Searching for a valid proof of the abc Conjecture

[Guest]

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

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

In a 'nutshell', for positive integers, A, B, and C we have have:

[tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}[/tex] [tex]\prod_{j_2 =1}^{l_2}[/tex] [tex]p_{j_2}[/tex] [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}[/tex]

= rad(ABC) = [tex](\frac{C}{\gamma})^{1/{\beta}}[/tex] = [tex](\frac{\prod_{j_3 =1}^{l_3}p_{j_3}^{k_{j_3}}}{\gamma})^{1/{\beta}}[/tex]

for some [tex]{\beta} > 1[/tex] such that:

Case 1: [tex]0 < \gamma < 1[/tex] implies an infinite set of triples, (A, B, C);

Case 2: [tex]\gamma > 1[/tex] implies an empty or finite set of triples, (A, B, C);

according to [tex]0 < \gamma < 2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta}[/tex]

such that gcd(A, B) = gcd(A, C) = gcd(B, C) = 1 with A < B < C = A + B.

-- Dave, https://www.researchgate.net/profile/David_Cole29

Notes:

[tex]k_{j_3}[/tex] represent constants (integers [tex]\ge 1[/tex]) while [tex]\beta \ge 1[/tex] is unrestricted (no upper bound) variable.

Case 1: [tex]0 < \gamma < 1[/tex] also implies [tex]k_{j_3} < 3 \beta[/tex] such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} < 1[/tex].

Case 2: [tex]\gamma > 1[/tex] also implies [tex]k_{j_3} \ge 3 \beta[/tex] which forces a upper bound on [tex]\beta[/tex].

[Guest]

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

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

In a 'nutshell', for positive integers, A, B, and C we have have:

[tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}[/tex] [tex]\prod_{j_2 =1}^{l_2}[/tex] [tex]p_{j_2}[/tex] [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}[/tex]

= rad(ABC) = [tex](\frac{C}{\gamma})^{1/{\beta}}[/tex] = [tex](\frac{\prod_{j_3 =1}^{l_3}p_{j_3}^{k_{j_3}}}{\gamma})^{1/{\beta}}[/tex]

for some [tex]{\beta} > 1[/tex] such that:

Case 1: [tex]0 < \gamma < 1[/tex] implies an infinite set of triples, (A, B, C);

Case 2: [tex]\gamma > 1[/tex] implies an empty or finite set of triples, (A, B, C);

according to [tex]0 < \gamma < 2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta}[/tex]

such that gcd(A, B) = gcd(A, C) = gcd(B, C) = 1 with A < B < C = A + B.

-- Dave, https://www.researchgate.net/profile/David_Cole29

Notes:

[tex]k_{j_3}[/tex] represent constants (integers [tex]\ge 1[/tex]) while [tex]\beta \ge 1[/tex] is unrestricted (no upper bound) variable.

Case 1: [tex]0 < \gamma < 1[/tex] also implies [tex]k_{j_3} < 3 \beta[/tex] such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} < 1[/tex].

Case 2: [tex]\gamma > 1[/tex] also implies [tex]k_{j_3} \ge 3 \beta[/tex] which forces a upper bound on [tex]\beta[/tex]
such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} \ge 1[/tex].

[Guest]

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

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

In a 'nutshell', for positive integers, A, B, and C we have have:

[tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}[/tex] [tex]\prod_{j_2 =1}^{l_2}[/tex] [tex]p_{j_2}[/tex] [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}[/tex]

= rad(ABC) = [tex](\frac{C}{\gamma})^{1/{\beta}}[/tex] = [tex](\frac{\prod_{j_3 =1}^{l_3}p_{j_3}^{k_{j_3}}}{\gamma})^{1/{\beta}}[/tex]

for some [tex]{\beta} > 1[/tex] such that:

Case 1: [tex]0 < \gamma < 1[/tex] implies an infinite set of triples, (A, B, C);

Case 2: [tex]\gamma > 1[/tex] implies an empty or finite set of triples, (A, B, C);

according to [tex]0 < \gamma < 2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta}[/tex]

such that gcd(A, B) = gcd(A, C) = gcd(B, C) = 1 with A < B < C = A + B.

-- Dave, https://www.researchgate.net/profile/David_Cole29

Notes:

[tex]k_{j_3}[/tex] represent constants (integers [tex]\ge 1[/tex]) while [tex]\beta \ge 1[/tex] is unrestricted (no upper bound) variable.

Case 1: [tex]0 < \gamma < 1[/tex] also implies [tex]k_{j_3} < 3 \beta[/tex] such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} < 1[/tex].

Case 2: [tex]\gamma > 1[/tex] also implies [tex]k_{j_3} \ge 3 \beta[/tex] which forces a upper bound on [tex]\beta[/tex]
such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} > 1[/tex].

[Guest]

Note: And when that upper bound on [tex]{\beta}[/tex] for case 2 is exceeded, there is a solution in case 1.

