# Searching for a valid proof of the abc Conjecture

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

Please refer to the following reference link for details:

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

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

Guest wrote: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.

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.

Guest

### 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.

Please review it at the following link.

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

### 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!

Please review it at the following link.

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

Next