TITLE: Unsolved problems in number theory
QUESTION [5 upvotes]: What are the most interesting examples  of unsolved problems in number theory which an 18 year can understand?

REPLY [3 votes]: 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 :)