Searching for a valid proof of the abc Conjecture

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Please refer to the following reference link for details:

https://www.researchgate.net/post/What_is_the_proof_of_the_ABC_Conjecture.

[Guest]

Please refer to the following reference link for details:

https://www.researchgate.net/post/What_is_the_proof_of_the_ABC_Conjecture.

See also the following link for a proof of the famous and important abc Conjecture.

Reference link:
https://plus.google.com/u/0/115467095957398372179/posts/dCTYHHAyY5e.

[Guest]

On the abc conjecture: Prof. Jeff Vaaler's abc conjecture lecture is excellent and highly recommended, and there's a video of it on youtube at link below.

https://www.youtube.com/watch?v=XYisYYhKKYA

[Guest]

On the abc Conjecture: The paper, 'The ABC-conjecture' by Frits Beukers is also excellent.

Please review it at the following link.

http://www.staff.science.uu.nl/~beuke106/ABCpresentation.pdf

[Guest]

'The paper, 'On the abc Conjecture and some of its consequences' by Michel Waldschmidt is also excellent! Amen!

Please review it at the following link.

https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcEnVI.pdf

[Guest]

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

"ABC-Conjecture (Masser-Oesterlé, 1985):

Let [tex]\beta[/tex] > 1. Then, with finitely many exceptions, we have C < rad[tex](ABC)^{\beta}[/tex] "

-- Dr. Frits Beukers, author of 'The ABC-conjecture', 2005.

We shall derive six relevant equations with nine distinct and relevant integer variables from the following statements.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

gcd(A, B) = gcd(A, C) = gcd(B, C) = 1;

C < rad[tex](ABC)^{\beta}[/tex] where [tex]\beta[/tex] > 1.

According to the Fundamental Theorem of Arithmetic, we define the positive integer variables, A, B, and C:

A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}^{k_{j_1}}[/tex];

B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}^{k_{j_2}}[/tex];

C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3}}[/tex].

Therefore, we have six relevant equations with nine distinct and relevant integer variables.

A * [tex]x_1[/tex] + B * [tex]y_1[/tex] = 1;

A * [tex]x_2[/tex] + C * [tex]y_2[/tex] = 1;

B * [tex]x_3[/tex] + C * [tex]y_3[/tex] = 1;

Note: The variables, [tex]x_1[/tex], [tex]x_2[/tex], [tex]x_3[/tex], [tex]y_1[/tex], [tex]y_2[/tex], and [tex]y_3[/tex] may be positive or negative integers.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

C = [tex]\gamma[/tex] * a * b * c * A * B * C where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1, respectively to [tex]\beta[/tex], A, B, and C.

Can we simplify the above system of equations?

What is the solution to the above system of equations?

[Guest]

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

"ABC-Conjecture (Masser-Oesterlé, 1985):

Let [tex]\beta[/tex] > 1. Then, with finitely many exceptions, we have C < rad[tex](ABC)^{\beta}[/tex] "

-- Dr. Frits Beukers, author of 'The ABC-conjecture', 2005.

We shall derive six relevant equations with nine distinct and relevant integer variables from the following statements.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

gcd(A, B) = gcd(A, C) = gcd(B, C) = 1;

C < rad[tex](ABC)^{\beta}[/tex] where [tex]\beta[/tex] > 1.

According to the Fundamental Theorem of Arithmetic, we define the positive integer variables, A, B, and C:

A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}^{k_{j_1}}[/tex];

B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}^{k_{j_2}}[/tex];

C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3}}[/tex].

Therefore, we have six relevant equations with nine distinct and relevant integer variables.

A * [tex]x_1[/tex] + B * [tex]y_1[/tex] = 1;

A * [tex]x_2[/tex] + C * [tex]y_2[/tex] = 1;

B * [tex]x_3[/tex] + C * [tex]y_3[/tex] = 1;

Note: The variables, [tex]x_1[/tex], [tex]x_2[/tex], [tex]x_3[/tex], [tex]y_1[/tex], [tex]y_2[/tex], and [tex]y_3[/tex] may be positive or negative integers.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

C = [tex]\gamma[/tex] * a * b * c * A * B * C where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1, respectively to [tex]\beta[/tex], A, B, and C.

Can we simplify the above system of equations?

What is the solution to the above system of equations?

Important Remainder: In the finite exceptional cases, we have [tex]\beta[/tex] > 1 and [tex]\gamma[/tex] > 1.

[Guest]

Few Comments...

This is a beautiful problem for algebraic geometry, number theory, and integer optimization. I cannot imagine how elementary tools of mathematics can solve our difficult problem here. But I will keep an open mind since the famous quote by the late great Alan Turing is quite insightful.

"Those who can imagine anything, can create the impossible."-- Alan Turing (https://futurism.com/images/turing/).

[Guest]

An Observation: All our constants except for the rational or irrational number, [tex]\beta[/tex], are positive rationals.

[Guest]

Few Comments...

