I don't know if this is the correct site to ask this question, but I felt it was off-topic on the Mathematics forum. I really like Number Theory and would like study some on my own. Which books should I read to get a good grasp on the subject?

I'm a 20 year old Software Engineering student. I've taken courses on introduction to Discrete Math, Linear Algebra, and Calculus I. I'm taking Calculus II and Graph Theory now. My Calculus lacks analysis of series/indefinite integrals though.

As for ability I wouldn't know how to answer. I've aced all those math subjects but they were fairly basic, they're designed for an engineering course after all. They're mostly introductions to the subjects

  • $\begingroup$ It would depend on your background: your age, the courses you have taken, your ability. $\endgroup$ – Joel Reyes Noche May 18 '18 at 13:03
  • $\begingroup$ Noted and edited, thanks $\endgroup$ – Francisco José Letterio May 18 '18 at 13:13
  • $\begingroup$ In particular, have you done proof writing in one or more of your previous courses? Most Number Theory texts would have that a prerequisite. $\endgroup$ – Gerald Edgar May 18 '18 at 13:20
  • $\begingroup$ I don't know what you mean by "proof writing". I've had some proof demonstrations both in class and as homework, but they were usually really simple, except for some cases, like proving the uniqueness of a formula for recurrence relations or proving that for prime numbers $p$, p divides all but two binomial coefficients pCk. Those are the hardest proofs I can remember $\endgroup$ – Francisco José Letterio May 18 '18 at 13:43
  • $\begingroup$ You might look at the Polya volumes on intuition. It is really more about problem solving, but he uses a lot of number theory examples. Not ideal volume, but pretty fun and accessible. Worthwhile as an adjunct. Other than that, my advice is to pick an easy, accessible book WITH answers. Ideally a Dover, but if not an old used cheap copy of some standard. $\endgroup$ – guest Jun 2 '18 at 17:54

I've heard very good reviews of the 2017 book, "An Illustrated Theory of Numbers" by MH Weissman.

The book's main site is here; a write-up, along with some reviews, by the American Mathematical Society can be found here. To quote from the latter (emphasis added by me):

An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history.

Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers.

Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition.

Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.

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As a software engineer, the book "concrete mathematics" by Graham, Knuth and Patashnik is a must. Chapter 4 is on number theory.

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