A Brief Guide toNumber Theory

A Brief Guide to Number Theory is a book that focuses on elementary number theory, specifically topics that are common in Olympiad mathematics. We start from the basics of modular arithmetic and progress to advanced concepts like quadratic reciprocity, Pell's equations, and density in number theory.

Number-theoretic functional equations are on the rise, with at least one appearing in the IMO number theory shortlist in recent years. It is quite surprising that there are so few comprehensive resources on this topic. Hence, we have dedicated an entire chapter to functional equations. The first subsection helps readers gain familiarity with regular algebraic techniques, while the next introduces number-theoretic ideas.

Another topic we have included is combinatorial number theory. Many problems in recent IMO number theory shortlists involve ideas that are combinatorial in nature. These problems often pose challenges to even the top ten countries at the IMO. Furthermore, very few resources are curated specifically for such problems, which is why we have written a section dedicated to them. This section focuses heavily on exercises and solved examples, though we do assume some basic knowledge of combinatorics. Prerequisites (for this section and the rest of the book in general) are mentioned on a separate page.

Each chapter consists of four major parts: theory, exercises, solved examples, and practice problems. Between the theory, we have included Exercise Problems to help readers gain a better understanding of the material. After each subchapter or section, we have added Solved Examples on the topic, along with complete solutions. We have also included Practice Problems where solutions are not provided, giving readers the opportunity to apply what they have learned independently.

For a long time, a student without extensive theoretical knowledge—but with strong fundamentals, a bit of creativity, and adequate preparation—could clear the INMO. However, in recent years, more and more problems have required a substantial theoretical background, especially in 2026, which featured solutions involving Zsigmondy's theorem, Catalan's conjecture, LTE, and Dirichlet's theorem. Furthermore, to do well in the IMOTC and at the IMO, having the theoretical prerequisites is a must. Hence, we have included most of the major concepts along with problems from various contests to help readers gain a deeper understanding of these topics.

In addition to the theory, we have carefully curated the problems for both the Solved Examples and Practice Problems. Rather than relying on problems that appear in most books and handouts, we chose to highlight lesser-known problems from recent years. For the first few chapters, almost all the Solved Examples are relatively new — some from not-so-well-known contests, and others from AoPS posts. One of the Solved Examples for Chapter 2 was sourced from a random AoPS post with a very short statement, yet ended up taking us hours just to write the solution. Some of the problems we have included are even from contests that we participated in last year, and a large chunk are from 2026 contests. In general, we aimed to make the Solved Examples decently challenging. By contrast, the Practice Problems are mostly from better-known contests and are somewhat simpler than the Solved Examples. This approach has helped readers engage with solutions to harder problems and practice with slightly easier ones, enhancing both understanding and problem-solving skills.

Our primary goal is to help readers understand and appreciate the beauty of number theory. We want students to gain intuition and a feeling for the subject instead of memorizing formulas and applying them blindly. Learning number theory requires patience and time; the journey is much more rewarding if you try to enjoy the subject rather than focus solely on the end results. The problems are undeniably hard, and you are strongly encouraged to take your time with them.