Number theory Divisibility and primes Theorem

Euclid's theorem

Builds on Prime number Existence of a prime divisor

Statement

There are infinitely many prime numbers. Equivalently, for each natural number $n$ there is a prime number $p$ with $p \ge n$ .

Remarks

Euclid's proof: let $p_1, \dots, p_k$ be a finite list of primes, and let $N = p_1 p_2 \cdots p_k + 1$ . A prime divisor of $N$ is not in the list, because each $p_i$ leaves the remainder $1$ when it divides $N$ . The divergence of the sum of the reciprocals of the primes is a stronger result. Dirichlet's theorem on arithmetic progressions gives infinitely many primes in each suitable arithmetic progression.

History

The theorem is Proposition 20 of Book IX of the Elements of Euclid.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility definition
Prime number definition
Existence of a prime divisor statement
Euclid's theorem statement

Where this leads

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Euclid's Elements

Tag	`0292EPPPT0`
URI	`m2://0292EPPPT0`
Aliases	`infinitude-of-primes`
Kind	statement, theorem
Writing	W1 stated
MSC	11A41
Status	This copy of the site has no build data. Run `m2 status 0292EPPPT0`.
Uses	Prime number, Existence of a prime divisor
Same as	Wikidata Q1506253 Wikipedia Euclid's theorem

Euclid's theorem

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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