This is a beautiful problem for algebraic geometry, number theory, and integer optimization. I cannot imagine how elementary tools of mathematics can solve our difficult problem here. But I will keep a open mind since the famous quote by the late great Alan Turing is quite insightful.

"Those who can imagine anything, can create the impossible."-- Alan Turing (https://futurism.com/images/turing/).

[Guest]

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

"ABC-Conjecture (Masser-Oesterlé, 1985):

Let [tex]\beta[/tex] > 1. Then, with finitely many exceptions, we have C < rad[tex](ABC)^{\beta}[/tex] "

-- Dr. Frits Beukers, author of 'The ABC-conjecture', 2005.

We shall derive six relevant equations with nine distinct and relevant integer variables from the following statements.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

gcd(A, B) = gcd(A, C) = gcd(B, C) = 1;

C < rad[tex](ABC)^{\beta}[/tex] where [tex]\beta[/tex] > 1.

According to the Fundamental Theorem of Arithmetic, we define the positive integer variables, A, B, and C:

A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}^{k_{j_1}}[/tex];

B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}^{k_{j_2}}[/tex];

C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3}}[/tex].

Therefore, we have six relevant equations with nine distinct and relevant integer variables.

A * [tex]x_1[/tex] + B * [tex]y_1[/tex] = 1;

A * [tex]x_2[/tex] + C * [tex]y_2[/tex] = 1;

B * [tex]x_3[/tex] + C * [tex]y_3[/tex] = 1;

Note: The variables, [tex]x_1[/tex], [tex]x_2[/tex], [tex]x_3[/tex], [tex]y_1[/tex], [tex]y_2[/tex], and [tex]y_3[/tex] may be positive or negative integers.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1, respectively to [tex]\beta[/tex], A, B, and C.

Can we simplify the above system of equations?

What is the solution to the above system of equations?

Important Remainder: In the finite exceptional cases, we have [tex]\beta[/tex] > 1 and [tex]\gamma[/tex] > 1.

[Guest]

Important Reminder: In the finite exceptional cases, we have β > 1 and γ > 1.

[Guest]

"It is always the people, who nobody imagines of who do the things who nobody can imagine."

-- Alan Turing (https://futurism.com/images/turing/).

[Guest]

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

"ABC-Conjecture (Masser-Oesterlé, 1985):

Let [tex]\beta[/tex] > 1. Then, with finitely many exceptions, we have C < rad[tex](ABC)^{\beta}[/tex] "

-- Dr. Frits Beukers, author of 'The ABC-conjecture', 2005.

We shall derive six relevant equations with nine distinct and relevant integer variables from the following statements.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

gcd(A, B) = gcd(A, C) = gcd(B, C) = 1;

C < rad[tex](ABC)^{\beta}[/tex] where [tex]\beta[/tex] > 1.

According to the Fundamental Theorem of Arithmetic, we define the positive integer variables, A, B, and C:

A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}^{k_{j_1}}[/tex];

B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}^{k_{j_2}}[/tex];

C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3}}[/tex].

Therefore, we have six relevant equations with nine distinct and relevant integer variables.

A * [tex]x_1[/tex] + B * [tex]y_1[/tex] = 1;

A * [tex]x_2[/tex] + C * [tex]y_2[/tex] = 1;

B * [tex]x_3[/tex] + C * [tex]y_3[/tex] = 1;

Note: The variables, [tex]x_1[/tex], [tex]x_2[/tex], [tex]x_3[/tex], [tex]y_1[/tex], [tex]y_2[/tex], and [tex]y_3[/tex] may be positive or negative integers.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1, respectively to [tex]\beta[/tex], A, B, and C.

Can we simplify the above system of equations?

What is the solution to the above system of equations?

Important Reminder: In the finite exceptional cases, we have [tex]\beta[/tex] > 1 and [tex]\gamma[/tex] > 1.

Relevant Resource Link:

'Finding ABC-triples using Elliptic Curves' by Johannes Petrus van der Horst', 2010,

https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/vanderhorstmaster.pdf.

[Guest]

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

"ABC-Conjecture (Masser-Oesterlé, 1985):

Let [tex]\beta[/tex] > 1. Then, with finitely many exceptions, we have C < rad[tex](ABC)^{\beta}[/tex] "

-- Dr. Frits Beukers, author of 'The ABC-conjecture', 2005.

We shall derive six relevant equations with nine distinct and relevant integer variables from the following statements.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

gcd(A, B) = gcd(A, C) = gcd(B, C) = 1;

C < rad[tex](ABC)^{\beta}[/tex] where [tex]\beta[/tex] > 1.

According to the Fundamental Theorem of Arithmetic, we define the positive integer variables, A, B, and C:

A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}^{k_{j_1}}[/tex];

B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}^{k_{j_2}}[/tex];

C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3}}[/tex].

Therefore, we have six relevant equations with nine distinct and relevant integer variables.

