Here you can find a collection of miscellanous documents that might be of some interest
Dr. Gangl is very fond of setting challenges during his lectures in Number Theory 3/4 (and Algebra & Number Theory 2). One such (very open ended) challenge was as follows.
Observe that \( 12^2 + 33^2 = 1233 \), find as many such pairs \( (x,y) \), with \( x^2 + y^2 = x \text{ concat } y \), as possible.
I found infinitely many solutions...
The article A curious cubic identity and self-similar sums of squares by van der Poorten, Thomsen and Wiebe contains an alternative explanation, a way generating examples, and some other cute observations. (And a more appealing name: self-similar sums of squares.)
The following solutions are hosted on github.
Solutions with \( y \) having the given number of digits:
Families generated by solution where \( y \) has given number of digits. Note: there is (probably a lot of) repetition of families. When prepending the repeating block to the initial solution, the initial solution may need \( x \) to be padded with a \( 0 \) to make \( x \), and \( y \) the same length, similarly for the repeating block.
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.
You can view a copy of my project work
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.
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.
I tried to extend 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.
Towards the end of my first PhD year, I had to write a progression report in order to continue. The department also asks for (shorter) reports mid-April each year.