NTIC Further Up and Further In

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.

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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 $x^{3} \equiv a (mod n) .$ (Chapter 7)
Knowing how to find the first nontrivial integer point on hard things like the Pell (hyperbola) equation $x^{2} - n y^{2} = 1 .$ (Chapter 15)
Writing a number not just in terms of a sum of squares, but a sum of cubes, or a sum like $x^{2} + 7 y^{2} .$ (Chapter 14)
The Prime Number Theorem, and finding ever better approximations to $π (x) .$ (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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Summary: Further Up and Further In

The final chapter in the book gives just a sense of possibly the most important open question in mathematics.

In Section 25.1 and Section 25.2, we begin the process of asking how to improve our estimates of primes.
The next section gives us enough background (and pictures!) to understand at least the gist of the Riemann Hypothesis, one of the Millennium Prize Problems.
Sections 25.4, 25.5, and 25.6 all lead up to seeing the Riemann explicit formula in Section 25.7.
The Epilogue reminds us that this book is just the beginning.

The Exercises lead you even further into the future of your number theory exploration!