[martyweissman]

The contents kind of evolved from everything I had taught in undergraduate number theory, and a few other courses (a 2-week course for high-school students, some work with K-12 teachers, etc.). First, I wanted to cover the core topics of an elementary number theory course: Euclidean algorithm, prime decomposition, multiplicative functions, modular arithmetic, quadratic reciprocity.

Add to that Gaussian/Eisenstein integers, because they're pretty, open the door to algebraic number fields, and might help the reader understand that uniqueness of prime decomposition is not obvious.

Add to that mediant fractions and Ford circles, because they give a really nice perspective on Diophantine approximation (the only approach which really stuck with me). They're also good for future K-12 teachers to better understand fractions.

For quadratic reciprocity, I like teaching with Zolotarev's proof... so add that. (I think I'll give a more traditional proof, in an extra few pages, in a future edition.)

Finally, Conway's topographs give a beautiful approach to binary quadratic forms, which are often not taught in a first course (outside of Pell's equation). Learning and teaching Conway's approach has influenced my own research, and it's beautiful and visual. That's the last part of the book.