martyweissman

Nope -- I didn't take an axiomatic approach, or give a set-theoretic construction of different sets of numbers. Instead, I take basic arithmetic and algebra and properties of real numbers as a starting point. I do dwell on things like division with remainder and prime decomposition.

I would like to expand the programming tutorials, though I doubt I'll go as abstract as Stepanov. I do provide some discussion of algorithms that generalize, e.g. the Euclidean algorithm and Pingala's algorithm for exponentiation. But I don't present these in such a generic form.

I've got a grant application out right now... if it goes through, I'll have some funding to support expansion of programming tutorials. I'd like to include more depth in both programming and number theory. On the programming side, I'd include classes, recursion, memoization, visualization. On the number theory side, I'd include Gaussian/Eisenstein/polynomials, Pollard rho and maybe SQUFOF for factorization.

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.

Yes -- that is a future option. It's a time commitment, but perhaps next summer I'll post a new section on the book webpage with more exercises and solutions to existing ones. (Those who have taught with the book asked for more problems too.)

That's cool -- you might like Underwood Dudley's book on Elementary Number Theory. It has a similar set of topics, and I think it has more exercises and contains some solutions. And it's inexpensive!

I hope you enjoy the book -- the AMS sale does give a great price! Part of the reason I went with the AMS was that they're a nonprofit and their prices are reasonable for a hardcover book printed (offset, not digital on-demand) in color.

As the author, I'll add a few remarks and answer some questions about the book.

1. I made a book webpage at illustratedtheoryofnumbers.com. The errata are there. Also you can find a series of programming tutorials, if you wish to learn number theory with Python. I go from programming basics to primality-testing, RSA, etc.

2. I didn't provide solutions in the book. :( But there's always online discussion boards. Someday I'll write many more exercises and provide some solutions.

3. It's been used as a textbook for undergraduate number theory, e.g. at Rice, UC San Diego, next semester at Georgia Tech I think, etc.

4. No full e-book version is planned. It's all very old-fashioned, but I spent a lot of time on page layout, optimizing for print, etc.

5. I took a stronger stance on zero being a natural number in an early draft. Now I just try to make it clear that it's the convention I choose. If it's good for Bourbaki, it's good for me.

Feel free to drop a note if you have more questions about the book. My email is not hard to find.

Happy holidays! Marty Weissman