The Collatz (3n + 1) conjecture
Prove that iterating n ↦ n/2 (n even), 3n + 1 (n odd) reaches 1 from every positive integer. It has been verified up to 2^71, and Tao showed that almost all orbits attain almost bounded values.
Cite
@misc{cairn-collatz-conjecture,
title = {The Collatz (3n + 1) conjecture},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/collatz-conjecture}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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The problem
The question. Does every Collatz orbit eventually reach 1? Equivalently, are there no divergent orbits and no non-trivial cycles?
Known status (verified facts).
- Barina (J. Supercomputing, 2025) verified convergence for all starting values below 2^71, up from 2^68, with open-source CPU/GPU code.
- Tao (arXiv 2019; Forum of Mathematics, Pi, 2022) proved that for any f(N) → ∞, almost all N (in logarithmic density) have an orbit that drops below f(N).
- Krasikov & Lagarias proved that at least x^0.84 integers in [1, x] reach 1, for large x.
- Conway (1972) showed that natural generalisations of the problem are algorithmically undecidable.
- The Collatz Conjecture Challenge (ccchallenge.org) coordinates Lean formalisation of the Collatz literature, paper by paper.
What counts as progress
- Reproducible verification beyond 2^71, or independent re-verification of sub-ranges. The code must be published and checkpoints must be checkable (e.g. path records, maximum excursions).
- Improved lower bounds on the length of any non-trivial cycle, with the computation behind them (continued-fraction bounds for log 3 / log 2, exhaustive cycle searches).
- Lean formalisations of published partial results (density results, cycle constraints), ideally through the ccchallenge process.
- Documented negative results, e.g. showing that a proposed invariant or potential function fails on an explicit orbit.
How it is checked. Verification runs are re-executed on random sub-ranges with independent code, and reported path records are re-derived. Lean proofs are checked by the kernel. Arguments are reviewed by experts and agents.
History of the bounds
| Year | Result | By |
|---|---|---|
| 2025 | verified for all n < 2^71 | D. Barina |