Primes of the form \( x^2 + ny^2 \)
Fourth Year Undergraduate Project
My fourth year project was on Primes of the form \( x^2 + ny^2 \), supervised by Dr Jens Funke. The project involved understanding how Class Field Theory can be used to derive conditions describing which primes can be represented by a given quadratic form. See Dr Funke's initial description of the topic.
Project work
You can view a copy of my project work
- Report
- Poster (packed full of information to get marks, but not good as a poster)
- Presentation and some notes I made in preparation
Rising Stars
I was invited to present a poster from the project at the Rising Stars 2012 symposium. A copy of the (much more readable and introductory) poster is available there, and is reproduced below.
SET awards
The project was entered for the 2012 SET awards, and was shortlisted to the final three in the mathematics category. Ultimately, though, it did not win. I prepared an expanded version of the poster as part of the entry.
More primes of the form
I've recently got interested in extending my fourth year project work a little further. Mainly going back to do things I didn't have time to write up at the time.
- Homogeneous splitting of primes in \( \mathbb{Q}(\sqrt[3]{7}) \), and other cubic fields with \( h_K = 3 \).
- Representation of primes by other cubic forms from \( \mathbb{Q}(\sqrt[3]{11}) \).
Gandalf talk
Shortly after startnig my PhD, I gave a much more extended talk about my project in the gandalf seminar. Details of the talk are listed on my talks page, and on the gandalf seminar 2012/13 page. The content is reproduced below.
- Slides of talk
- Notes of talk
- Maple worksheets implementing some of the criteria discussed for: \( x^2 + y^2 \), \( x^2 + 2y^2 \), \( x^2 + 3y^2 \), \( x^2 + 5y^2 \), \( x^2 + 17y^2 \), \( x^2 - 2y^2 \), \( x^2 - 142y^2 \)
- Maple worksheet implementing the criteria for the cubic form \( a^3 + 11b^3 + 121c^3 - 33abc \)
- Maple worksheet to count the number of solutions to \( N = x^2 + 5y^2 \) and \( N = 2x^2 + 2xy + 3y^2 \)
Abstract: Fermat's observation about which primes can be written as the sum of two squares motivates the question: which primes does a given quadratic form represent? After relating quadratic forms with ideals in quadratic fields, we show how Class Field Theory can be applied to construct general criteria describing these primes.