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 "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 text Calculus, 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: Fields, their properties, and uniqueness The topology of [itex]\mathbb{R}^n[/itex] Something from Analysis 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. Any suggestions are greatly appreciated and I can provide additional information if needed. Thanks!