantee progress towards the final decision.


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First Characterization:

Theorem: Beal-Nieves.

If: A = B

When: C = A. 1n A

n3

Such that: 1n A

Then Beal's conjecture: Ax + By = Cz is true

If and only If: x =

And so: y = (n+1)

Where: z= n

And therefore: An+Bn+1 = Cn

Remarks: This theorem allows to proof that effectively (A; B; C) have a common

factor equal to: A as establishes the Beal's conjecture which can be prime or composite

and if it is composite then the common factor is a prime number belonging to the

factorial decomposition of: A

Examples:

A = 31

n = 5

C = 62

B = 31

Ax + By = Cz = 315 + 316 = 625

A = 26

n = 3

C = 78

B = 26

Ax + By = Cz= 263 + 264 = 783


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A = 127

n = 7

C = 254

B = 127

Ax + By = Cz= 1277 + 1278 = 2547

A = 63

n = 3

C = 252

B = 63

Ax + By =Cz= 633 + 634 = 2523

First canonical equation generating the Beal's triples:

Let: (n - 1) n + (n - 1) n +1 = ((n - 1). A)n

To all: a 2

And all n 2

Second characterization:

Let: Ax + By = Cz be the equation of the Beal's conjecture.

For: A = B orA B

When: C = c

If: an + bn = c

And also: x = n

y = n

z = n+1

n3

Then: A = (a.c)

When: B = (b.c)

Such that: (a.c)n + (b.c)n = cn+1 Ax

+ By

= Cz


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Generalization

Let: [a.( an

+ bn)]

n+ [b.( a

n+ b

n)]

n= [a

n+ b

n]

n+1

When: a 2

And besides: b 2

To all: n 2

Second canonical equation generating the Beal's triples:

Let: (a.c)n + (b.c)n = cn+1

Only when: an + bn = c

For all: a1

For all: b1

And all: n2

Remarks: To all prime number of the form: 4.m+1 = a2

+ b2

= c is proved that: a=1

and also: A = C = c and likewise: b2

= 4.m. This is guaranteed by the Fermat's proof to

the conjecture of: Girard

Examples:

a = 3

b = 5

n = 3

C = c = 152

A = 456

B = 760

4563

+ 7603

= 1524 A

x+ B

y= C

z


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a = 2

b = 3

n = 3

C = c = 35

A = 70

B = 105

703

+ 1053

= 354 A

x+ B

y= C

z

a = 1

b = 4

n = 2

C = c = 17

A = 17

B = 68

172+ 68

2= 17

3 A

x+ B

y= C

z

a = 3

b = 3

n = 4

C = c = 162

A = 486

B = 486

4864

+ 4864

= 1625 A

x+ B

y= C

z


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Criterion and Nieves discriminant on Beal's Conjecture

In order this to be true: Ax + By = Cz

When: Ax + By = CzisBeal's equation

Only is necessary and sufficient that:

[A-x

/ C-z

] - [A-x

/ B-y

] =1

[Cz

/ Ax] - [B

y/ A

x] =1

[Cz

/ By] - [A

x/ B

y] =1

[Ax

/ Cz] + [B

y/ C

z] =1

Examples:

[1-1

/ 3-2

] - [1-1

/ 2-3

] = 1

[32

/ 11] - [2

3/ 1

1] = 1

[32

/ 23] - [1

1/ 2

3] = 1

[11

/ 32] + [2

3/ 3

2] = 1

Remarks: the prior example is related to Catalan's Conjecture presented by Eugene

Charles Catalan in 1884 and proved by Preda Mihailescu in 2002.

[2-5

/ 3-4

] - [2-5

/ 7-2

]= 1

[34

/ 25] - [7

2/ 2

5]= 1

[34

/ 72] - [2

5/ 7

2]= 1

[25

/ 34

] + [72

/ 34

]= 1

Remarks: the above example allows to visualize one of the conditions when the three

terms do not have a common factor then at least one of the exponents is equal to: 2 andbesides the term: Cz = 34 can be traded by: CZ = 92


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Verification of the corresponding values to Beal's triples on the following chart

through Nieves's Discriminant

Transformations

of: Cz

Cz

= AX

+ By

Transformations

of: Bz

z x y

82= 2

6=4

3=64

1 8

2 = 39

1 + 5

2

5

2= 25

12;6;3;1 1 2;1

Discriminant or Nieves's Identity

[A-x

/ C-z

] - [A-x

/ B-y

]= 1

[Cz

/ Ax] - [B

y/ A

x]= 1

[Cz

/ By] - [A

x/ B

y]= 1

[Ax

/ Cz] + [B

y/ C

z]= 1

Verification:

