Hard and interesting problems (especially in Number Theory) to research for a high-schooler 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.
 A: Ok, I do not know what your experience and knowledge are, but here are my thoughts and recommandations:
$1.$ 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 shoudl innovate and create "tricks" yourself.
$2.$ I am glad you are interested in maths and if you want to srudy this beautiful science at a high level, you might need help. With all our recommandations, 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 graduately.
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 challange 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 theoremes 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.
