{"id":"twin-prime-conjecture","domain":"number-theory","type":"Conjecture","title":"Twin Prime Conjecture","slug":"twin-prime-conjecture","url":"/mathematics/number-theory/twin-prime-conjecture/","posed":"1849, Alphonse de Polignac (general form)","status":"open","summary":"Infinitely many primes p where p+2 is also prime. Zhang 2013 proved bounded gaps exist (70M); refined to 246 (Maynard/Tao/Polymath8). Gap-of-2 remains open.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics, 179(3), 1121-1174."},{"tier":1,"citation":"Maynard, J. (2015). Small gaps between primes. Annals of Mathematics, 181(1), 383-413."},{"tier":2,"citation":"Goldston, D.A., Pintz, J. and Yıldırım, C.Y. (2009). Primes in tuples I. Annals of Mathematics, 170(2), 819-862."},{"tier":3,"citation":"Kolata, G. (2013). Solving a Riddle of Primes. The New York Times."}],"relationships":[{"type":"built_on","target":"prime-number","label":"Prime Number"},{"type":"related_to","target":"goldbach-conjecture","label":"Goldbach Conjecture"},{"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"}