## Elegant Math: Infinite Primes

3/24/2011

This is the second installment in my Elegant Math series.  Today we'll look at the proof that there are an infinite number of primes.

We discussed prime numbers in a previous post.  And we showed how they are the building blocks for all other counting numbers in Tuesday's proof.  So we understand how important prime numbers are, but now the big question:  How many prime numbers are there?

It seems like prime numbers go on forever.  We keep finding bigger ones.  But they also seem to spread out as they get higher.  This makes sense because the higher the number, the more prime numbers there are below it that could divide into it.  If I pick a number, and there are a hundred prime numbers smaller than it, what are the chances that none of them is a factor?

Wouldn't we eventually get to a point, if we looked at big enough numbers, that every number is composite because at least one lower number divides into it?

It turns out that this can't happen.  Prime numbers do go on forever.  But how do we know for sure?  Here's a proof.  Don't worry, it's mostly in plain English, and once again I've written my comments in italics so the math-phobic needn't be frightened.
3/24/2011 07:54:16 am

I love it! Now as soon as I figure out my equation for prime number generation I'll be all set.

Dustin S.
4/25/2017 06:15:10 pm

Thank you for posting a "plain English" version of the proof of infinite primes. I especially appreciate you getting into the detail of why we add 1 to our "assumed complete" list of primes. I was confused about this until I stumbled onto your website. I hope you write more!