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/.