[39-1

/ 8-2

] - [39-1

/ 5-2

]= 1

[82

/ 391] - [5

2/ 39

1]= 1

[82

/ 52] - [39

1/ 5

2]= 1

[391

/ 82] + [5

2/ 8

2]= 1

Transformation:

[39-1

/ 4-3

] - [39-1

/ 5-2

] = 1

[26

/ 391] - [5

2/ 39

1] = 1

[43

/ 251] - [39

1/ 5

2] = 1

[391

/ 641] + [5

2/ 4

3] = 1


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Main Theorem:

All even numbers can be expressed by the difference of two squares

12-0

2= 1

22-1

2 = 3

32-2

2 = 5

42-3

2 = 7

32-0

2 = 5

2-4

2 = 9

62-5

2 = 11

72-6

2 = 13

42-1

2 = 8

2-7

2 = 15

92-8

2 = 17

102-9

2 = 19

52-2

2 = 11

2-10

2 = 21

122-11

2 = 23

52-0

2 = 13

2-12

2 = 25

62-3

2 = 14

2-13

2 = 27

152-14

2 = 29

162-15

2 = 31

72-4

2 = 17

2-16

2 = 33

62

-12

= 182

-172

= 3519

2-18

2 = 37

82-5

2 = 20

2-19

2 = 39


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Theorem 1:

All even numbers can be expressed by the difference of two consecutive squares

12-0

2= 1

22-1

2 = 3

32-2

2 = 5

42-3

2 = 7

52-4

2 = 9

62-5

2 = 11

72-6

2 = 13

82-7

2 = 15

9

2

-8

2

= 1710

2-9

2 = 19

112-10

2 = 21

122-11

2 = 23

132-12

2 = 25

142-13

2 = 27

152-14

2 = 29

162-15

2 = 31

172-16

2 = 33

182-17

2 = 35

192-18

2 = 37

202

-192

= 39

Theorem 2:

Two consecutive numbers are always co-primes

Theorem 3:

If two any numbers do not divide the sum of both. Then both are co-primes to it

Theorem 4:

All nth power whose base is an odd prime number or composite will always be an

odd number


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Corollary 1:If the bases of the three terms of Beal's equation are co-primes and

also one of such bases is an odd number. Then at least one of the exponents is equal

to two

Cz

= Ax

+ By

12

= 1

+ 02

22 = 3

+ 1

2

32 = 5 + 2

2

42 = 7 + 3

2

52 = 9 + 4

2 3

2= 9

62 = 11 + 5

2

72 = 13 + 6

2

82 = 15 + 7

2

92 = 17 + 8

2

102 = 19 + 92

112 = 21 + 10

2

122 = 23 + 11

2

132 = 25 + 12

2 5

2= 25

142 = 27 + 13

2 3

3= 27

152 = 29 + 14

2

162 = 31 + 15

2

172 = 33 + 16

2

182 = 35 + 17

2

192 = 37 + 18

2

202 = 39 + 19

2

CZ

= Ax

+ BY

300429072 = 96222

3+ 43

8

210629382 = 76271

3+ 17

7

1222 = 11

4+ 3

5

712 = 17

3+ 2

7

712 = 3

4+ 2

5

132 = 5

2

+12

2

This corollary is proved by the main theorem and theorem 1 and it allows to prove

one of the conditions of the Beal's conjecture.


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Application of the above theorems:

Cz

= Ax

+ By

4

2

= 7

1

+ 3

2

252 = 7

2+ 24

2

1722 = 7

3+ 171

2

12012 = 7

4+ 1200

2

Proof:

Ax

= Cz

- By

71

= 42

- 32

72

= 252 - 24

2

73

= 1722 - 171

2

74

= 12012 - 1200

2

Since: Axis always odd if: A is odd to all: x Then by the main theorem Beal's

conjecture is true by corollary: 1. And besides: Ax

is always odd to all: x if and

only if: A is an odd prime number and so: x is also odd, and the transformations

are possible when: Ax

is a power of a prime number or free square.

