Proposition 20 of Book IX of Euclid's Elements, and one of the oldest known proofs still taught essentially unchanged today. Euclid's argument is a proof by contradiction adjacent to what modern mathematics calls reductio ad absurdum: assume a finite complete list of primes exists, multiply them all together and add one, and the result must either itself be prime, or have a prime factor, and either way that prime cannot be on the original list, a contradiction. No largest prime can therefore exist.
Facts
Disputed
Proof YearDated to the compilation of Euclid's Elements; no more precise date for the individual proposition survives. Classification
Statement Form StatementThere are infinitely many prime numbers. 1 Connections
Associated With
The atlas's own theorem proving there are infinitely many primes.
Source Mathematics Atlas Long-Form Articles, First Edition
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source MacTutor History of Mathematics Archive
Additional Source Euclid's Theorem (Wikipedia)Opening section
Proved By
Source MacTutor History of Mathematics Archive
Additional Source Euclid's Theorem (Wikipedia)Opening section
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsHistory Topics: Prime numbersQuote, History Topics: Prime numbers
In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers.
Euclid's Theorem (Wikipedia)
Wikimedia FoundationProof
Euclid offered a proof in his work Elements (Book IX, Proposition 20).
Proved By: Euclid, Opening section
It was first proven by Euclid in his work Elements.
In Branch: Number Theory, Opening section
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.
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