# Revisiting Jacob Lurie’s Breakthrough Prize lecture

Last night I asked my older son what he what topics were being covered in his math class at school.  He said that they were talking about different kinds of numbers -> natural numbers, integers, rational numbers, and irrational numbers.   I asked him if he thought it was important to learn about the different kinds of numbers and he said that he thought it was but didn’t know why.

I decided share Jacob Lurie’s Breakthrough Prize lecture with the boys this morning since he touches on the study of different kinds of number systems.  The first 12 or so minutes of the lecture are accessible to kids:

Near the beginning of Lurie’s talk he mentions that the equation $x^2 + x + 1 = y^3 - y$ has no integer soltuions. I stopped the video here to what the boys thought about this problem. It took two about 10 minutes for the boys to think through the problem, but eventually they got there. It was fun to watch them think through the problem.

Here’s part 1 of that discussion:

and part 2:

The next problem that we discussed from the video was Lurie’s reference that all primes of the form $4n + 1$ can be written as the sum of two squares. I checked that the boys understood the problem and then switched to a problem that would be easier for them to tackle -> No prime of the form $4n + 3$ can be written as the sum of two squres.

Finally, to finish up, we began by discussing Lurie’s question about whether or not numbers were real things or things that were made up by mathematicians. Then we wrapped up by looking at why 13 is not prime when you expand the integers to include complex numbers of the form $A + Bi$ where $A$ and $B$ are integers.

There aren’t many accessible public lectures from mathematicians out there. I’m happy that part of Lurie’s lecture is accessible to kids. It is nice to be able to use this lecture to help the boys understand a bit of history and a bit of why these different number systems are interesting to mathematicians.