The problem shows a neat connection between number theory and geometry -> what is the average number of ways to write an integer as the sum of (exactly) two squares?

We’ve looked this problem previously, but it was so long ago that I’m pretty sure that they boys didn’t remember it:

I started by introducing the problem and then having the boys check the number of ways to write some small integers as the sum of two squares:

In the last video we found that the number 3 couldn’t be written as the sum of two squares. I asked the boys to find some others and they found 11 and 6. My older son then conjectured that numbers of the form couldn’t be written as the sum of two squares. We explored that conjecture.

My son’s conjecture was such an interesting idea that I decided to take a little detour and explore squares mod 4.

Slightly unluckily we were time constrained this morning, so the diversion in the last part left me with a tough choice about how to proceed. I decided to show them a sketch of Gauss’s proof fairly quickly. Don’t know if that was the right decision, but they did find the ideas and result to be amazing!

Even though I had to rush at the end, I’m really happy with how this project went. It is fun to see kids making number theory conjectures! It is also really fun to see gets really excited about amazing results in math!

If you’d like to see another fun (and similar) connection between number theory and geometry, Grant Sanderson did an amazing video about pi and primes:

Here are our the two projects that we did based on that video.