Proof of Collatz Conjecture

[Guest]

Remark: For our previous remarks, we assume and we refer to the ordered set of all positive even integers, E = {2, 4, 6, 8, 10, 12, ..., [tex]\infty[/tex]}.

Remark: The distribution(s) of maximum divisors, [tex]2^{x}[/tex], of even integers, between 2 and [tex]\infty[/tex], inclusively, is flat/uniform/identical/infinite.

[Guest]

Final Remarks: Let's imagine the largest positive even integer, [tex]48 < t < \infty[/tex], in any appropriate Collatz sequence.

The maximum divisor, 2, occurs 50% (.5) or close to 50% (.5) of the time, [tex]\frac{t}{2}[/tex], between 2 and t, inclusively.

Or equivalently, the maximum divisor, 2, occurs about [tex]\frac{t}{4}[/tex].

That fact is central to our proof of the Collatz Conjecture because it explains why there is growth in the Collatz sequence, and it also explains why the Collatz sequence always converges to one.

[Guest]

FYI: "The are still some 'diehards' (mathematicians) who still believe the Collatz Conjecture is unproven. Well, my proof of the Collatz Conjecture still stands. And hopefully, they will accept it in the near future. Meanwhile, let's pray for them. Amen! " -- David Cole, the Rodney Dangerfield of mathematics since he receives little or no respect/credit for his work.

Relevant Reference Link:

'Mathematicians Are So Close to Cracking This 82-Year-Old Riddle',

https://news.yahoo.com/mathematicians-close-cracking-82-old-180000581.html.

[Guest]

Relevant Reference Link:

"A Brief Analysis of the Collatz Conjecture",

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

[Guest]

FYI: "The are still some 'diehards' (mathematicians) who still believe the Collatz Conjecture is unproven. Well, my proof of the Collatz Conjecture still stands. And hopefully, they will accept it in the near future. Meanwhile, let's pray for them. Amen! " -- David Cole, the Rodney Dangerfield of mathematics since he receives little or no respect/credit for his work.

Relevant Reference Link:

'Mathematicians Are So Close to Cracking This 82-Year-Old Riddle',

https://news.yahoo.com/mathematicians-close-cracking-82-old-180000581.html.

FYI:
"The Simple Math Problem We Still Can’t (Oops!) Solve.
Despite recent progress on the notorious Collatz conjecture, we still don’t know (Oops!) whether a number can escape its infinite loop."

[Guest]

Missing Link for Previous Post:

https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/.

[Guest]

\rightarrow\gamma

Relevant Reference Link:

"A Brief Analysis of the Collatz Conjecture",

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

Some Food for Thought:

Hmm. We have extracted our important equation ("A Brief Analysis of the Collatz Conjecture"):

[tex]n_{t } = r_{t } * \prod_{j=1}^{t }(\frac{3}{2^{i_{j }}}) = r * \prod_{m=1}^{k }(\frac{3}{2^{m}})^{\frac{t}{2^{m}}}[/tex] with [tex]r_{t } > r[/tex] where k = Floor( [tex]\frac{log(e_{max })}{log(2)}[/tex])

and where [tex]e_{max }[/tex] is the maximum positive even integer in a Collatz sequence.

The set of all possible maximum divisors for the Collatz sequence of positive even integers is {[tex]2^{0}, 2^{1}, 2^{2}, ..., 2^{k}[/tex]}.

Remark: The existence of the maximum divisor, [tex]2^{0}[/tex], implies [tex]n_{t } = 2^{0} = 1[/tex].

So, the probability that the maximum divisor, [tex]2^{0}[/tex], does not occur is roughly [tex](\frac{k}{k+1})^{\gamma}[/tex] for any closed cycle in a Collatz sequence of positive even integers.

Remark: "Roughly" indicates a crude approximation.

But as [tex]\gamma \rightarrow \infty[/tex], [tex](\frac{k}{k+1})^{\gamma} \rightarrow 0[/tex].

Therefore, [tex]n_{t } = 1[/tex]. And we conclude the Collatz conjecture is true!

Remark: Our "food for thought" (analysis) may be over the 'top'... It's Ok!

[Guest]

Hah! One is odd!

And your math is weird! It's over the 'top' for sure!

But life is strange too! Haha...

[Guest]

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

Some Food for Thought: (Update)

Hmm. We have extracted our important equation ("A Brief Analysis of the Collatz Conjecture"):

[tex]n_{t } = r_{t } * \prod_{j=1}^{t }(\frac{3}{2^{i_{j }}}) = r * \prod_{m=1}^{k }(\frac{3}{2^{m}})^{\frac{t}{2^{m}}}[/tex] with [tex]r_{t } > r[/tex] where k = Floor( [tex]\frac{log(e_{max })}{log(2)}[/tex])

and where [tex]e_{max }[/tex] is the maximum positive even integer in a Collatz sequence.

The set of all possible maximum divisors for the Collatz sequence of positive even integers is {[tex]2^{1}, 2^{2}, ..., 2^{k}[/tex]}.


So, the probability that the maximum divisor, [tex]2^{1}[/tex], does not occur is roughly [tex](\frac{k-1}{k})^{\gamma}[/tex] for any closed cycle in a Collatz sequence of positive even integers.

