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I am a homeschooled rising senior in high school, and I would like to research an open problem in mathematics. I have taken a number of undergraduate-level mathematics courses, including single-variable calculus, multi-variable calculus, linear algebra, discrete mathematics, and number theory.

My current interests include number theory, graph theory, algorithms, and topology, and I would be willing to read extensive existing material on a subject before I begin my research. Can anyone suggest some open problems that might interest me but would be within my grasp?

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If by 'research' you mean solve an open problem, this is very unlikely. On the other hand, there are many open problems that you would be able to understand and gain some insight for very easily here –  Daniel Littlewood Jun 29 at 21:54
    
I was actually looking for open problems that could, theoretically, be solved by high-achieving high school students. –  ntomlin1996 Jun 29 at 22:00
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That's almost certainly not going to happen, but if it were then one of your best starting points is probably the answer I linked. If you want a project, pick an area (for instance, you mention graph theory) and pick up a textbook. If you can't understand it, work out what you're missing and fix it. Eventually (and this will take a very long time) you might hit something new. –  Daniel Littlewood Jun 29 at 22:36

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Casually working in the twin prime conjecture, Fermat's last theorem, and poly-time integer factorization led me to learn a tremendous amount about number theory, abstract algebra, cryptography, computational complexity theory, and a number of other more specific topics.

I'd recommend it. Each topic is easily accessible and with enough determination you could uncover and learn a lot. Just don't get into the mind set that you will definitely solve them as it will obviously take a while given the even the pros have only solved one. But do get frustrated! Get frustrated that you are frustrated! And learn how to handle yourself while you learn the math behind the problems .

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I suggest that you look at the research of the professors at the university where you took courses, identify the professors that work in areas that interest you, and then ask them for problems. You will not only get to know the professor well by doing this (which is very beneficial), but you will likely contribute to a larger project while being mentored by a professional. A worthwhile professor will be happy to hand you some problems and let you run with them.

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I second this response: There are known cases of HS students proving nontrivial theorems (I know 3, but surely there are many more), but, to my knowledge, they all required individual supervision from senior mathematicians. One additional suggestion is to take graduate courses as well, one typically cannot do serious research in pure math with just undergrad education. –  studiosus Jun 29 at 23:43

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