What are Some Complex Mathematical Questions to Explore for a Research Essay?

[greenglasses]

This isn't a question, per se.

I have to write a 2500 word + research essay on a mathematical concept for school. It's expected that I look into university-level concepts; game theory and abstract algebra are among common subjects.

I proposed some ideas to my math teacher, but they were shot down because they were looking at a mathematical concept instead of a mathematical question.

So basically, I was wondering if anyone had any suggestions for famous (or not), complicated questions that I could possibly stretch out a solution and explanation for in around 2500 words.

I do enjoy Calculus, so I was looking into there but couldn't find anything complex enough, so I'm asking you guys. Of course, if you know any sort of early-university-level questions in a different subject that could work I'd love to hear about it too.

 

[RLBrown]

p-adic numbers

This topic is rich with interesting and surprising ideas.
Keep it fun, keep it simple.

Did you know that...
...333333334 = 2/3 in 10-adic numbers?
Try this product:
...333333334*3 = 2

Some fun places to start are...
MF90: Fractions and p-adic numbers
Rounding p-adic Rationals
Golden Integers