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.
I know you probably don’t want to expose solutions to the exercises so they can be used in a classroom, but for people like me, I don’t want to create an online discussion to every problem I attempt just to check my work.
I prefer to read textbooks that have solutions, so I can know for sure my answers are correct.
I don’t really buy the “you know when your solutions are correct”. The beginner can easily fool themself into thinking their solutions are correct, when they aren’t.
Due to that, I won’t be buying your book.
I'm on the same boat as you. I might still buy this book, it seems beautiful.
In the days of online autodidactic zeitgeists, it's a shame not to include the solutions. I understand the author wants to monetize his hard work and ensure it'll make money as a textbook and I believe textbooks can provide some income as long as they become the mainstay of a particular subjects.
That being said, I wonder if there is a future option for creating a problem/solution booklet exclusive for the teachers and opening up the solutions to this book so the entire world could possibly learn from this book.
The table of contents looks interesting — how did you choose which topics to cover? Were they influenced by what would be easy to illustrate? My knowledge of number theory is limited to the (very) early chapters of Burton's Elementary Number Theory or Niven & Zuckerman's book, so I'm wondering how this book differs in its choice of contents. (The presentation of course is outstanding.)
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.
For example, Stepanov starts with the problem of multiplying two integers n * m using the Ancient Egyptian Multiplication algorithm from the Rhind Papyrus (which runs in log n if i remember correctly), then observes the only property needed for the algorithm is associativity, so that it can run on any semi-group: mult(n,s) is n repetitions of the semi-group element s, using the semi-group operation. Useful and surprising applications are (1) matrix exponentiation to solve systems of linear recurrences in log n steps (no stupid Fibonacci implementation here!), and (2) encode a graph in your matrix, with elements in a tropical semi-ring, then matrix exponentiation solves shortest path. Not too bad for a 4,000 year old multiplication algorithm.
A second example is the euclidean algorithm, which he extends first to polynomials following Stevin, then to Gaussian integers, then to euclidean domains. A surprising application was to permutation algorithms managing memory.
Anyhow, I think it would be really cool if you showed these kind of applications of number-theoretic algorithms as well as the cryptography stuff. Unfortunately basically all of the modern algorithms literature seems to avoid even the tiniest hint of abstraction; it makes the subject so much harder to hold in your head! So providing a new set of examples that is more accessible than Stepanov would be doing the world a great service, i think.
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.
> 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.
It is very strange choice for a book written in 2017.
Table of contents (pdf):
Sample Chapter (pdf):
EDIT: Oh, the sample chapter actually mentions this exact point in a footnote. I guess I can still be clever by saying, "Sets of numbers are equivalent if they're isomorphic." :)
Niven - The Theory of Numbers. Contains hints for some of the proof based exercises, and answers for many of the computational exercises. I used it for my undergrad course, I remember it being reasonably beginner friendly.
Pressley - Elementary Differential Geometry. Has terse answers to every exercise! The subject is a nice mixture of concrete and abstract, calculation and proofs, and there's some interesting work using differential geometry in CS via computational geometry. Also i think it's used in robotics and a few other things? This book is very beginner friendly, it will get you about 1/2 way to General Relativity.
I think if I were you I'd try math.stackexchange.com.
It's nice that they got that right, on the first page!
The author's own book discusses the 1st prime, in chapter 0, which begins on page 1. How is this natural?
Problem 0.4 Add the odd numbers between 1 and 30."
Why would you say 1 and 30 instead of 0 and 30, since 0 is natural?
Getting hung up on notation and arguing over conventions will hold you back from learning math.
Offsets (0) and ordinals
(1) aree both useful in arrays. The only thing wrong* is not saying which you mean.
If something is a matter of taste, it really doesn't make sense to say one taste is "right" over another. It's more important to define your terms and be consistent for the context you're working in.