Oct 10, 2019
Why do prime numbers make these spirals?
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Why do prime numbers make these spirals?
A Talk on Dirichlet’s Theorem
Today, I gave my first maths talk ever. I spoke for 20 minutes in front of about 15 people, including a few lecturers and students at Sheff Uni, about some of the cool things I discovered about Dirichlet’s Theorem and its proof.
I was told it was apparently loads better than a lot of the audiences’ first maths talks (!) and that I knew exactly what I was doing and that my voice carried well.
The stuff I was talking about was very nice, although I have no idea how I’d ever come up with anything like that on my own.
I haven’t even mentioned the IMO lecture yesterday!
I’m hoping to get some of the content from the summer project up on this blog in the next few days.
I’ve been working on a project on Dirichlet’s theorem recently. This involves trying to prove the theorem from a road map of an elementary proof given in the book ‘A Prime Puzzle’ by Martin Griffiths.
It doesn’t get too difficult to understand, despite some of the content being 3rd-year university material. Such a simple question to state: does every arithmetic progression of integers with coprime step and starting integer contain infinitely many primes?
I’ve found myself just copying out lines I understand when I read them, but not remembering them. I think to finish, I’m going to have to prove the whole thing by myself, using the road map, to show I’ve learned from this project.
The proof is really advanced and really quite elegant, although it’s a little on the fiddly side. To prove the sum we’re interested in tends to infinity, we need to prove it is log x + O(1). The way the book does this involves splitting the sum into two parts, a nice part (log x multiplied by a constant) and a nasty part. The nasty part is chipped away at, piece by piece, and every piece that is chipped off is O(1) until we are left with something that we can also prove is O(1). This chipping and rearranging to knock off a bit more is fiddly and complicated.
I present the project in exactly two weeks’ time. Hopefully, I’ll be ready by then!
my textbook refers me to another textbook.