Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:
Infinite Primes via Fibonacci numbers by John Cook
Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:
Dave Radcliffe’s Amazing Fibonacci GCD post
That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:
Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:
We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.
Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci function and a Prime function, so the computer exploration was fairly easy.
One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.
This project made me really happy 🙂 If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.