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'Euclid's proof of the infinitude of primes'
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| Title of object: |
Euclid's proof of the infinitude of primes |
| Canonical Name: |
ProofThatThereAreInfinitelyManyPrimes |
| Type: |
Proof |
| Created on: |
2002-06-04 15:02:01 |
| Modified on: |
2004-03-23 05:00:43 |
| Classification: |
msc:11A41 |
Revision comment (for changes between this and next version):
| Changes for correction #6856 ('slight grammar error'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
| If there was only a finite amount of primes then there would be some largest prime $p$. However $p!+1$ is not divisible by any number $1<n\leq p$, since $p!$ is, so $p!+1$ cannot be factored by the primes we already know, but every integer greater than one is divisible by at least one prime, so there must be some prime greater than $p$ by which $p!+1$ is divisible. |
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