Research Proposal:
Continued Fractions and Their Interpretations

Chris Hanusa
Advisor: Francis Su

        For my thesis, I would like to explore Continued Fractions, and all the current interpretations of them. After this exploration, I hope to find a correlation between them, or at least use them to prove various known results from Number Theory. Not only will this be an exploration into Continued Fractions, but also into methods of proofs, most likely processes that include Combinatorics.
        I have prior interests in continued fractions, because I learned about them at a Number Theory camp, after which I thought they looked fun. Then, early last year, I found a connection between work in the combinatorial area of tiling by Professor Benjamin and Su co-authored last year. I hope to expand upon this paper in the process of exploring for my thesis.
        I am also interested in using, learning, and developing different proof techniques as I work on my thesis. I wish to do this because one of my beliefs for as long as I have done rigorous mathematics is that "if you're going to do something right, you might as well do it right in several ways", to show your understanding of the topic. I enjoy the prospect of learning a new way to prove things, which I would be able to apply in working on my thesis.
        I think that this topic is a good strengthening possibility for me, because I plan to explore continued fractions and their implications fully, increase my Number Theory foundations, and possibly learn new techniques that can help me in the future as well.
 

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