A * [tex]x_1[/tex] + B * [tex]y_1[/tex] = 1;

A * [tex]x_2[/tex] + C * [tex]y_2[/tex] = 1;

B * [tex]x_3[/tex] + C * [tex]y_3[/tex] = 1;

Note: The variables, [tex]x_1[/tex], [tex]x_2[/tex], [tex]x_3[/tex], [tex]y_1[/tex], [tex]y_2[/tex], and [tex]y_3[/tex] may be positive or negative integers.

A = [tex]\lambda[/tex] * B where 0 < [tex]\lambda[/tex] < 1;

A + B = C;

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1, respectively to [tex]\beta[/tex], A, B, and C.

Note:

a * A = [tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}[/tex];

b * B = [tex]\prod_{j_2 =1}^{l_2}[/tex][tex]p_{j_2}[/tex];

c * C = [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}[/tex].

Can we simplify the above system of equations?

What is the solution to the above system of equations?

Important Reminder: In the finite exceptional cases, we have [tex]\beta[/tex] > 1 and [tex]\gamma[/tex] > 1.

Relevant Resource Link:

'Finding ABC-triples using Elliptic Curves' by Johannes Petrus van der Horst', 2010,

https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/vanderhorstmaster.pdf.

[Guest]

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1.

The above equation implies [tex](A + B)^{\beta - 1}[/tex] * [tex]{ (A * B)}^{\beta}[/tex] = [tex]{(\gamma *{(a * b * c)}^{\beta})}^{-1}[/tex].

[Guest]

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1.

The above equation implies [tex](A + B)^{\beta - 1}[/tex] * [tex]{ (A * B)}^{\beta}[/tex] = [tex]{(\gamma *{(a * b * c)}^{\beta})}^{-1}[/tex].

Wow! That latter equation above strongly suggests that the ABC-conjecture is true since rational numbers are dense in the real numbers! Of course, there's more work needed to confirm our hunch!

[Guest]

Stage 2:

Suppose there exists [tex]\beta_{e }[/tex] > 1 where [tex]\beta_{e }[/tex] is a rational number such that there are only finitely many (or n) ABC-triples which satisfy our system of equations when [tex]\gamma[/tex] < 1. And let S be the ordered set ( from smallest ordinate [tex]B_{1}[/tex] of ([tex]A_{1}[/tex], [tex]B_{1}[/tex], [tex]C_{1}[/tex]) to the largest ordinate [tex]B_n[/tex] of ([tex]A_{n}[/tex], [tex]B_{n}[/tex], [tex]C_{n}[/tex]) ) of those ABC-triples which solve our system of equations....

[Guest]

C = [tex]\gamma[/tex] *[tex]{(a * b * c * A * B * C)}^{\beta}[/tex] where 0 < [tex]\gamma[/tex] < 1, 0 < a [tex]\leq[/tex] 1, 0 < b [tex]\leq[/tex] 1, and 0 < c [tex]\leq[/tex] 1.

The above equation implies [tex](A + B)^{\beta - 1}[/tex] * [tex]{ (A * B)}^{\beta}[/tex] = [tex]{(\gamma *{(a * b * c)}^{\beta})}^{-1}[/tex].

Wow! That latter equation above strongly suggests that the ABC-conjecture is true since rational numbers are dense in the real numbers! Of course, there's more work needed to confirm our hunch!

An Interesting Observation:

[tex]{( (A + B) * (A * B) )}^{\beta}[/tex] = [tex]{(A +B)/ (\gamma *{(a * b * c)}^{\beta})}[/tex].

[Guest]

Stage 2:

Suppose there exists [tex]\beta_{e }[/tex] > 1 where [tex]\beta_{e }[/tex] is a real number such that there are only finitely many (or n) ABC-triples which satisfy our system of equations when [tex]\gamma[/tex] < 1 and when [tex]\gamma[/tex] is also a real number . And let S be the ordered set ( from smallest ordinate [tex]B_{1}[/tex] of ([tex]A_{1}[/tex], [tex]B_{1}[/tex], [tex]C_{1}[/tex]) to the largest ordinate [tex]B_n[/tex] of ([tex]A_{n}[/tex], [tex]B_{n}[/tex], [tex]C_{n}[/tex]) ) of those ABC-triples which solve our system of equations....

Now, we shall consider very carefully the following important equation one when [tex]\beta_{e }[/tex] > 1 and when [tex]\gamma[/tex] < 1:


1. [tex](A + B) * (A * B)[/tex] = [tex]{( (A + B)/ \gamma )^{1 / \beta_{e}}} * (a * b * c)^{-1}[/tex].

Once coprimes, A, B, and C = (A + B) are computed. The numbers, a, b, and c are rational constants. So, we need only to compute the existence or nonexistence of [tex]\gamma[/tex] to prove or disprove, respectively, the ABC-conjecture for any ABC_triples outside of S.

Our previous hutch is very close to being right since the ABC-conjecture is most likely true! But our work to prove or disprove the conjecture is not complete...

Dave, aka the antworker123 or primework123

https://www.researchgate.net/profile/David_Cole29