A neat idea about primes I saw from Dave Richeson

Saw a neat tweet from Dave Richeson earlier this week:

His post is about a “new to me” proof that there are an infinite number of primes. Same warning as in Dave’s blog post – don’t click the link unless you want to know the punchline!

“Prime Number Generator” on Futility Closet’s website

Here’s the problem without giving away the punchline:

Take the set consisting of the first n prime numbers – { 2, 3, 5, \ldots, p_n }. Divide the set into two (non-intersecting) sets, and let A and B be the product of the numbers in each of those two sets. What can you say about A + B, and A – B?

I thought it would be fun to talk through the idea with the boys today – it really did seem like a fun way to talk through some introductory ideas about mathematical proofs.

By coincidence, about a year ago we also talked about a “new to me” proof that there are an infinite number of primes. That project was also inspired by a twitter post (!) and is here:

“A “new to me” proof that there are infinitely many primes

I began today’s project by first asking the kids to talk a little about prime numbers and then I introduced the idea from Futility Closet’s post.

The first thing they noticed was that exactly one of A and B would be even, and both A + B and A – B would be odd.

 

Next we looked at the example where the original set of primes was {2, 3, 5, 7}. Playing around with this set let to a few more conjectures – some of which we were able to show were not true, but (despite a little clumsiness by me) one of the ideas about finding new primes looked interesting.

 

For the next part of the project we looked more carefully at why we seemed to be getting new primes when we looked at the two numbers A + B and A – B. There are a couple of neat to talk through here. The one we focus on here is about divisibility or modular arithmetic. This idea helps us understand why we are seeing new primes:

 

A second idea that we need to understand is more subtle. This idea is about the difference between “an integer is prime” and “an integer is not divisible by any of the first n primes”. The kids began to understand this idea and then I got a big surprise – working together the boys came to see that the ideas here would lead us to conclude that there were infinitely many primes!

So, when that idea appeared on the table I ran with it 🙂

 

So, a great introductory proof project, and a fun problem to talk through with kids in general. There are lots of math ideas that are so close to the ideas here, and so many different directions that the talk could go that I think you’d find new ideas with every group of kids working on this problem.

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