{"id":"collatz-conjecture","domain":"named-results","also_in_domain":"number-theory","type":"Conjecture","title":"Collatz Conjecture","slug":"collatz-conjecture","url":"/mathematics/named-results/collatz-conjecture/","statement":"For any positive integer, repeatedly applying n->n/2 (even) or n->3n+1 (odd) always eventually reaches 1.","proposed":"c. 1937, Lothar Collatz","status":"unsolved","summary":"Verified computationally to ~2^68; unproven in general. Tao (2019) proved 'almost all' orbits attain almost bounded values — significant partial progress, not full proof.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Lagarias, J.C. (ed.) (2010). The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society."},{"tier":1,"citation":"Tao, T. (2022). Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi, 10, e12."},{"tier":2,"citation":"Conway, J.H. (1972). Unpredictable iterations. Proceedings of the 1972 Number Theory Conference, 49-52."},{"tier":3,"citation":"Lagarias, J.C. (1985). The 3x+1 problem and its generalizations. American Mathematical Monthly, 92(1), 3-23."}],"relationships":[{"type":"related_to","target":"godels-incompleteness-theorems","label":"Gödel's Incompleteness Theorems — undecidability speculation"},{"type":"related_to","target":"integer","label":"Integer"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}