We seem to have covered a great deal of ground in this work, but we have left considerably more unsaid. Much of the delight of number theory derives from its boundless breadth and depth, allowing even non-professionals the scope to make discoveries and, more importantly, to enjoy its endless fascination. This, and a natural human desire to solve puzzles, explains why so many people are willing to devote so much time studying some aspect of it, even though it may seem to have little or no practical application. However, despite many years of very significant progress, there are still several important questions whose answers elude us. In this final chapter we will discuss a few of these.