# Collatz Conjecture

**Type:** Conjecture  
**Domain:** Named Results · Number Theory  
**Proposed:** c. 1937, Lothar Collatz  
**Status:** Unsolved  
**Codex URL:** /mathematics/named-results/collatz-conjecture/  
**Entry status:** Live — v1.0 (2026-05-27)

## Statement
For any positive integer, repeatedly applying n→n/2 (if even) or n→3n+1 (if odd) always eventually reaches 1.

## Summary
Verified computationally to ~2⁶⁸; unproven in general. Tao (2019/2022) proved "almost all" orbits attain almost bounded values — significant partial progress, not a full proof.

## Also known as
3n+1 problem, Ulam conjecture, Kakutani problem, Thwaites conjecture, Syracuse problem — independently rediscovered by many mathematicians.

## Erdős's comment
"Mathematics is not yet ready for such problems."

## Tao's 2019/2022 result
Proved almost all Collatz orbits (logarithmic density sense) attain almost-bounded values — celebrated partial progress, explicitly not extendable to a full proof by Tao's own assessment.

## Undecidability speculation
Conway (1972) proved a broader class of "Collatz-like" functions contains provably undecidable problems — lends indirect plausibility to speculation about Collatz itself, without resolving it.

## Sources
### Tier 1
- Lagarias, J.C. (ed.) (2010). *The Ultimate Challenge: The 3x+1 Problem*. AMS.
- Tao, T. (2022). *Forum of Mathematics, Pi*, 10, e12.
### Tier 2
- Conway, J.H. (1972). Unpredictable iterations. Proc. 1972 Number Theory Conf.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*