Proof of Collatz Conjecture

Re: Proof of Collatz Conjecture

Postby Guest » Sat Jul 04, 2020 12:06 am

Guest wrote: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
 

Re: Proof of Collatz Conjecture

Postby Guest » Sat Jul 04, 2020 7:55 pm

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.
Attachments
A Collatz Sequence.jpg
"The Collatz Conjecture is true!" -- David Cole.
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Re: Proof of Collatz Conjecture

Postby Guest » Wed Aug 12, 2020 1:59 am

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
 

Re: Proof of Collatz Conjecture

Postby Guest » Sun Sep 20, 2020 11:32 pm

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
 

Re: Proof of Collatz Conjecture

Postby Guest » Fri Oct 30, 2020 4:21 pm

Guest wrote: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
 


Re: Proof of Collatz Conjecture

Postby Guest » Sat Nov 14, 2020 4:25 pm

\rightarrow\gamma
Guest wrote: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
 

Re: Proof of Collatz Conjecture

Postby Guest » Sat Nov 14, 2020 4:36 pm

Hah! One is odd! :shock:

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

But life is strange too! Haha... :)
Guest
 

Re: Proof of Collatz Conjecture

Postby Guest » Sat Nov 14, 2020 4:45 pm



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
 

Re: Proof of Collatz Conjecture

Postby Guest » Sat Nov 14, 2020 4:50 pm

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

Re: Proof of Collatz Conjecture

Postby Guest » Sat Nov 14, 2020 5:58 pm

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
 

Re: Proof of Collatz Conjecture

Postby Guest » Thu Nov 19, 2020 4:38 pm

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/.
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