# Searching for a valid proof of the abc Conjecture

### Searching for a valid proof of the abc Conjecture

https://www.researchgate.net/post/What_is_the_proof_of_the_ABC_Conjecture
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### Re: Searching for a valid proof of the abc Conjecture

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

### Re: Searching for a valid proof of the abc Conjecture

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.

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### Re: Searching for a valid proof of the abc Conjecture

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

http://www.staff.science.uu.nl/~beuke106/ABCpresentation.pdf
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### Re: Searching for a valid proof of the abc Conjecture

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

https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcEnVI.pdf
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### Re: Searching for a valid proof of the abc Conjecture

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

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

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

-- 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 = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

C < rad$$(ABC)^{\beta}$$ where $$\beta$$ > 1.

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

A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}^{k_{j_1}}$$;

B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}^{k_{j_2}}$$;

C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}^{k_{j_3}}$$.

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

A * $$x_1$$ + B * $$y_1$$ = 1;

A * $$x_2$$ + C * $$y_2$$ = 1;

B * $$x_3$$ + C * $$y_3$$ = 1;

Note: The variables, $$x_1$$, $$x_2$$, $$x_3$$, $$y_1$$, $$y_2$$, and $$y_3$$ may be positive or negative integers.

A = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

Can we simplify the above system of equations?

What is the solution to the above system of equations?
Guest

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

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

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

-- 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 = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

C < rad$$(ABC)^{\beta}$$ where $$\beta$$ > 1.

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

A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}^{k_{j_1}}$$;

B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}^{k_{j_2}}$$;

C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}^{k_{j_3}}$$.

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

A * $$x_1$$ + B * $$y_1$$ = 1;

A * $$x_2$$ + C * $$y_2$$ = 1;

B * $$x_3$$ + C * $$y_3$$ = 1;

Note: The variables, $$x_1$$, $$x_2$$, $$x_3$$, $$y_1$$, $$y_2$$, and $$y_3$$ may be positive or negative integers.

A = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

C = $$\gamma$$ * a * b * c * A * B * C where 0 < $$\gamma$$ < 1, 0 < a $$\leq$$ 1, 0 < b $$\leq$$ 1, and 0 < c $$\leq$$ 1, respectively to $$\beta$$, 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 $$\beta$$ > 1 and $$\gamma$$ > 1.
Guest

### Re: Searching for a valid proof of the abc Conjecture

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/)
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Guest

### Re: Searching for a valid proof of the abc Conjecture

An Observation: All our constants except for the rational or irrational number, $$\beta$$, are positive rationals.
Guest

### Re: Searching for a valid proof of the abc Conjecture

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

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:
Guest wrote:Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

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

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

-- 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 = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

C < rad$$(ABC)^{\beta}$$ where $$\beta$$ > 1.

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

A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}^{k_{j_1}}$$;

B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}^{k_{j_2}}$$;

C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}^{k_{j_3}}$$.

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

A * $$x_1$$ + B * $$y_1$$ = 1;

A * $$x_2$$ + C * $$y_2$$ = 1;

B * $$x_3$$ + C * $$y_3$$ = 1;

Note: The variables, $$x_1$$, $$x_2$$, $$x_3$$, $$y_1$$, $$y_2$$, and $$y_3$$ may be positive or negative integers.

A = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

C = $$\gamma$$ *$${(a * b * c * A * B * C)}^{\beta}$$ where 0 < $$\gamma$$ < 1, 0 < a $$\leq$$ 1, 0 < b $$\leq$$ 1, and 0 < c $$\leq$$ 1, respectively to $$\beta$$, 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 $$\beta$$ > 1 and $$\gamma$$ > 1.
Guest

### Re: Searching for a valid proof of the abc Conjecture

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

### Re: Searching for a valid proof of the abc Conjecture

"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

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:
Guest wrote:
Guest wrote:Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

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

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

-- 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 = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

C < rad$$(ABC)^{\beta}$$ where $$\beta$$ > 1.

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

A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}^{k_{j_1}}$$;

B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}^{k_{j_2}}$$;

C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}^{k_{j_3}}$$.

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

A * $$x_1$$ + B * $$y_1$$ = 1;

A * $$x_2$$ + C * $$y_2$$ = 1;

B * $$x_3$$ + C * $$y_3$$ = 1;

Note: The variables, $$x_1$$, $$x_2$$, $$x_3$$, $$y_1$$, $$y_2$$, and $$y_3$$ may be positive or negative integers.

A = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

'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

### Re: Searching for a valid proof of the abc Conjecture

Formulating a Proof or Disproof of the ABC-Conjecture:

Stage 1:

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

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

-- 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 = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

C < rad$$(ABC)^{\beta}$$ where $$\beta$$ > 1.

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

A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}^{k_{j_1}}$$;

B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}^{k_{j_2}}$$;

C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}^{k_{j_3}}$$.

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

A * $$x_1$$ + B * $$y_1$$ = 1;

A * $$x_2$$ + C * $$y_2$$ = 1;

B * $$x_3$$ + C * $$y_3$$ = 1;

Note: The variables, $$x_1$$, $$x_2$$, $$x_3$$, $$y_1$$, $$y_2$$, and $$y_3$$ may be positive or negative integers.

A = $$\lambda$$ * B where 0 < $$\lambda$$ < 1;

A + B = C;

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

Note:

a * A = $$\prod_{j_1 =1}^{l_1}$$$$p_{j_1}$$;

b * B = $$\prod_{j_2 =1}^{l_2}$$$$p_{j_2}$$;

c * C = $$\prod_{j_3 =1}^{l_3}$$$$p_{j_3}$$.

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 $$\beta$$ > 1 and $$\gamma$$ > 1.

'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

### Re: Searching for a valid proof of the abc Conjecture

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

The above equation implies $$(A + B)^{\beta - 1}$$ * $${ (A * B)}^{\beta}$$ = $${(\gamma *{(a * b * c)}^{\beta})}^{-1}$$.
Guest

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:C = $$\gamma$$ *$${(a * b * c * A * B * C)}^{\beta}$$ where 0 < $$\gamma$$ < 1, 0 < a $$\leq$$ 1, 0 < b $$\leq$$ 1, and 0 < c $$\leq$$ 1.

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

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

### Re: Searching for a valid proof of the abc Conjecture

Stage 2:

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

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:
Guest wrote:C = $$\gamma$$ *$${(a * b * c * A * B * C)}^{\beta}$$ where 0 < $$\gamma$$ < 1, 0 < a $$\leq$$ 1, 0 < b $$\leq$$ 1, and 0 < c $$\leq$$ 1.

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

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:

$${( (A + B) * (A * B) )}^{\beta}$$ = $${(A +B)/ (\gamma *{(a * b * c)}^{\beta})}$$.
Guest

### Re: Searching for a valid proof of the abc Conjecture

Guest wrote:Stage 2:

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

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

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

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 $$\gamma$$ 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
Guest

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