You are here

An Introduction to the Theory of Numbers

G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman
Oxford University Press
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This 6th edition is a modest update to the 1979 5th edition, which was itself a modest update to the 1960 4th edition, so we're essentially dealing with a new publication of a 50-year old book. It's still a wonderful book, and well worth buying if you don't already have a copy. It even has an index now!

The book has been completely re-typeset for this edition, giving it a more attractive look, with more white space and with more of the formulas displayed. But the typesetting introduced an alarming number of typographical errors: I counted 23 typos in this first printing, some serious, and I wasn't even looking very hard.

The body of each chapter is essentially unchanged, except for the correction of some typos in the 5th edition and some footnotes pointing out now-inaccurate statements or obsolete terminology. The chapter endnotes have been expanded, updating progress since the last edition. There is a completely new chapter (by Joseph Silverman) on Elliptic Curves. Although this is well done, it has a very different character from the rest of the book: it is a survey, proving some results but quoting many. The rest of the book is narrowly focused: If it quotes a result, it proves it (with few exceptions).

Hardy & Wright is a good example of Mark Twain's definition of a classic: A book which people praise and don't read. It's worthwhile to summarize some of the more unusual items included: construction of the regular 17-gon, proof of the transcendence of π and e, the Lucas-Lehmer test for primality of Mersenne numbers, Bertrand's postulate (actually a theorem: between every number and its double is a prime), and a complete elementary proof of the Prime Number Theorem. The discussion of orders of magnitude of arithmetical functions is very thorough, covering average, normal, minimal, and maximal orders of most common arithmetic functions (another good book for that subject, that goes into even more depth, is Gerald Tenenbaum's An Introduction to Analytic and Probabilistic Number Theory). The exposition is very clear, and the proofs have just the right amount of detail: concise, but not too concise to follow. The chapters on continued fractions and on algebraic numbers, two notoriously confusing subjects, are especially clear.

It's also worth comparing Hardy & Wright (here abbreviated HW) against another heavyweight in the introductory number theory textbook arena: Niven, Zuckerman, and Montgomery's An Introduction to the Theory of Numbers (abbreviated here as NZM). This book is itself 18 years old (the 5th edition was in 1991) but in many ways it is much more modern than Hardy & Wright. The most conspicuous difference is that HW has no exercises; it is that peculiar thing, an introductory textbook aimed at mathematicians. NZM is packed densely with exercises and is clearly aimed at undergraduates. HW is deep rather than broad; it ignores many topics of elementary number theory, but goes into great detail on the ones it tackles. NZM has coverage of some newer topics not covered in HW, for example Schnirelmann density, factorization methods, and public key cryptography. NZM also covers some older topics omitted from HW, such as quadratic forms and looking at some number theory theorems as special cases of abstract algebra theorems.

Bottom line: If you like number theory, you should own (and read) of Hardy & Wright, but if you already have a copy, you probably don't need to buy this edition.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Preface to the sixth edition, by Andrew Wiles
Preface to the fifth edition
1. The Series of Primes (1)
2. The Series of Primes (2)
3. Farey Series and a Theorem of Minkowski
4. Irrational Numbers
5. Congruences and Residues
6. Fermat's Theorem and its Consequences
7. General Properties of Congruences
8. Congruences to Composite Moduli
9. The Representation of Numbers by Decimals
10. Continued Fractions
11. Approximation of Irrationals by Rationals
12. The Fundamental Theorem of Arithmetic in k(l), k(i), and k(ρ)
13. Some Diophantine Equations
14. Quadratic Fields (1)
15. Quadratic Fields (2)
16. The Arithmetical Functions ø(n), µ(n), d(n), σ(n), r(n)
17. Generating Functions of Arithmetical Functions
18. The Order of Magnitude of Arithmetical Functions
19. Partitions
20. The Representation of a Number by Two or Four Squares
21. Representation by Cubes and Higher Powers
22. The Series of Primes (3)
23. Kronecker's Theorem
24. Geometry of Numbers
25. Elliptic Curves, by Joseph H. Silverman
List of Books
Index of Special Symbols and Words
Index of Names
General Index