Number theory

A book in 21 chapters about the whole numbers: from "which numbers divide which" to the Riemann hypothesis, elliptic curves and the proof of Fermat's last theorem. Number theory is full of questions a child can ask and the best mathematicians cannot yet answer; this book shows which is which. Each chapter starts from a question, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Slower help. The first chapters point to the matching lessons on the Topics shelf, which teach the same ideas more gently, for younger readers.
• Compute. A calculator or a few lines of Python turn every example into an experiment; most chapters show how.

Contents

Part	Chapter	Level
I. Whole numbers	1. Divisibility	beginner
	2. Primes and unique factorization	beginner
	3. GCD and Euclid's algorithm	beginner to intermediate
II. Modular arithmetic	4. Congruences	intermediate
	5. Fermat, Euler and Wilson	intermediate
	6. The Chinese remainder theorem and multiplicative functions	intermediate
	7. Primitive roots	intermediate to advanced
	8. Quadratic residues and reciprocity	advanced
III. Equations and approximation	9. Diophantine equations	advanced
	10. Continued fractions	advanced
	11. Sums of squares	advanced
	12. Primality, factoring and RSA	advanced
IV. Analytic number theory	13. How primes are spread	advanced
	14. The zeta function and the prime number theorem	advanced to expert
	15. Primes in progressions	expert
	16. Partitions	advanced
V. Modern number theory	17. Algebraic number theory	expert
	18. p-adic numbers	expert
	19. Elliptic curves	expert
	20. Modular forms and Fermat's last theorem	expert
	21. Where number theory goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• Starting out: read chapters 1 to 6. They are the core every later chapter uses.
• For codes and computers: chapters 3 to 7 and 12, then the cryptography book.
• You know some algebra and calculus: read chapters 13 to 15 for the primes and the Riemann hypothesis, and 17 to 21 for the modern subject. The Algebra book's chapters on groups, rings and fields are the companion.