[Guest]

An Update:

ABC-Conjecture (Masser-Oesterlé, 1985):

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

Proof of the ABC Conjecture:

In a 'nutshell', for positive integers, A, B, and C we have:

[tex]\prod_{j_1 =1}^{l_1}[/tex][tex]p_{j_1}[/tex] [tex]\prod_{j_2 =1}^{l_2}[/tex] [tex]p_{j_2}[/tex] [tex]\prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}[/tex]

= rad(ABC) = [tex](\frac{C}{\gamma})^{1/{\beta}}[/tex] = [tex](\frac{\prod_{j_3 =1}^{l_3}p_{j_3}^{k_{j_3}}}{\gamma})^{1/{\beta}}[/tex]

for some [tex]{\beta} > 1[/tex] such that:

Case 1: [tex]0 < \gamma < 1[/tex] implies an infinite set of triples, (A, B, C);

Case 2: [tex]\gamma > 1[/tex] implies an empty or finite set of triples, (A, B, C);

according to [tex]0 < \gamma < 2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta}[/tex]

such that gcd(A, B) = gcd(A, C) = gcd(B, C) = 1 with A < B < C = A + B.



Notes:

The exponent, [tex]k_{j_3}[/tex], represent constants (integers [tex]\ge 1[/tex]) while [tex]\beta > 1[/tex] is unrestricted (no upper bound) real variable.

Case 1: [tex]0 < \gamma < 1[/tex] also implies [tex]k_{j_3} < 3 \beta[/tex] such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} < 1[/tex].

Case 2: [tex]\gamma > 1[/tex] also implies [tex]k_{j_3} \ge 3 \beta[/tex] which forces a upper bound on [tex]\beta[/tex]
such that [tex]2 * \prod_{j_3 =1}^{l_3}[/tex][tex]p_{j_3}^{k_{j_3} - 3\beta} > 1[/tex].

And when that upper bound on [tex]{\beta}[/tex] for case 2 is exceeded, there is a solution in case 1.

David Cole.

Relevant Reference Links:

'Searching for a valid proof of the abc Conjecture',

https://www.math10.com/forum/viewtopic.php?f=63&t=1793;

'What is the proof of the ABC Conjecture?',

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

[Guest]

FYI: 'Does ABC (conjecture) implies Fermat's last theorem?'

https://math.stackexchange.com/questions/1157932/does-abc-implies-fermats-last-theorem.

Since the abc Conjecture is true, then the following conjectures or theorems are also confirmed:

Modified Szpiro conjecture;

Beal conjecture ("I am still waiting for that million-dollar award for proving the Beal Conjecture..." -- David Cole);

Fermat–Catalan conjecture;

Roth's theorem;

Tijdeman's theorem.

Relevant Reference Link:

'The abc Conjecture',

https://en.wikipedia.org/wiki/Abc_conjecture.

[Guest]

On the Beal Conjecture:

I am still waiting for that million-dollar award for proving the Beal Conjecture directly or via the proof of the abc Conjecture.

Sincerely,

David Cole,

openmind123omega@gmail.com

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

P.S. Please do not spam me. Thank you!

[Guest]

On the Beal Conjecture:

I am still waiting for that million-dollar award for proving the Beal Conjecture directly or via the proof of the abc Conjecture.

Sincerely,

David Cole,

openmind123omega@gmail.com

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

P.S. Please do not spam me. Thank you!

I, David Cole, have made some significant contributions to the mathematical sciences here and elsewhere, and I have not received any significant recognition nor any significant award for my works thus far. And therefore, I shall stop sharing my work...

May Lord God bless me and help me in this very miserable so-called human world,

David Cole.

[Guest]

FYI: 'Mochizuki's inter-universal Teichmüller proof has been published (Update).'

https://phys.org/news/2020-04-mochizuki-inter-universal-teichmller-proof-published.html.

[Guest]

FYI: 'Mochizuki's inter-universal Teichmüller proof has been published (Update).'

https://phys.org/news/2020-04-mochizuki-inter-universal-teichmller-proof-published.html.

FYI: 'On the abc Conjecture and some of its consequences',

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

Hmm. This great but "redundant" link! (I may have posted it earlier. I don't recall when... Please forgive if I have.) Thank you!

[Guest]

\int

On the Beal Conjecture:

I am still waiting for that million-dollar award for proving the Beal Conjecture directly or via the proof of the abc Conjecture.

Sincerely,

David Cole,


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

https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/.

Thank you!

I, David Cole, have made some significant contributions to the mathematical sciences here and elsewhere, and I have not received any significant recognition nor any significant award for my works thus far...

May Lord God bless me and help me.

David Cole.


P.S. FIGHT RACISM IN THE SCIENCES INCLUDING MATHEMATICS! THANK YOU!

