Associated Topics || Dr. Math Home || Search Dr. Math

### Are Prime Numbers Infinite?

```
Date: 06/18/98 at 23:22:24
From: Vincent
Subject: Prime numbers

How do we know the number of primes is infinite?

Thank you!
```

```
Date: 06/19/98 at 07:56:33
From: Doctor Allan
Subject: Re: Prime numbers

Hi Vincent,

The proof depends on the fact that every natural number has a unique
factorisation into primes. I assume that you are aware of this fact.

Now assume that there is only a finite number of primes and let's call
these p_1, p_2, ... , p_m.

Look at the number (p_1)*(p_2)*...*(p_m) + 1

Because of the unique factorisation, this number can be factored into
our primes p_1, ... , p_m or it is prime itself.

You should now convince yourself that neither of these options is
possible. Therefore our assumption that there is only a finite number
of primes is incorrect, proving that there is indeed an infinite
number of primes.

Sincerely,

-Doctor Allan,  The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search