# 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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### 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
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