The Collatz Equation that supports the Collatz Conjecture

The Collatz Equation that supports the Collatz Conjecture

Postby Guest » Thu Oct 15, 2020 3:38 pm

The Collatz equation that supports the truth of the Collatz conjecture:

[tex]r * \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}} = 1[/tex]

where [tex]r = r(n_{1 })[/tex] is a positive real number and where t is the number of trials it takes the Collatz sequence of odd positive integers to converge to one.

Remark: [tex]n_{1 }[/tex] is any (initial) positive odd integer greater than one.

Remark: [tex]k = \lfloor \frac{log (e_{max })}{log(2)} \rfloor[/tex] where [tex]e_{max }[/tex] is the maximum positive even integer in the Collatz sequence.

Remark: We assume the algorithm for the Collatz conjecture.


Example: If we let [tex]n_{1} = 57[/tex], then we compute [tex]e_{max} = 196[/tex], [tex]k = 7[/tex], and [tex]t = 10[/tex].

Therefore, [tex]r = r(57) = 1/.0841394 = 11.8850384[/tex].

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

Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 3:49 pm

Dave's Update wrote:The Collatz equation that supports the truth of the Collatz conjecture:

[tex]n_{t } = r * \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}} = 1[/tex]

where [tex]r = r(n_{1 })[/tex] is a positive real number and where t is the number of trials it takes the Collatz sequence of odd positive integers to converge to one.

Remark: [tex]n_{1 }[/tex] is any (initial) positive odd integer greater than one.

Remark: [tex]k = \lfloor \frac{log (e_{max })}{log(2)} \rfloor[/tex] where [tex]e_{max }[/tex] is the maximum positive even integer in the Collatz sequence.

Remark: We assume the algorithm for the Collatz conjecture.


Example: If we let [tex]n_{1} = 57[/tex], then we compute [tex]e_{max} = 196[/tex], [tex]k = 7[/tex], and [tex]t = 10[/tex].

Therefore, [tex]r = r(57) = 1/.0841394 = 11.8850384[/tex].

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

Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 6:18 pm

Example: If we let [tex]n_{1 } = 85[/tex], then compute [tex]e_{max } = 256[/tex], [tex]k = 8[/tex], and [tex]t = 1[/tex].

Therefore, [tex]r = r(85) = 1/.767285 = 1.303296689[/tex].

Relevant Reference Link:

https://www.wolframalpha.com/input/?i=prod+%283.%2F2%5Ei%29%5E%281%2F2%5Ei%29+i%3D1+to+8.
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 6:31 pm

Is [tex]r = r(n_{1 }) =[/tex] O[tex](t)[/tex] or [tex]r = ct[/tex] for some real number, [tex]c > 1[/tex]?
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 6:40 pm

Example: If we let [tex]n_{1 } = 1,398,101[/tex], then compute [tex]e_{max } = 4,194,304[/tex], [tex]k = 22[/tex], and [tex]t = 1[/tex].

Therefore, [tex]r = r(1,398,101) = 1/.750003 = 1.333328[/tex].

Relevant Reference Link:

https://www.wolframalpha.com/input/?i=prod+%283.%2F2%5Ei%29%5E%281%2F2%5Ei%29+i%3D1+to+22.[/quote]
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 6:51 pm

Dave wrote:Is [tex]r = r(n_{1 }) =[/tex] O[tex](t)[/tex] or [tex]r = ct[/tex] for some real number, [tex]c > 1[/tex]?


Dave's Conjecture: [tex]r = r(n_{1 }) =[/tex] O[tex](t)[/tex] or [tex]r = ct[/tex] for some real number, [tex]c > 1[/tex].

Dave.
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 7:19 pm

We should let [tex]n_{0}[/tex] be our initial positive odd integer greater one...

Dave.
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 7:52 pm

Dave wrote:
Dave's Conjecture: [tex]r = r(n_{1 }) =[/tex] O[tex](t)[/tex] or [tex]r = ct[/tex] for some real number, [tex]c > 1[/tex].



Hmm. We may want to consider [tex]c = c(k) > 1[/tex]...
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 8:46 pm

Dave's Conjecture: [tex]r = r(n_{0 }, t) =[/tex] O[tex](t)[/tex] or [tex]r = c_{t } * t[/tex] for some real number, [tex]c_{t } > 1[/tex]. (Update)

Example: If we have [tex]t = 1[/tex] for some [tex]n_{0 }[/tex], then as [tex]k \rightarrow \infty[/tex],

[tex]r = \frac{1}{ \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}}} \rightarrow \frac{4}{3 } = c_{1 }[/tex].

Therefore, [tex]r = r(n_{0 }, 1) \le \frac{4}{3}[/tex].
Guest
 

Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 9:15 pm

What are the Collatz constants, [tex]c_{t }[/tex], for [tex]t >1[/tex]?

Dave.
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 10:52 pm

Dave wrote:
Dave's Conjecture: [tex]r = r(n_{0 }, t) =[/tex] O[tex](t)[/tex] or [tex]r = c_{t } * t[/tex] for some real number, [tex]c_{t } > 1[/tex]. (Update)

Example: If we have [tex]t = 1[/tex] for some [tex]n_{0 }[/tex], then as [tex]k \rightarrow \infty[/tex],

[tex]r = \frac{1}{ \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}}} \rightarrow \frac{4}{3 } = c_{1 }[/tex].

Therefore, [tex]r = r(n_{0 }, 1) \le \frac{4}{3}[/tex].


[tex]r = r(n_{0 }, 1) \approx \frac{4}{3}[/tex] for infinitely many positive odd integers, [tex]n_{0} > 1[/tex].
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 11:29 pm

Remark: The Collatz conjecture is a deep and interesting math conjecture. :)
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Thu Oct 15, 2020 11:31 pm

Remark: The Collatz conjecture is true! :D
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Fri Oct 16, 2020 6:28 pm

Guest wrote:What are the Collatz constants, [tex]c_{t }[/tex], for [tex]t >1[/tex]?

Dave.


Observation: We compute the Collatz constant, [tex]c_{2} = \frac{8}{9}[/tex], but we have not identified any [tex]n_{0 }[/tex] for that constant with t = 2.

If we do identify [tex]n_{0 }[/tex], then our conjecture (Dave's Conjecture) must be revised.
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Re: The Collatz Equation that supports the Collatz Conjectur

Postby Guest » Fri Oct 16, 2020 6:41 pm

Guest wrote:
Guest wrote:What are the Collatz constants, [tex]c_{t }[/tex], for [tex]t >1[/tex]?

Dave.


Observation: We compute the Collatz constant, [tex]c_{2} = \frac{8}{9}[/tex], but we have not identified any [tex]n_{0 }[/tex] for that constant with t = 2.

If we do identify [tex]n_{0 }[/tex], then our conjecture (Dave's Conjecture) must be revised.


Tentatively, we have [tex]c_{10 } = 1.775773127[/tex] since [tex]n_{0 } = 57[/tex]. However, we need much more data, [tex]n_{0 }[/tex], to confirm that result.
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