\rightarrow\gamma

Some Food for Thought:

Hmm. We have extracted our important equation ("

A Brief Analysis of the Collatz Conjecture"):

[tex]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}}}[/tex] with [tex]r_{t } > r[/tex] where k = Floor( [tex]\frac{log(e_{max })}{log(2)}[/tex])

and where [tex]e_{max }[/tex] 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 {[tex]2^{0}, 2^{1}, 2^{2}, ..., 2^{k}[/tex]}.

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

So, the probability that the maximum divisor, [tex]2^{0}[/tex],

does not occur is

roughly [tex](\frac{k}{k+1})^{\gamma}[/tex] for any closed cycle in a

Collatz sequence of positive even integers.

Remark: "Roughly" indicates a crude approximation.

But as [tex]\gamma \rightarrow \infty[/tex], [tex](\frac{k}{k+1})^{\gamma} \rightarrow 0[/tex].

Therefore, [tex]n_{t } = 1[/tex]. And we conclude the Collatz conjecture is true!

Remark: Our "food for thought" (analysis) may be over the 'top'... It's Ok!