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David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple question, and then he builds a whole machinery to answer it. And to be precise here's what he says on page 1:

This leads to the basic question of the whole book, which we formulate as follows:

Basic Question 0.1. Given a positive integer $n$, which primes $p$ can be expressed in the form $$p=x^2+ny^2$$ where $x$ and $y$ are integers?

We will answer this question completely, and along the way we will encounter some remarkably rich areas of number theory.

I think such way is extremely great, it gives the person the motivation to continue the whole book just by the existence of that basic question. So, are there any books in the spirit of David A. Cox "Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." ?

J W
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user153330
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3 Answers3

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John Conway's book: The sensual (quadratic) form is excellent and roughly fits the pattern you describe. Each chapter is about solving a new problem about quadratic forms, the machinery gets more and more sophisticated.

Brennan.Tobias
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4

Knuth's book: Surreal numbers builds Conways surreal number system up from the very beginning and steps through the very interesting proofs of it's basic properties. It's a bit more elementary and has a story element to it but there is certainly good mathematics in it too. Highly recommended.

Brennan.Tobias
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3

Romik's The Surprising Mathematics of Longest Increasing Subsequences. A book about the problem that deals with finding out the distribution of the longest increasing subsequence of a random permutation. It is research-level, however.

Review can be found at MAA Reviews.

twnly
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