Consider a Collatz trajectory of some number n, such as:
nx0R0(n)x1R(n0)x2R(n1)x3…xmR(nm−1)
The Exponential Canonical Form (ECF) is a canonical representation of this trajectory, without necessarily depending on n. ECF is defined as a list, with elements ordered in ascending order:
{x0,x0+x1,…,x0+x1+…+xm}
Notice that elements are in accumulated sums as we iterate the list. Alternatively, we can write the trajectory as:
The ECF of this trajectory is written a lot cleaner this way:
{p0,p1,p2…,pm}
Prefix
For example, the sequences of 3 and 11 are as follows:
Bijective Mapping
ECF of a reduced trajectory 3031541 looks like {0,1,5}.
Consider ECFs of two trajectories, {p0,p1,…,pm} and {q0,q1,…,qk}. The prefix of these trajectories (or ECFs) is defined as:
{ρ0,ρ1,…,ρt}
where ∀i:0≤i≤t we have pi=qi=ρi for largest t. In simpler terms, if we think of ECFs as strings, the prefix is the common prefix in both of these strings. Prefix can be an empty list, in which case we say that there is no prefix. Prefix itself is also an ECF!
3031541 with ECF {0,1,5}.
110111172133541 with ECF {0,1,3,6,10}.
Their prefix is thus {0,1}.
Any ECF {p0,p1,…,pm} can be represented by a unique positive integer. They are defined as follows:
2p0+2p1+…+2pm
The sequence of 3 was shown by ECF {0,1,5}, which can be represented by 35=20+21+25.