Theorems
Infinitude of Primes
Also Known As Euclid's Theorem
Number Theory
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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. StatementThere are infinitely many prime numbers. 1 Cross-Tradition Connections
Associated With
The atlas's own theorem proving there are infinitely many primes.
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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 FoundationProofQuote, Proof
Euclid offered a proof in his work Elements (Book IX, Proposition 20).
Euclid's Theorem (Wikipedia)
Wikimedia FoundationProved By: Euclid, Opening sectionQuote, Proved By: Euclid, Opening section
It was first proven by Euclid in his work Elements.
Euclid's Theorem (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Opening sectionQuote, 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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