> For the complete documentation index, see [llms.txt](https://erhany96.gitbook.io/collatz-prefixes/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://erhany96.gitbook.io/collatz-prefixes/theory/operation-table/testing-an-ecf.md).

# Testing an ECF

A simple ECF test using the table

We have seen a relationship of numbers in [different rows within a column](/collatz-prefixes/theory/operation-table/navigating.md#row-skipping) 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 $$n$$. 

{% hint style="danger" %}
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.**
{% endhint %}