{"id":"prime-number","domain":"number-theory","type":"Concept","title":"Prime Number","slug":"prime-number","url":"/mathematics/number-theory/prime-number/","summary":"A natural number > 1 with no divisors other than 1 and itself. The atoms of arithmetic — every integer factors uniquely into primes (FTA). There are infinitely many (Euclid, c. 300 BCE). Distribution governed by Prime Number Theorem; fine structure linked to Riemann Hypothesis (unsolved).","formal_definition":"p is prime if p > 1 and the only positive divisors of p are 1 and p.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Hardy, G.H. and Wright, E.M. (2008). An Introduction to the Theory of Numbers. 6th ed. Oxford University Press.","chapters":"1–3"},{"tier":1,"citation":"Davenport, H. (2000). Multiplicative Number Theory. 3rd ed. Springer (GTM 74)."},{"tier":2,"citation":"Crandall, R. and Pomerance, C. (2005). Prime Numbers: A Computational Perspective. 2nd ed. Springer."},{"tier":3,"citation":"Derbyshire, J. (2003). Prime Obsession. Joseph Henry Press."}],"relationships":[{"type":"proved_infinite_by","target":"euclids-elements","label":"Euclid IX.20"},{"type":"related","target":"natural-number","label":"Natural Number"},{"type":"open_problem","target":"riemann-hypothesis","label":"Riemann Hypothesis — controls distribution"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.2"}