Alright, this topic may be a little out of place in this particular forum, but I figured that I would probably get the best response from posters here. My dilemma: I need help selecting a topic for an essay of 2000-4000 words (equations don't count) that explores and evaluates a particular aspect of mathematics. For reference, some previous titles for this assignment are "(adsbygoogle = window.adsbygoogle || []).push({}); How many square numbers are also triangular numbers, where are they, and what other problems lead to Pell's equation?" and "How well can [itex]\pi[/itex], [itex]e[/itex], [itex]\sqrt{2}[/itex] and other irrationals be approixmated by rational numbers?".

Some background: I'm a high school senior, have recently completed Spivak's textCalculus, and my current math class is exploring the topology of [itex]\mathbb{R}^n[/itex] (open and closed sets, compatness, etc.). Consequently, given my mathematical background, the topic certainly can't be too deep. From my very limited experience with math, I would like to write my essay on one of several topics:

However, since I'm not certain whether or not these topics will lend themselves to achievable questions (and even if they do, I still need help formulating a reasonably specific question). Additionally, it would be best if the questions lent themselves to some individual thought on my behalf.

- Fields, their properties, and uniqueness
- The topology of [itex]\mathbb{R}^n[/itex]
- Something from Analysis

Any suggestions are greatly appreciated and I can provide additional information if needed. Thanks!

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