What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?
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A problem I particularly like is Broccard's problem. I investigated this one by myself some time ago and broke the Berndt&Galway's $2000$ record by a $100x$ factor with an ordinary computer and ordinary computer code. Of course it is not an interesting result because I did not provided better algorithms or theory, just $13$-years better technology.
There are lots of Unsolved problems in number theory and some of them are not paid enough attention. Try it if you want to!
P.S. And give special thanks to the guy who implemented legendre() in the GMP library :)
The title of your question is also the title of a fantastic book by Richard Guy. Get that book, and you'll never run out of good problems to think about.
I personally would go with Goldbach conjecture:
Recent developments of the subject is very interesting. There is a paper very recently the claimed the proof of Goldbach weak conjecture:
Terrence Tao proved in 2012 that "Every odd number greater than 1 is the sum of at most five primes". Also look at this: https://plus.google.com/114134834346472219368/posts/8qpSYNZFbzC
I think nearly everything one does in number theory is understandable if you followed the story far enough. Even the most difficult to understand theorem usually makes sense in special cases if you understood the great theorems that came before it.
But yeah there is a wealth of immediately understandable theorems in number theory that have gone unsolved so far. One that hasn't been mentioned yet is the twin prime conjecture that states there are infinitely many pairs of primes that are $2$ apart, such as $(3,5),(5,7),(11,13),...$.
There are also problems that have been solved...assuming other results that haven't been proved. My favourite of these is the congruent number problem. This asks which positive integers can be areas of right angled triangles with rational sides. For example $(3,4,5)$ has area $5$ yet no such triangle exists with area $1$ (not obvious).
In studying this problem you get to be led through a rich area of number theory upto modern day research on elliptic curves and modular forms. As mentioned above this problem has been solved...assuming a really interesting but technical result called the Birch Swinnerton-Dyer conjecture, an extremely difficult problem that is worth a million dollars to whoever solves it!