by 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/.
[b]The Collatz equation that supports the truth of the Collatz conjecture[/b]:
[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 [b]t[/b] is the number of trials it takes the [i]Collatz sequence of odd positive integers[/i] to converge to one.
[b]Remark[/b]: [tex]n_{1 }[/tex] is any (initial) [i]positive odd integer greater than one[/i].
[b]Remark[/b]: [tex]k = \lfloor \frac{log (e_{max })}{log(2)} \rfloor[/tex] where [tex]e_{max }[/tex] is the[i] maximum positive even integer[/i] in the Collatz sequence.
[b]Remark[/b]: We assume the algorithm for the Collatz conjecture.
[b]Example[/b]: 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,
[url]https://theory-of-energy.org/2020/09/17/a-brief-analysis-of-the-collatz-conjecture/[/url].