Senior Thesis Topic in Number Theory

[IAmAbelian]

As the title suggests, I am looking for a topic to do my senior thesis on. It doesn't have to be original work, but it does have to be rather math intensive. Also, I have to do it on a topic in Number Theory for reasons beyond my control. I have the typical senior level background(e.g. Modern Algebra, Topology, Complex and Real Analysis, etc.) and I have also taken a class in Number Theory. I have done a research paper in the past(for a different mathematics course) on the Collatz Conjecture, but I really don't want to try to write a 15-20 page paper on that(which is the relative length required of my thesis).

I would greatly appreciate any input anyone can offer on the matter, as I really want to do my thesis on a subject which I find beautiful and interesting. Thanks!

 

[R.Mesri]

One possible topic is the theory of continued fractions. There are many interesting topics to explore in this field, such as the properties and behavior of continued fractions, theorems about approximations of irrational numbers by rationals, and the relationship between algebraic numbers and continued fractions. You could also look into topics related to the Pell equation, which is closely related to continued fractions, and involves some interesting number theoretic concepts. Another option is the theory of quadratic forms. This involves the study of certain types of polynomials in two or more variables, particularly those that are quadratic in each variable. Here you could investigate topics such as the classification of quadratic forms, the representation of primes using quadratic forms, and the local-global principle for quadratic forms.