Cz

= Ax

+ By

52

= 91

+ 42

412 = 9

2+ 40

2

3652 = 9

3+ 364

2

32812 = 9

4+ 3280

2

Proof:

Ax

= Cz

- By

32=9

1= 5

2- 4

2

34=9

2= 41

2 - 40

2

36=9

3= 365

2 - 364

2

38=9

4= 3281

2 - 3280

2


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Transformation of terms:

CZ

= Ax

+ BY

z x y

1

2

= 1

1

+ 0

2

2 1 22

2 = 3

1+ 1

2 2 1 2

32 = 5

1+ 2

2 2 1 2

42 = 7

1+ 3

2 2 1 2

32 = = 3

2+ 0

2 2 2 2

62 = 11

1+ 5

2 2 1 2

72 = 13

1+ 6

2 2 1 2

42 = = 15

1+ 1

2 2 1 2

92 = 17

1+ 8

2 2 1 2

102 = 19

1+ 9

2 2 1 2

52 = = 21

1+ 2

2 2 1 2

122

= 231

+ 112

2 1 213

2 = 5

2+ 12

2 2 2 2

62 = = 3

3+ 3

2 2 3 2

152 = 29

1+ 14

2 2 1 2

162 = 31

1+ 15

2 2 1 2

72 = = 33

1+ 4

2 2 1 2

182 = 35

1+ 17

2 2 1 2

192 = 37

1+ 18

2 2 1 2

82 = = 39

1+ 5

2 2 1 2

Analysis of the possible transformations in general:

Transformations of:CZ CZ

= Ax

+ BY

z x y

12 = 1

1+ 0

2 2 1 2

22 = 3

1+ 1

2 2 1 2

32 = 5

1+ 2

2 2 1 2

42

= 24= 16

1 4

2 = 7

1+ 3

2 2 1 2

32 = = 3

2+ 0

2 2 2 2

62 = 11

1+ 5

2 2 1 2

72 = 13

1+ 6

2 2 1 2

42

= 24

42 = = 15

1+ 1

2 2 1 2

92 = 34 = 811 92 = 171 + 82 2 1 2

102 = 19

1+ 9

2 2 1 2

52 = = 21

1+ 2

2 2 1 2

122 = 23

1+ 11

2 2 1 2

132 = 5

2+ 12

2 2 2 2

62 = = 3

3+ 3

2 2 3 2

152 = 29

1+ 14

2 2 1 2

162

= 44

= 28

162 = 31

1+ 15

2 2 1 2

72 = = 33

1+ 4

2 2 1 2

182 = 35

1+ 17

2 2 1 2

19

2

= 37

1

+ 18

2

2 1 28

2= 2

6=4

3=64

1 8

2 = = 39

1+ 5

2 2 1 2


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Transformation with trivial examples:

CZ C

Z C

Z= A

x+ B

Y

12 = 11 + 022

2 = 3

1+ 1

2

32 = 5

1+ 2

2

42 = 7

1+ 3

2

32 = 9

1+ 0

2

62 = 11

1+ 5

2

72 = 13

1+ 6

2

42 = 15

1+ 1

2

92 = 17

1+ 8

2

102 = 19

1+ 9

2

52 = 21

1+ 2

2

122 = 231 + 112

52 = = 25

1+ 0

2

62 = 27

1+ 3

2

152 = 29

1+ 14

2

162 = 31

1+ 15

2

72 = 33

1+ 4

2

62 = 35

1+ 1

2

192 = 37

1+ 18

2

82 = 39

1+ 5

2

Cz

= Ax

+ By

12

= 1

+ 02

22 = 3

+ 1

2

32 = 5 + 2

2

42 = 7 + 3

2

52 = 9 + 4

2

62 = 11 + 5

2

72 = 13 + 6

2

8

2

= 15 + 7

2

9

2 = 17 + 8

2

102 = 19 + 9

2

112 = 21 + 10

2

122 = 23 + 11

2

132 = 25 + 12

2

142 = 27 + 13

2

152 = 29 + 14

2

162 = 31 + 15

2

172 = 33 + 16

2


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Results:

On the equation belonging to the Beal's conjecture necessarily at least one of the three

terms is an odd prime number or a composite number. And since the sum is even

necessarily the other two terms either are both odd or both even and if both are even the

common factor is two and if both are odd or at least one is odd the sum is even or odd

respectively and if it is odd then the Beal's conjecture is true by all the above.

Conclusion:

And as the four criteria above are identities this allows the generalization of the results

leading us toward the definite proof of the Beal's conjecture and to ensure that it is true.

References:

[1] K. Raja Rama Gandhi; Reuven Tint. Proof of Beals conjeture; Bulletin of

Mathematical Sciences & Applications.Vol. 2 ; N 3. United States 2013. pp. 61-64.

[2] Nieves R. Rodolfo Demostracin de una conjetura presentada en el Quinto

Congreso Internacional de Matemticas en 1.912. Memorias XIX Jornadas Tcnicas de

Investigacin y III de Postgrado. Ed, Horizontes. Venezuela, 2011. pp. 123-128.

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On the Characterization of Two Canonical Equations Generating Triples Terms Belonging to Beal's Conjecture