[Nobody Knows]

full proof -> https://www.researchgate.net/publication/351347153_COLLATZ_CONJECTURE_-THE_PROOF
Images that can help to process and understand this proof (they are NOT part of the proof)

[Giovanni Di Savino]

Please refer to the following reference link for details:

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

1. Michele, my nephew, is keen to point out that he is not 3 but almost 4 years old and yesterday Sunday he came to visit his grandparents and, even if he is not talkative, he shows off the new things he learned in nursery school. During one of his stories, I seemed absent to him and, to get my attention, he began to spell out and repeat the word pyramid. The new word said by Michele struck me and made me think of Thales' well-known measurement of the pyramids (1). Explained below, I am proposing it as a solution to the ABC conjecture and I hope you agree with me that the credit for this discovery goes to Michele who: suggested "pyramid" to me and which, he would like to point out, does not have 3 but has almost 4 years. 1.1 Thales measures the height of the inaccessible pyramid of Cheops by comparing two shadows on the ground and, knowing the height of a rod generating a shadow, he was able to determine the height of the pyramid generating the other shadow and from that measurement it is known and demonstrated that everything that can be reported on the plan can be measured. Triplets of natural numbers which on the plane are defined with the name of abscissa (x), ordinate (y) and quota (z), can represent and measure: a) all numbers that are the sum of two numbers, zc=xa+ yb; b) all numbers that are the product of two numbers, zc=xa*yb. All the natural numbers are represented on the plane but we will never have the time, space and computing power necessary to know how many the natural numbers represented are and how they are generated; we will never be able to claim to have verified how all the natural numbers represented are generated, also because, for each number, there is always the next number which will never have been verified but which exists, how they exist and may not be known: all prime numbers ≤ zc and all the numbers "c" and "d" which with c=a+b and the numbers d=rad(abc) determine by how much and when c<>d :"the solution of the abc conjecture". Thales, in subsequent days, at the same time, with the same rod or with the same one that measured the pyramid of Cheops, could have generated the shadows again to measure all the pyramids that were as high as the one already measured and would have been able to measure , also, pyramids and artefacts of different heights as long as their heights were divisible by the same measure which, today with the Fundamental Theorem of Arithmetic. we know, it can only be a prime factor of the comparison shadow generating rod. If an rod could compare and measure two artifacts of different heights, the number "c" and the number "d", which are two integer and different numbers of the abc conjecture, can be compared and measured with a measure that is equal and present in both "c" and "d"; this measurement is one of the factors of the number c. Only in this way, even if the processing times will not allow us to know the result, it is known when, how and by how much c<>d. The numbers c and numbers d can be of any size, they are represented on the plane, they exist, they are measurable and comparable with a prime number that exists even if it is not known. That prime number present in "c" and in "d", is the factor prime number rad(c) present in "c" which is =a+b, and present in "d" which is = rad(abc). The rarely reported in the statement has a well-defined mathematical meaning........It has been demonstrated that all the natural numbers are represented on the plane, including all the prime numbers and the numbers of the abc conjecture which are: the number a, the number b, the number "c", which is the sum of the numbers a and b , and the number "d" which is the result of the product of the factors^1 of the number a, the number b and the number c and it is known that to know what the c<>d numbers are we cannot verify all the infinite numbers reported on the floor. The number "zc" is a number that is the sum of two numbers, xa+yb that have no factors in common, but zc is a number generated by factors, a prime number^n≥1 or even by the product of two or all the first known^n≥1; the number "d" is a number that is the product of the factors^1 of the number xa, with the factors^1 of the number yb and the factors^1 of the number xc. When and why the sum of two numbers can be <> the product of the factors of the two numbers and factors of their sum we cannot verify among all the numbers reported on the plane. The determinations of this work are to be considered a continuation of the work: "the abc conjecture resolved by giving a mathematical value to the 'rarely' which determines when and by how much c=a+b ≠ d=rad(abc) " reported in (2) . With "almost" 4 years old Michele and almost 78 I, we are both out of quota for the prizes but it is nice if you tell us if we have and where we have not done a good job.

[Guest]

Why are people posting purported proofs of the abc conjecture here?

Create your a new topic and post your work....

[Guest]

Why are people posting purported proofs of the abc conjecture here?

Create a new topic and post your work....

[Giovanni Di Savino]

Why are people posting purported proofs of the abc conjecture here?

Create a new topic and post your work....

https://www.math10.com/forum/viewtopic.php?f=42&t=11294

[Guest]

On the Beal Conjecture:

I am still waiting for that million-dollar award for proving the Beal Conjecture directly or via the proof of the abc Conjecture.

Sincerely,

David Cole,

openmind123omega@gmail.com

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

P.S. Please do not spam me. Thank you!

I, David Cole, have made some significant contributions to the mathematical sciences here and elsewhere, and I have not received any significant recognition nor any significant award for my works thus far. And therefore, I shall stop sharing my work...

May Lord God bless me and help me in this very miserable so-called human world,

David Cole.