A neat unsolved problem in number theory that kids can explore

Yesterday I saw a really neat thread on the Collatz conjecture from Alex Kontorovich

In that thread is a blog post by Alex’s friend Igor Park and Park’s blog post as a link to a neat set of lecture notes by Barry Mazur. AND, in Mazur’s notes is this “new to me” unsolved problem in number theory:


Instead of continuing on our journey through Mosteller’s 50 Challenging Problems in Probability, I decided to explore this problem with the boys today.

Here’s the introduction to the problem and a bit of playing around with a few of the small cases:

In the last video the boys thought that the squares would all have to be odd and the primes would have to be odd. Here we explored both of those conjectures. That exploration led to a discussion of why odd numbers always have squares that are congruent to 1 mod 8:

Now we continued the discussion from last video and investigated the primes that could appear in this problem. We started by showing that 2 could never appear and then eventually found that only primes of the form 4k + 1 could appear:

Next we moved to the computer to explore more cases of the conjecture. This was mainly an exercise into writing a simple program in Mathematica, but it led to an interesting discussion as well as an idea for further exploration:

Finally, we modified our program to explore the number of different solutions to the problem for each number. The modification to the program was actually really easy and the histogram was fascinating to see:

It is really fun to be able to explore an unsolved problem with kids. I especially love unsolved problems that allow kids to get in some secret arithmetic practice will getting a bit of exposure to some advanced ideas in math. Seeing this problem yesterday and getting to explore it today with the boys was a real treat!