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Chapter 25 Further Up and Further In

If you survived this book, hooray! You made it. You did a great job making it through a whole arc of number theory accessible at the undergraduate level.
Although we really did see a lot of the problems out there, there are many we did not see all the way through. We were able to prove some things about them. Here are just a few problems we started touching on.
  • Solving higher-degree polynomial congruences, like \(x^3\equiv a\text{ (mod }n)\text{.}\) (Chapter 7)
  • Knowing how to find the first nontrivial integer point on hard things like the Pell (hyperbola) equation \(x^2-ny^2=1\text{.}\) (Chapter 15)
  • Writing a number not just in terms of a sum of squares, but a sum of cubes, or a sum like \(x^2+7y^2\text{.}\) (Chapter 14)
  • The Prime Number Theorem, and finding ever better approximations to \(\pi(x)\text{.}\) (Chapter 21)
It’s this last one we will focus on in this extended postscript, for it takes us to the very frontiers of the deepest questions about numbers.