# Looking at Mazur and Stein’s new book about primes part 2

Yesterday we looked at a new book by Mazur and Stein I saw through a Steven Strogatz tweet:

That project is here:

Talking Prime Numbers with Kids

Last night I asked the kids what they wanted to do for today’s math project and got a fun surprise – they wanted to look at more sections from the book!

To start off today’s project I asked them why they wanted to learn more from this book and we picked three sections to study. One of the reasons, it turns out, was that my younger son was disappointed that the section on music and primes didn’t really talk about primes. He wanted to learn more about the word “spectrum” 🙂

The first thing we talked about was “Named Prime Numbers” – this section discussed Mersenne Primes and Fermat Primes. The book gives a problem that is probably a bit too challenging for kids – prove that if $2^n - 1$ is prime, show that $n$ has to be prime.

I simplified the challenge a bit and asked them to show that $n$ couldn’t be even if $2^n - 1$ is prime and $n$ was bigger than 2.

Next we looked at the “Good Approximation” section. This section actually stands on its own, but the examples mainly come from a discussion in the previous chapter about Gauss finding a nice approximation to the number of primes less than a given number. Guass approximated the number of primes less than 3,000,000 as 216,960. It turned out he was off by 154 (and sorry, I said 225 in the video which came from reading the example in the prior section too quickly). The question is was Gauss’s approximation a good one or not?