Testing an ECF

A simple ECF test using the table

We have seen a relationship of numbers in different rows within a column in the previous section. This actually provides a rather handy method to test whether some ECF may be valid for some trajectory that starts with a given number.

Let us re-write the formula from before:

$$\beta(n + k, \rho) = \beta(n, \rho) +k\frac{3^{m}}{2^{p_m}}$$

Now, let us reinterpret this by rewriting $n=1$ and $k=n-1$ as follows:

$$\beta(n, \rho)=\beta(1, \rho)+(n-1)\frac{3^m}{2^{p_m}}$$

Recall that if ECF belongs to the Collatz sequence of a number, then the terminating number would be 1. So, we could assume $\beta(n, \rho)=1$ and then rewrite the equation above:

$$1=\beta(1, \rho)+(n-1)\frac{3^m}{2^{p_m}}$$

$$n=\frac{2^{p_m}}{3^m}\left(1-\beta(1,\rho)\right)+1$$

If this equation holds, then $\rho$ is indeed the ECF of the Collatz sequence for $n$.

This method is does not make sense until we find an efficient way to calculate $\beta(1,\rho)$. This part of the research remains open.