# Difference between revisions of "Euclid's proof of the infinitude of primes"

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− | '''Euclid's proof of the infinitude of primes''' is a [[proof]] by [[Euclid]] that | + | '''Euclid's proof of the infinitude of primes''' is a classic and well-known [[proof]] by the Greek mathematician [[Euclid]] that there are infinitely many [[prime number]]s. |

==Proof== | ==Proof== | ||

− | We proceed by [[proof by contradiction | contradiction]]. Suppose there are only | + | We proceed by [[proof by contradiction | contradiction]]. Suppose there are in fact only finitely many prime numbers, <math>p_1, p_2, p_3, \ldots, p_n</math>. Let <math>N = p_1 \cdot p_2 \cdot p_3 \cdots p_n + 1</math>. Since <math>N</math> leaves a remainder of 1 when divided by any of our prime numbers <math>p_k</math>, it is not divisible by any of them. However, the [[Fundamental Theorem of Arithmetic]] states that all positive integers have a unique prime factorization. Therefore, <math>N</math> must have a prime factor (possibly itself) that is ''not'' among our set of primes, <math>\{p_1, p_2, p_3, \ldots, p_n\}</math>. This means that <math>\{p_1, p_2, p_3, \ldots, p_n \}</math> does not contain all prime numbers, which contradicts our original assumption. Therefore, there must be infinitely many primes. |

==See Also== | ==See Also== | ||

− | *[[Number theory]] | + | * [[Number theory]] |

− | *[[Euclid]] | + | * [[Prime]] |

+ | * [[Euclid]] | ||

[[Category:Proofs]] | [[Category:Proofs]] | ||

[[Category:Number theory]] | [[Category:Number theory]] |

## Latest revision as of 20:47, 18 March 2008

**Euclid's proof of the infinitude of primes** is a classic and well-known proof by the Greek mathematician Euclid that there are infinitely many prime numbers.

## Proof

We proceed by contradiction. Suppose there are in fact only finitely many prime numbers, . Let . Since leaves a remainder of 1 when divided by any of our prime numbers , it is not divisible by any of them. However, the Fundamental Theorem of Arithmetic states that all positive integers have a unique prime factorization. Therefore, must have a prime factor (possibly itself) that is *not* among our set of primes, . This means that does not contain all prime numbers, which contradicts our original assumption. Therefore, there must be infinitely many primes.