# Infinite number of primes :)

In today’s post , I will present a very well known proof. What makes this proof especially appealing is that it is not too complex. Even so, it is very **powerful.** This theorem was first presented inÂ Euclid’sÂ Elements (Book IX, Proposition 20).

Theorem: **There are an infinite number of primes.**

Proof:

(1) Assume that there is only a finite number of primes.

(2) Then, there exists a primeÂ p_{n}Â that is the largest prime.

(3) LetÂ p_{1}, p_{2}, …, p_{n}Â be the list of all primes that exist.

(4) LetÂ x = p_{1}*p_{2}*…*p_{n}Â + 1.

(5) By the fundamental theorem of arithmetic (see Theorem 3,Â here), we know there is at least one prime that dividesÂ x. Let us call this primeÂ p*.

(6) But none of the primesÂ p_{1}Â … p_{n}Â divideÂ xÂ sinceÂ x â‰¡ 1 (mod p_{i})Â for any of the primesÂ p_{1}Â … p_{n}

(7) Therefore, we have a contradiction. We have a primeÂ p*Â that is not in the complete list of primes.

(8) So, we reject our assumption in step#1 and conclude that there are an infinite number of primes that exist.

QED

Posted on Tuesday,July 3, 2012, in Uncategorized and tagged Euclid, euclid elements, infinite primes, number theory, pr, primes. Bookmark the permalink. 28 Comments.

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