Unsolved problems in number theory What are the most interesting examples  of unsolved problems in number theory which an 18 year can understand?

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!
