Euclidean prime number theorem

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The set of prime numbers is infinite (Euclid's Elements, Book IX, Prop. 20). The Chebyshev theorems on prime numbers and the asymptotic law of the distribution of prime numbers provide more precise information on the set of prime numbers in the series of natural numbers.


The proof of the Euclidean theorem is simple. Suppose there exist only finitely many prime numbers $p_1,\ldots,p_k$. Consider the number $N=p_1\ldots p_k+1$. Since $N>1$ it must be divisible by a prime number $p$, which equals some $p_i$ due to the finiteness of the amount of prime numbers. Hence $p=p_i$ divides $N=p_1\ldots p_i\ldots p_k+1$, and thus $p_i$ divides 1. This contradiction shows that there must be infinitely many prime numbers.

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Euclidean prime number theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.M. Voronin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article