Help finding a topic in Number Theory

[MostlyHarmless]

First, I'm in need of a topic to write a paper about as part of my degree requirements. The paper is supposed to be 10-15 pages. (I only mention this because it limits the scope of the paper and thus the topic.)

I took a course in number theory last semester and really enjoyed it, so I'm really looking for something in that area, although I do enjoy Abstract Algebra as well.

So far, I've tried p-adic numbers and Gaussian Integers.

There didn't seem to be enough about Gaussian integers that was accessible to write a paper about, also I'm not a fan of Complex Analysis.

P-adic numbers seems very promising, but when reading about them, I didn't have that ah-hah! moment.

I've spent a few hours on google already, and intend to spend a few more, I would just be remiss if I didn't ask on here.

I'm open to any suggestions, links, or otherwise.EDIT: Cryptography might be another thing I try. We spent a lot of time on it last semester, and it seems like it has its roots in Number/Group Theory. And being an application, I feel it would make a more readable paper.

 

[Stephen Tashi]

You'll get better advice if you explain (perhaps quote) the requirements for the paper. Is it to be rigorous and formal? - expository? pure math? applied math?

 

[MostlyHarmless]

The requirements haven't really been laid out for us, other than a "research/analytic paper". We aren't expected to present "new math". I think the point of the paper is to expose us to "math writing" and to give us experience finding and citing papers etc..

I think we are pretty free as far as the intent of the paper. We have people presenting on math education which seems to be more informative/persuasive. We have people writing on applied math, and pure math.

 

[Stephen Tashi]

Some unconventional suggestions:

1. Apply Richardson extrapolation to something. https://en.wikipedia.org/wiki/Richardson_extrapolation (Might it be applied to some sequence in number theory?)

2. I think this guy's math is legitimate: http://www.gallup.unm.edu/~smarandache/FlorentinSmarandache.htm Can you explain some of it? (I don't know about his philosophy.)

 

[reuben]

Cryptography would be a great topic if you want to wave around some number theory and abstract algebra! You'd probably have to choose one cryptosystem if you want to go into the math in any depth, but it's all tons of fun. I don't know how much you know already, but "An Introduction to Mathematical Cryptography" (Hoffstein) is a pretty well written, self contained undergrad text. Would be a good starting point if you don't know much, or useful for reference if you do.

Also - I bet the recent developments with the twin prime conjecture is a really overdone topic, but maybe you could focus on the developments on related problems it's spurred? In particular there's an interesting polynomial analogue to the twin prime conjecture that new work has been done on. And the idea of crowdsourcing in math is big and new and exciting if you could write on that.

Idk if your paper has to be on anything recent, but both of these topics are super relevant right now, and you could include a lot of 'what next?' type discussion, which sounds fun.

 

[MostlyHarmless]

Thank you for that, I think I'm going to stick with p-adic numbers, a few of my classmates have chosen to do cryptography.