Finding a complex mathematical question for a 2500-word research essay

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greenglasses
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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.
 
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