I have thought of a single simple rule for Colatz conjecture that is slightly different to the normal one. Here it is:
$$f:(a,b)\rightarrow (3a+\gcd(a,b),b)$$
Then the Colatz conjecture can be written as:
$$\forall_{n:\mathbb{N}}\exists_{m:\mathbb{N}}\exists_{p:\mathbb{N}}\{ f^p(n,2^m) = (2^m,2^m)\}$$
Examples:
the orbit $5\rightarrow 16\rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$ is written: $$(5,16) \rightarrow (16,16)$$ The orbit $17,52,26,13,20,10,5,16,8,4,2,1$ becomes: $$(17,512)\rightarrow (52,512) \rightarrow (160,512) \rightarrow (512,512)$$
Is this known? And is this an improvement or just makes things worse?