Working through a neat problem from Martin Weissman’s An Illustrated Theory of Numbers

I just got back from a wo7rk trip to Sydney and I’m going to blame jet lag for goofing up the videos. Because I forgot to zoom out after zooming in during the first video, this is really more of an audio project than a video one!

Today we returned to Martin Weissman’s An Illustrated Theory of Numbers. Flipping through the chapter on prime numbers (which is incredible!) I ran across a problem dealing with the set of numbers {1, 4, 7, 10, 13, \ldots } and thought it would be a great one to talk through with the boys.

It was really fun as you will see hear . . .

I stared by introducing the problem and also making it impossible to see what we were doing:

Next we started playing with the first part of the problem. What we talk through here is this idea from number theory: If two numbers A and B are in our set, and A = B*C, then C is also in the set.

The boys looked at a few examples initially and noticed that lots of numbers in the set didn’t factor in the set. Then they noticed that the problem was really a problem about modular arithmetic.

The next part of the problem we played with was going through an exercise similar to the “Sieve of Eratosthenes” procedure to find the “primes” in our set:

Finally, we took at look at the part of the problem that caught my attention -> find elements of our set that factor into irreducible elements in non-unique ways.

My older son found one example -> 100 = 10*10 = 25*4.

The property of our set shows that the integers factoring into primes in a unique way is actually a pretty special property.

Sorry for the filming screw up – fortunately the visuals for this project were quite a bit less important than average. I’m excited to play around in the project chapter this week – I really love this book!