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I'm a high school student (or the equivalent in England). I love beautiful maths problems, and I'd love some suggestions of problems to put some research into, difficult, but not ones with "trick" solutions, like Olympiads, by which I mean there is some short solution to a synthetic problem.

I am particularly interested in Number Theory, so I'd love it if you could suggest something new for me to research.

Bill Dubuque
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Roskiller
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    Something like prove that $\sum \limits_{ k \leq n} \frac{1}{k} $ is never integer for $n >1$ , or a book of questions about number theory ?! – Ahmad Oct 09 '20 at 16:07
  • If you don't like "trick" solutions, then you are stuck to "straightforward" problems. All interesting problems require some sort of trick, so that you gain great joy after solving them. – WhatsUp Oct 09 '20 at 16:09
  • @Ahmad I can see how to solve this already, and the problem doesn't really seem investigative enough. I'd love some general equation to study the solutions of. – Roskiller Oct 09 '20 at 16:10
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    Take some $n \in \mathbb{N}$. Find all primes of the form $p = x^2+ny^2$ for $x,y \in \mathbb{Z}$. – Qi Zhu Oct 09 '20 at 16:11
  • @WhatsUp It's not so much that I don't want nice tricks, but more I'm looking for a 10 page investigation into some property, not a single page solution of a single problem. I'm looking for something more general, if that makes sense. – Roskiller Oct 09 '20 at 16:11
  • 250 Problems in Elementary Number Theory - Sierpinski (1970), would be a good start, I enjoyed solving it, but I don't know if this is what you look for. – Ahmad Oct 09 '20 at 16:13
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    Then you should take a textbook and start reading, e.g. algebraic number theory. But then you need some preparations, e.g. from abstract algebra etc. – WhatsUp Oct 09 '20 at 16:13
  • @QiZhu And then find that it's the title of a book. – WhatsUp Oct 09 '20 at 16:13
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    I came across Hardy and Wright's book on number theory at your age. Very old skool, but a nice introduction to university level mathematics. There is a wealth of books on 'elementary number theory', that would be where to begin. – DanLewis3264 Oct 09 '20 at 16:15
  • (I agree with all of the other commenters btw. Elementary number theory can be misleading as in its deceptively difficult problems can be understood by every middle schooler but there are only few people who can really appreciate them. The only way to do so is to start reading books on number theory, algebra, geometry, etc etc.) – Qi Zhu Oct 09 '20 at 16:25
  • https://math.stackexchange.com/questions/215424/ – Bart Michels Oct 09 '20 at 17:06
  • To my taste, the "secret" about number theory is that very few things can be accomplished "directly", even if they appear to be about elementary middle-school algebra. Some abstract algebra (to aid in understanding "modular arithmetic", and much more), and some complex analysis, are really irreducible minimums for being able to do almost anything at all. Fermat, Euler, Lagrange, and such people had amazing intuition for proto-complex-analysis and proto-abstract algebra, even if not formalized. But/and nowadays "anyone" can learn those techniques! :) – paul garrett Oct 09 '20 at 17:48

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Ok, I do not know what your experience and knowledge are, but here are my thoughts and recommendations:

$1.$ You will never get anywhere without those tricks and experience. Usually, when a mathematician tries to study a conjecture or another result they always study every single thing someone else did on that topic. To do research you must have a heck lot of knowledge and know very many "tricks". Moreover, all interesting problems have beautiful ideas, which I really do not want to be called tricks and I don't even want to mention that in 99% of the cases, in research you should innovate and create "tricks" yourself.

$2.$ I am glad you are interested in maths and if you want to study this beautiful science at a high level, you might need help. With all our recommendations, you might need an actual professor. (I, for example, am a student too)

$3.$ When you reach a very high level in mathematics, you will see everything is linked. You cannot just study one area and do research. Search, for example, for the Green Tao theorem, an exquisite result in number theory, but which has a proof that includes statistics, combinatorics, algebraic number theory, etc.

To conclude this section, take it gradually.

Now to dive into books:

$1.$ "$250$ Problems in Elementary Number Theory" - Waclaw Sierpinski

$2.$ "Problems from the book" - Titu Andreescu, Gabriel Dospinescu

$3.$ Any book provided by some university out there in England (I am sure there are plenty of handouts)

$4.$ If you want to read about unsolved problems: "Unsolved problems in number theory" - Richard K Guy

$5.$ "$104$ Number theory problems (from the training of the IMO USA team)" - Titu Andreescu, Dorin Andrica, Zuming Feng

$6.$ "Number theory concepts" - Titu Andreescu, Gabriel Dospinescu, Oleg Mushkarov

Finally, I want to challenge you to solve $2$ problems. One of them, exactly the way you want it to be, no tricks or ideas, plain and straightforward hard work and usage of theorems, and the other, no results, just beautiful ideas. Post an answer to your own thread, over here, with solutions or questions about them. Other people, please do not answer those (but have fun solving them!).

$1.$ (the no idea, only work problem)

Let $i=\sqrt{-1}$. Prove that $$\prod_{k=1}^{\infty}(k^2+i)$$ is not a real number

$2.$ (the idea problem)

Suppose $\mathcal{P}$ is a polynomial with integer coefficients such that for every integer $n$, the sum of the decimal digits of $|\mathcal{P}(n)|$ is not a Fibonacci number. Must $\mathcal{P}$ be constant?

I understand you think research is not about "silly" (but very beautiful) olympiad problems, but those represent the complex and creative thinking needed in research. Let me know which one you enjoyed the most.

Neo Gan
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  • Another book suggestion: Unsolved Problems in Plane Geometry and Number Theory by Klee and Wagon (1991). Of course, a web search for "unsolved problems number theory" gives a variety of leads. – Blue Oct 09 '20 at 17:09
  • The appearance of $\prod_{k=1}^{\infty}(k^2+i)$ vaguely reminded me of a question I asked here a few years ago (see also this question and this other question), which might be of interest to @Roskiller in that it (and the comments and answers) show how such questions can sometimes involve several seemingly different mathematical areas. – Dave L. Renfro Oct 09 '20 at 17:31
  • Indeed. What do yout hink about the second problem? I am particularly attracted to that one for its uniqueness. –  Oct 09 '20 at 19:02
  • If you're asking me (regarding the second problem), then I have no idea. I don't involve myself in number theory much, in spite of the question I asked, and what interests I have in number theory tend to be those involving irrational numbers, normal numbers, and metric number theory issues. – Dave L. Renfro Oct 09 '20 at 19:20