{"id":"p-vs-np","domain":"named-results","type":"Open Problem","title":"P vs NP","slug":"p-vs-np","url":"/mathematics/named-results/p-vs-np/","posed":"1971, Stephen Cook (independently Leonid Levin)","status":"open","summary":"Can every problem whose solution is quickly verifiable also be quickly solved? Clay Millennium Prize. Most experts believe P!=NP but it remains unproven.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Cook, S.A. (1971). The complexity of theorem-proving procedures. STOC 1971, 151-158."},{"tier":1,"citation":"Karp, R.M. (1972). Reducibility among combinatorial problems. Complexity of Computer Computations, 85-103."},{"tier":2,"citation":"Fortnow, L. (2009). The status of the P versus NP problem. Communications of the ACM, 52(9), 78-86."},{"tier":3,"citation":"Fortnow, L. (2013). The Golden Ticket. Princeton University Press."}],"relationships":[{"type":"defined_via","target":"big-o-notation","label":"Big-O Notation"},{"type":"related_to","target":"graph-theory","label":"Graph Theory"},{"type":"tracked_in","target":"proof-status","label":"Proof Status Board"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.3"}