{"id":"hilberts-23-problems","domain":"history-philosophy","type":"Concept","title":"Hilbert's 23 Problems","slug":"hilberts-23-problems","url":"/mathematics/history-philosophy/hilberts-23-problems/","presented":"8 August 1900, International Congress of Mathematicians, Paris","presenter":"David Hilbert","summary":"23 problems shaping 20th-century mathematics. Roughly half resolved; several remain open including Problem 8 (Riemann Hypothesis).","selected_problems":[{"number":1,"topic":"Continuum Hypothesis","status":"Independent of ZFC (Gödel 1940, Cohen 1963)"},{"number":8,"topic":"Riemann Hypothesis","status":"Unsolved"},{"number":10,"topic":"Diophantine equation algorithm","status":"Proven impossible (Matiyasevich 1970)"},{"number":18,"topic":"Kepler Conjecture (sphere packing)","status":"Solved (Hales 1998)"}],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society, 8(10), 437-479."},{"tier":1,"citation":"Gray, J.J. (2000). The Hilbert Challenge. Oxford University Press."},{"tier":2,"citation":"Yandell, B.H. (2002). The Honors Class. A K Peters."},{"tier":3,"citation":"Derbyshire, J. (2003). Prime Obsession. Joseph Henry Press."}],"relationships":[{"type":"problem_8_is","target":"riemann-hypothesis","label":"Riemann Hypothesis"},{"type":"problem_2_connects_to","target":"godels-incompleteness-theorems","label":"Gödel's Incompleteness Theorems"},{"type":"modern_successor","target":null,"label":"Clay Millennium Prize Problems (2000)"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.6"}