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Chapter 25 Further Up and Further In

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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.

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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)

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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.

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