Numberphile released a really nice video about the Goldbach Conjecture today:
I thought it would make an excellent project with the boys even though some of the ideas involving logarithms might be over their head. So, we watched the movie and then talked about some of the ideas that caught their eye.
Next we moved on to the individual ideas. The first one was the chart that David Eisenbud made at the beginning of the video. Drawing and then filling in this chart is a nice little arithmetic activity for a kid in elementary school.
Next we talked about logarithms. I started with an idea I learned from Jordan Ellenberg’s book “How Not to be Wrong” – the “flogarithm”. That idea is to oversimplify the logarithm by defining it to be the number of digits in the number. That simple (and genius) idea really opens the door to kids thinking about logarithms.
With that short introduction I explained what the natural logarithm was and moved on to some of the properties of primes that Eisenbud mentioned in the video (after fumbling with the calculator on my phone for a minute . . . .).
(Also, I noticed watching the video just now that I forgot to divide by 2 at one point – sorry about that.)
Finally, we checked a specific example – how many ways were there to take two primes and add up to 50? This part is about as far away from the complexity of logarithms as you can get – just some nice arithmetic practice for kids.
To warp up I asked them if they knew any other unsolved problems about primes. My older son mentioned something about twin primes. I showed the boys a simple argument (fortunately quite similar to the one Eisenbud gave in the movie for why there are lots of ways two primes can add to be a given even number) for why there ought to be infinitely many twin primes.
I think that kids are going to be naturally curious about primes. The Goldbach conjecture is one of the few unsolved problems that kids can understand. It was fun to share this video with the boys tonight.