I'm a grade 12 student interested in Number Theory, Graph Theory and Combinatorics and I am currently looking for ideas for an original research project/paper in mathematics. I was hoping that someone could recommend me some areas or topics of research which are challenging yet do not exceed my capabilities.

I have done lots of competitive (Olympiad) level mathematics, especially number theory, combinatorics and inequalities. Geometry is a bit weak though. I've taken basic single variable calculus courses as well. I don't know much abstract algebra (if anyone can recommend some good places to start learning, that'd be great!) or statistics/probability. I can program in python and work in LaTeX.

I really appreciate all help! Thanks!


closed as off-topic by Will Jagy, Mark Fantini, user147263, Chappers, Mario Carneiro Apr 1 '15 at 0:48

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  • "This question is not about mathematics, within the scope defined in the help center." – Will Jagy, Mark Fantini, Community, Chappers, Mario Carneiro
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  • $\begingroup$ Try doing this question using elementary number theory. It's quite challenging and a thing of research too. $\endgroup$ – user196761 Mar 31 '15 at 6:15
  • $\begingroup$ @Broly What do you mean by "a thing of research"? $\endgroup$ – Kshe Mar 31 '15 at 7:51
  • $\begingroup$ By "thing of research", I was referring to the fact that many areas of number theory haven't been explored yet. It is surprising that the theorems and problems which Fermat solved in his time was done using only elementary methods. Moreover, apart from just one incorrect hypothesis (i.e. numbers of the form $2^{2^n} +1$ are primes), all of his generalizations have been proved to be true today. There must be some sort of theory that he had developed, using only elementary methods, which still haven't been found out yet. $\endgroup$ – user196761 Apr 17 '15 at 11:07
  • $\begingroup$ And since the question I mentioned is from one of Fermat's letters, still lacking an elementary solution, it'd be an intriguing experience for you to solve it. Also, he was particularly fond of his method of infinite descent, which is also a topic scarcely explored. So, it might be a possibility that an ingenious aid to the method of infinite descent will be able to solve the toughest of problems and expand the horizons of mathematics. Therefore, it surely is a thing of research. $\endgroup$ – user196761 Apr 17 '15 at 11:14