# Proof of Collatz Conjecture

### Re: Proof of Collatz Conjecture

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, ..., $$\infty$$}.

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

### Re: Proof of Collatz Conjecture

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

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

Or equivalently, the maximum divisor, 2, occurs about $$\frac{t}{4}$$.

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.
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"The Collatz Conjecture is true!" -- David Cole.
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Guest

### Re: Proof of Collatz Conjecture

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.

'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

"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

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.

'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

Guest

### Re: Proof of Collatz Conjecture

\rightarrow\gamma

"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"):

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

and where $$e_{max }$$ 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 {$$2^{0}, 2^{1}, 2^{2}, ..., 2^{k}$$}.

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

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

Remark: "Roughly" indicates a crude approximation.

But as $$\gamma \rightarrow \infty$$, $$(\frac{k}{k+1})^{\gamma} \rightarrow 0$$.

Therefore, $$n_{t } = 1$$. 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

Hah! One is odd!

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

But life is strange too! Haha...
Guest

### Re: Proof of Collatz Conjecture

Some Food for Thought: (Update)

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

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

and where $$e_{max }$$ 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 {$$2^{1}, 2^{2}, ..., 2^{k}$$}.

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

Remark: "Roughly" indicates a crude approximation.

But as $$\gamma \rightarrow \infty$$, $$(\frac{k-1}{k})^{\gamma} \rightarrow 0$$.

Therefore, $$n_{t } = 1$$. 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

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

### Re: Proof of Collatz Conjecture

A Clarification:

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

### Re: Proof of Collatz Conjecture

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

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

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

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