Earlier in the week I saw a nice exchange on twitter between Michael Pershan and Justin Lanier. Sorry for the cumbersome tweet linking, but here’s the part of the exchange that caught my attention:
After seeing this exchange I wanted to come up with a project that (i) gave the boys a peek at some modern math, and (ii) showed a connection to something that they already understood. Prime numbers were on my mind since my younger son is working through Art of Problem Solving’s Introduction to Number Theory, and some amazing recent papers from a Terry Tao led team and James Maynard also have been in the back of my mind, too,. So the topic I chose was a walk through how mathematicians have come to understand prime numbers.
We began by looking at two results known in ancient Greece. The first result we look at is Euclid’s proof that the number of prime numbers is infinite. I have talked about this result with both of the kids previously and they are familiar with the sketch of the proof. A key detail that I take for granted is that every integer can be factored into prime numbers in a unique way, but you have to start somewhere in an overview.
The second idea from ancient Greece that we discuss is the construction of prime numbers via the Sieve of Eratosthenes. This topic is also familiar to both of the boys from their Introduction to Number Theory book.
So, 2500 years ago we know that there are an infinite number of prime numbers and we knew how to construct them. Even if the way of constructing them is not super efficient, these two facts together are a pretty good starting point for kids to learn more about primes.
Now fast forward to the 1800s. In this century mathematicians began to think about how to approximate the number of primes less than a given integer. The eventual result was the so-called Prime Number theorem proven by Jacques Hadamard and Charles Jean de la Vallee-Poussin in 1896. With this work mathematicians extended Euclid’s idea that the number of primes is infinite to be able to say roughly how the prime numbers are distributed in the integers.
To get our arms around the work here we need to have a basic understanding of the logarithm function. For simplicity I used the “flogarithm” idea from Jordan Ellenberg’s book How not to be Wrong. That idea is simply that the number of digits of a given number is a good enough approximation to the logarithm for purposes of thinking about these prime number theorems. I do also mention the difference between log base 10 and log base e, but this is not a point I wanted to dwell on.
Next we moved into the first half of the 1900s. The idea that mathematicians began to think about during this time was how to approximate number of prime numbers that divide into a given number. For purposes of these theorems you count repeated primes numbers multiple times (so 18 = 2*3*3 has three prime divisors). In the early 1900’s Hardy and Ramanujan found that the expected number of prime divisors of a given integer n was about log (log n). In the mid 1900’s Erdos and Kac improved this estimate with a beautiful theorem describing the distribution of the number of prime divisors of the integers. Instead of discussing the normal distribution used in the theorem, though, I used the binomial distribution from Pascal’s triangle as a picture because it was more familiar to the kids.
One piece of particularly fun math from this video is at the end when we talk about the incredible and bizarre connection between the prime numbers and random numbers. Thanks to Eratosthenes we have an explicit way to construct all of the prime numbers – so the prime numbers are not random at all. However, the beautiful description of the number of prime divisors of a given number from the Erdos-Kac theorem arises from ideas about random numbers. How can ideas from random numbers work so well to describe numbers that are not random at all? I think that the answer to this question remains a mystery.
Finally we jump to the developments in understanding prime numbers that have been made in the last two years. The first thing we discuss is Zhang’s paper about prime gaps and the subsequent work that was done around the world after the publication of his paper. Next we discuss an incredible coincidence from this year when a group led by Terry Tao proved another theorem about prime gaps one day ahead of the Canadian mathematician James Maynard announcing a different proof of the same theorem. Incredibly, this problem about prime numbers that had two different solutions posted one day apart had been unsolved for the previous 75 years!
Today’s Family Math was really fun to work through. While I was uploading the videos onto the computer my younger son came in to ask a question about logarithms (using the number of digits definition). He said that he thought that no matter how big a number was taking the log of that number, then the log of the result, and so on would always get you down to 1. Once you got to one, though, you’d be stuck since 1 has 1 digit. Nice to see him thinking about these ideas 🙂
So, thanks to Justin Lanier and Michael Person for the inspiration for this Family Math. It was definitely a fun morning.