# The Collatz Equation that supports the Collatz Conjecture

### The Collatz Equation that supports the Collatz Conjecture

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

$$r * \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}} = 1$$

where $$r = r(n_{1 })$$ 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: $$n_{1 }$$ is any (initial) positive odd integer greater than one.

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

Remark: We assume the algorithm for the Collatz conjecture.

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

Therefore, $$r = r(57) = 1/.0841394 = 11.8850384$$.

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

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

$$n_{t } = r * \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}} = 1$$

where $$r = r(n_{1 })$$ 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: $$n_{1 }$$ is any (initial) positive odd integer greater than one.

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

Remark: We assume the algorithm for the Collatz conjecture.

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

Therefore, $$r = r(57) = 1/.0841394 = 11.8850384$$.

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

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

Therefore, $$r = r(85) = 1/.767285 = 1.303296689$$.

https://www.wolframalpha.com/input/?i=prod+%283.%2F2%5Ei%29%5E%281%2F2%5Ei%29+i%3D1+to+8.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

Is $$r = r(n_{1 }) =$$ O$$(t)$$ or $$r = ct$$ for some real number, $$c > 1$$?
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

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

Therefore, $$r = r(1,398,101) = 1/.750003 = 1.333328$$.

https://www.wolframalpha.com/input/?i=prod+%283.%2F2%5Ei%29%5E%281%2F2%5Ei%29+i%3D1+to+22.[/quote]
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

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

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

Dave.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

We should let $$n_{0}$$ be our initial positive odd integer greater one...

Dave.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

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

Hmm. We may want to consider $$c = c(k) > 1$$...
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

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

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

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

Therefore, $$r = r(n_{0 }, 1) \le \frac{4}{3}$$.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

What are the Collatz constants, $$c_{t }$$, for $$t >1$$?

Dave.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

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

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

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

Therefore, $$r = r(n_{0 }, 1) \le \frac{4}{3}$$.

$$r = r(n_{0 }, 1) \approx \frac{4}{3}$$ for infinitely many positive odd integers, $$n_{0} > 1$$.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

Remark: The Collatz conjecture is a deep and interesting math conjecture.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

Remark: The Collatz conjecture is true!
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

Guest wrote:What are the Collatz constants, $$c_{t }$$, for $$t >1$$?

Dave.

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

If we do identify $$n_{0 }$$, then our conjecture (Dave's Conjecture) must be revised.
Guest

### Re: The Collatz Equation that supports the Collatz Conjectur

Guest wrote:
Guest wrote:What are the Collatz constants, $$c_{t }$$, for $$t >1$$?

Dave.

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

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

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