Remark: "Roughly" indicates a crude approximation.

But as [tex]\gamma \rightarrow \infty[/tex], [tex](\frac{k-1}{k})^{\gamma} \rightarrow 0[/tex].

Therefore, [tex]n_{t } = 1[/tex]. And we conclude the Collatz conjecture is true!

Remark: Our "food for thought" (analysis) may be over the 'top'... It's Ok!

[Guest]

The previous post is more acceptable. And one is an odd number. Hehehe....

[Guest]

A Clarification:

The probability that the maximum divisor, [tex]2^{i}[/tex], (assuming [tex]2^{i}[/tex] appears in the Collatz sequence of positive even integers) does not occur is roughly [tex](\frac{k-1}{k})^{\gamma}[/tex] for any closed cycle in a Collatz sequence of positive even integers...

[Guest]

Collatz Equation: [tex]n_{t } = r * \prod_{m=1}^{k }(\frac{3}{2^{m}})^{\frac{t}{2^{m}}}[/tex] = 1.

Remark: As [tex]t \rightarrow \infty[/tex], [tex]\prod_{m=1}^{k }(\frac{3}{2^{m}})^{\frac{t}{2^{m}}} \rightarrow 0[/tex] and [tex]r \rightarrow \infty[/tex].

Relevant Reference Link:

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

[Guest]

FYI: "The are still some 'diehards' (mathematicians) who still believe the Collatz Conjecture is unproven. Well, my proof of the Collatz Conjecture still stands. And hopefully, they will accept it in the near future. Meanwhile, let's pray for them. Amen! " -- David Cole, the Rodney Dangerfield of mathematics since he receives little or no respect/credit for his work.

Relevant Reference Link:

'Mathematicians Are So Close to Cracking This 82-Year-Old Riddle',

https://news.yahoo.com/mathematicians-close-cracking-82-old-180000581.html.

FYI:
"The Simple Math Problem We Still Can’t (Oops!) Solve.
Despite recent progress on the notorious Collatz conjecture, we still don’t know (Oops!) whether a number can escape its infinite loop."

FYI: "Collatz Conjecture-the simplest impossible problem (Oops!)"

Please see attached file for some details of this flawed opinion...

Wordpress Link: https://wordpress.com/read/feeds/36951305/posts/3012037143.

You may need a wordpress account to read the complete paper.

[Guest]

Final Remark (Hmm): The [tex]3n_{0} + 1[/tex] Problem (Collatz Conjecture) has a [tex]n_{0 }[/tex] ≡ 1 mod 4 solution!

Go Blue!

Relevant Reference Link:

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

[Guest]

Update:

Final Remark (Hmm): The [tex]3n_{0} + 1[/tex] Problem (Collatz Conjecture) has a [tex]n_{0 } + 1[/tex] ≡ 0 mod 4 solution!

Go Blue!

Relevant Reference Link:

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

[Guest]

Oops! We got it wrong! We revert back to a previous post. And our final remark assumes infinitely solutions for the Collatz Problem beginning with [tex]n_{0}[/tex]. The next solution is [tex]4n_{0} + 1[/tex] and so on. And we urge caution too.

Update:

Final Remark (Hmm): The [tex]3n_{0} + 1[/tex] Problem (Collatz Conjecture) has a [tex]n_{0 }[/tex] ≡ 1 mod 4 solution!

Go Blue!

Relevant Reference Link:

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

[Guest]

We assume [tex]n_{0} \to 4n_{0} + 1[/tex] for our previous post.

Dave.

[Guest]

We assume [tex]n_{0} \to 4n_{0} + 1[/tex] for our previous post.

...

Hmm. If the above assumption makes sense, then please accept it.

But we think it is best to assume [tex]n_{0} \gets 4n_{0} + 1[/tex] for a previous post where [tex]n_{0}[/tex] is a positive odd integer greater than one.

We apologize for the confusion and the sloppy (flawed) reasoning...

Dave.

Final Remark (Hmm): The [tex]3n_{0} + 1[/tex] Problem (Collatz Conjecture) has a [tex]n_{0 }[/tex] ≡ 1 mod 4 solution!

Go Blue!

Relevant Reference Link:

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

[Guest]

FYI: 'Bakuage Offers Prize of 120 Million JPY to Whoever Solves Collatz Conjecture, Math Problem Unsolved for 84 Years',

https://mathprize.net/posts/collatz-conjecture/index.html.

The Collatz conjecture is true! Amen!

"Hmm. Life seems so unfair... And when life (existence) ends, all is well. Amen!"

[Guest]

FYI: 'Bakuage Offers Prize of 120 Million JPY to Whoever Solves Collatz Conjecture, Math Problem Unsolved for 84 Years',

https://mathprize.net/posts/collatz-conjecture/index.html.

The Collatz conjecture is true! Amen!

"Hmm. Life seems so unfair... And when life (existence) ends, all is well. Amen!"

Hmm. You claim the Collatz conjecture is true. And that your work supports your claim.

I suggest you refer to the latest work ('THE 3x + 1 PROBLEM: AN OVERVIEW', https://arxiv.org/pdf/2111.02635.pdf) by a world-renowned authority on the Collatz conjecture who states the conjecture is still unproven. Do you agree?