Guest wrote:Please refer to the following reference link for details:
https://www.researchgate.net/post/What_is_the_proof_of_the_ABC_Conjecture.
Guest wrote: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 wrote: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 wrote:Guest wrote: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 wrote:Guest wrote:Guest wrote: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.
Guest wrote: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 wrote:Guest wrote: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 wrote: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....
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