An Introduction to the Theory of Numbers

Citation

Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.; D. R. Heath-Brown & J. H. Silverman, Eds.). Oxford University Press.

Chapter Summary

The sixth edition has 25 chapters, an appendix, a list of books and indexes, with a preface by Andrew Wiles and prefaces to earlier editions. Hardy and Wright's 1938 preface calls the book not a systematic treatise but a series of introductions to many sides of number theory, with plenty of variety and little depth, built around topics the authors found congenial. Chapters 1–24 are the authors' text as revised by Wright for the fifth edition (1979); Chapter 25, on elliptic curves, is new in the sixth edition. The 1938 preface says the first eighteen chapters assume nothing beyond school mathematics and the last six presuppose a little more, but nothing beyond simple university courses; each chapter ends with notes.

Chapters 1–2: The Series of Primes (1); The Series of Primes (2)

The first two of three chapters on prime numbers. The third, Chapter 22, comes much later in the book.

Chapters 3–4: Farey Series and a Theorem of Minkowski; Irrational Numbers

Chapter 3 introduces Farey series together with one of Minkowski's fundamental theorems; Chapter 4 turns to irrational numbers.

Chapters 5–8: Congruences and Residues; Fermat's Theorem and its Consequences; General Properties of Congruences; Congruences to Composite Moduli

Four chapters on congruences. A reviewer notes that results such as the existence of primitive roots are sometimes used long before they are proved.

Chapters 9–11: The Representation of Numbers by Decimals; Continued Fractions; Approximation of Irrationals by Rationals

These chapters cover decimal representation, continued fractions and the approximation of irrationals by rationals. Mordell's 1939 review groups decimals and continued fractions with transcendental numbers under the heading of irrational numbers.

Chapters 12–15: The Fundamental Theorem of Arithmetic in k(1), k(i), and k(ρ); Some Diophantine Equations; Quadratic Fields (1); Quadratic Fields (2)

This group extends factorization to simple arithmetical fields and applies it to Diophantine equations before treating quadratic fields in two chapters.

Chapters 16–18: The Arithmetical Functions φ(n), μ(n), d(n), σ(n), r(n); Generating Functions of Arithmetical Functions; The Order of Magnitude of Arithmetical Functions

These chapters cover the familiar arithmetical functions, their generating functions (Dirichlet series and power series), and their orders of magnitude.

Chapters 19–21: Partitions; The Representation of a Number by Two or Four Squares; Representation by Cubes and Higher Powers

Partitions and the representation of numbers as sums of squares, cubes and higher powers. Mordell notes that quaternions appear in the middle of Chapter 20.

Chapters 22–24: The Series of Primes (3); Kronecker's Theorem; Geometry of Numbers

Chapter 22 returns to primes; from the third edition on it includes Selberg's elementary proof of the prime number theorem, as T. A. A. Broadbent noted. Chapter 23 treats Kronecker's theorem and Chapter 24 the geometry of numbers, the last of the authors' chapters.

Chapter 25: Elliptic Curves

Joseph H. Silverman's new chapter, which the editors say connects with the original chapter on Diophantine equations and treats a subject that has become central to number theory; the Oxford description links it to modular elliptic curves and the proof of Fermat's Last Theorem.

Key Concepts

Chapters 1–15: Primes, congruences, irrationals and fields

• Prime numbers: three chapters, 1, 2 and 22.
• Farey series and the theorem of Minkowski: Chapter 3.
• Congruences: Chapters 5–8.
• Continued fractions and approximation of irrationals: Chapters 10–11.
• Quadratic fields and unique factorization in simple fields: Chapters 12 and 14–15.

Chapters 16–25: Arithmetical functions, additive problems and geometry

• Arithmetical functions and generating functions: Chapters 16–18.
• Partitions: Chapter 19.
• Sums of squares and higher powers: Chapters 20–21.
• Geometry of numbers: Chapter 24.
• Elliptic curves: Chapter 25.

Critical Analysis

Strengths

• Range of topics at an accessible level. In Birkhoff's 1939 review the discussion is elementary so far as technical knowledge is presupposed but professional in insight and thoroughness. The authors' own preface says the first eighteen chapters should be comparatively easy for an intelligent university student.
• Notes that follow the subject. Mordell called the end-of-chapter notes particularly valuable both historically and in putting the subject in perspective; the fifth edition's notes were revised by Wright, and the sixth edition expands them with the help of specialists so that they lead toward current research.
• A readable, individual style. Mordell describes a refreshing and attractive volume in a pleasant and readable style, to be read from cover to cover or dipped into. Ian Anderson, reviewing the sixth edition in The Mathematical Gazette, calls it very readable and clear.
• A careful sixth edition. Anderson explains that the editors kept the fifth edition's text unchanged and added footnotes, a foreword by Wiles, the elliptic curves chapter, and a general index, which earlier editions lacked.

Limitations

• Not a systematic course. The authors say that the book has no very definite plan and omits the theory of quadratic forms. Mordell adds that topics are placed without great regard to order, that results are sometimes used long before they are proved, and that some proofs are ad hoc.
• Depth varies. The preface describes plenty of variety but very little depth, so a reader who wants a full treatment of one area will need a specialist book.
• Updates live in the notes. Because the main text is kept as in 1979, recent developments appear in footnotes and end-of-chapter notes rather than in the argument.

Where It Fits in the Course

This is the classic reference-style text in a list that otherwise offers books written as courses. For a first course Joseph Silverman's A Friendly Introduction to Number Theory, David Burton's Elementary Number Theory and Kraft and Washington's text (with cryptography applications) are organized as step-by-step teaching books with exercises, and Robin Wilson's Number Theory: A Very Short Introduction gives a quick overview. Hardy and Wright works well after such a course, or alongside it as a source of proofs and historical remarks. Ireland and Rosen's A Classical Introduction to Modern Number Theory goes deeper into the algebraic and arithmetic-geometric side, which Hardy and Wright reach only at the end with elliptic curves; the two complement each other.

How to Use This Book

The book suits mathematics students from the first year upwards, as Oxford puts it, who want a rich selection of classical topics, and it serves as a reference for number theorists. The authors' preface says the first eighteen chapters need nothing beyond school mathematics.

• Core for a reading course: Chapters 1–3, 5–11 and 16–18 (primes, Farey series, congruences, irrationals, continued fractions, arithmetical functions).
• Selective extensions: Chapters 12–15 for quadratic fields, 19–21 for partitions and sums of powers, 23–24 for Kronecker's theorem and the geometry of numbers, and Chapter 25 for elliptic curves.
• Reading the notes: read each chapter's notes after the chapter, since they give references and later developments; for problems and graded exercises, pair the book with one of the course-style texts above.