Number Theory: Concepts and Problems

"Several books in number theory are my favorite. 1: Introduction to Number Theory by AOPS, 2) Elementary Number Theory by Burton, 3) Number Theory: Concepts and Problems by Andreescu et.al, and 4) An Introduction to The Theory of Numbers by Hardy and Wright.

By listing the books in this particular order, I acknowledge that Titu Andreescu’s book is not for the absolute beginners. In my opinion, it is the right book to study after learning basics number theory from some easier books (e.g. the first one on the list). On the other hand, I put Andreescu’s book before Hardy’s classic text in term of difficulty. While Hardy’s text is harder and broader in theories and concepts, it does not target math Olympiad training. Andereescu’s book is perfect for Math Olympiad e.g. USA(J)MO, IMO.

The highlights of the book are two aspects: first, it contains the right amount of concepts, from the divisibility, GCD, LCM, to congruence, Chinese Remainder Theorem, Eurler’s Theorem, and to more advanced topics like p-Adic. The mathematically language in the text is very formal. Concepts are well explained and theorems are rigorously proved.

The second highlight is the powerful collection of problems. In each chapter, you will see many example problems from major math competitions around the world: USAMO, Romania TST, USA TST, China TST, IMO Short List, IMO, Putnam, Mathematical Reflections, etc. There are total approximately 300 practice problems at the end of each chapter. They are also from major national/international Olympiads and the author’s own creations. The last chapter (approximately 220 pages) contains detailed solutions for each of the practice problem.

In summary, Titu Andreescu’s Number Theory book is a must-have for national / international level math Olympiad. To my knowledge, no book serves this purpose better than this one."

The book covers much of Number Theory and is a great book to learn information with lots of practice problems, located at the end of every chapter. One particular chapter that was extremely beneficial was Congruence involving prime numbers, specifically the subsection on Fermat's little theorem. Despite the formula looking rather simple, its proof and applications are widespread in a variety of number theory problems. Fermat's Little theorem was not only applied to proofs of other formulas, but also to high-level math problems that look challenging. Fermat's Little Theorem has helped me on many number theory proofs, and the book's examples taught me how to use the theorem and when to apply it which is useful in many competitions.

Number Theory: Concepts and Problems is one of my favorite books covering topics in elementary number theory. I really liked how clear this book explained hard theoretical concepts which is often hard to find in other textbooks. My favorite part of this book are the practice problems because they are very challenging and lead to intriguing results. Also the book has solutions to every practice problem which explain every step very thoroughly to succeed in getting the right answer. I personally bought this book to prepare for math competitions and the topics covered such as Diophantine equations and Euler’s theorem helped me prepare. I fully recommend this book to anyone looking to better understand introductory number theory even if you don’t have any prior experience with this subject.

This is a gem of a book on number theory. The book covers a broad range of important topics that start from basic divisibility and congruences and move to quadratic residues, p-adic valuations and systems of congruences. The pace is fast, but the exposition in each chapter is detailed. The authors prove lots of theorems and they make sure they also provide lots of examples to drive home the theory. The examples themselves come from IMOs, math competitions, or even math journals (for example, the unsurpassed, but extinct, Russian Kvant or the Romanian journal Gazeta Matematica). At the end of each chapter there are many problems for practice and the authors designate to which topic in the chapter the problems relate. Again, the source of the problems are mostly math contests from around the world. Solutions are provided for all problems, while for a few solutions the authors point to other sources for further investigation or remark how they are related to other topics. The book provided me with new insight and knowledge in this field, as some of the material I had not encountered before. The problems do make you think and they successfully reinforce the theory. The authors are well known in the area of mathematical problem solving and they have access to a rich set of material and this shows everywhere in the book. Highly recommended for the topics chosen and the quality of the exposition.

"This book provides a large variety of problems. I really like this book because the problem in this book are challenging and up-to-date. It also covers advanced topics like p-adic and theorem about prime number. It is my most favorite Number Theory book I ever had since it has all range (by difficulty scale) of the problem in every topics. There are a lot of problems that I have never seen before, and a lot of problems that I attempted to do and got stuck halfway. Moreover, the problems at the end of each chapter are really fun to solve and may take some time but it is definitely worth a try."