# Reviews of G H Hardy and E M Wright's *Theory of Numbers*

G H Hardy and E M Wright's

*An Introduction to the Theory of Numbers*has run to six editions, with the most substantial changes in the later editions being to the notes which give developments which have occurred since the deaths of the authors. We present below publishers information, extracts from prefaces and extracts from some reviews of these editions. There is some information about each of the six editions. We treat the new editions as separate works in our numbering below.

#### 1. An Introduction to the Theory of Numbers (1938), by G H Hardy and E M Wright.

**1.1. From the Preface.**

This book has developed gradually from lectures delivered in a number of universities during the last ten years, and, like many books which have grown out of lectures, it has no very definite plan.

It is not in any sense (as an expert can see by reading the table of contents) a systematic treatise on the theory of numbers. It does not even contain a fully reasoned account of any one side of that many-sided theory, but is an introduction, or a series of introductions, to almost all of these sides in turn. ... There is plenty of variety in our programme, but very little depth; it is impossible, in 400 pages, to treat any of these many topics at all profoundly.

There are large gaps in the book which will be noticed at once by any expert. The most conspicuous is the omission of any account of the theory of quadratic forms. This theory has been developed more systematically than any other part of the theory of numbers, and there are good discussions of it in easily accessible books. We had to omit something, and this seemed to us the part of the theory where we had the least to add to existing accounts.

We have often allowed out personal interests to decide out programme, and have selected subjects less because of their importance (though most of them are important enough) than because we found them congenial and because other writers have left us something to say. Our first aim has been to write an interesting book, and one unlike other books. We may have succeeded at the price of too much eccentricity, or we may have failed; but we can hardly have failed completely, the subject-matter being so attractive that only extravagant incompetence could make it dull.

The book is written for mathematicians, but it does not demand any great mathematical knowledge or technique. In the first eighteen chapters we assume nothing that is not commonly taught in schools, and any intelligent university student should find them comparatively easy reading. The last six are more difficult, and in them we presuppose a little more, but nothing beyond the content of the simpler university courses. The title is the same as that of a very well-known book by Professor L E Dickson (with which ours has little in common). We proposed at one time to change it to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book. A number of friends have helped us in the preparation of the book. Dr H Heilbronn has read all of it both in manuscript and in print, and his criticisms and suggestions have led to many very substantial improvements, the most important of which are acknowledged in the text. Dr H S A Potter and Dr S Wylie have read the proofs and helped us to remove many errors and obscurities. They have also checked most of the references to the literature in the notes at the ends of the chapters. Dr H Davenport and Dr R Rado have also read parts of the book, and in particular the last chapter, which, after their suggestions and Dr Heilbronn's, bears very little resemblance to the original draft. We have borrowed freely from the other books which are catalogued on pp. 417-419, and especially from those of Landau and Perron. To Landau in particular we, in common with all serious students of the theory of numbers, owe a debt which we could hardly overstate.

**1.2. Review by: Eric Temple Bell.**

*Bulletin of the American Mathematical Society*

**45**(1939), 507-509

As the authors have taken pains to describe - too modestly - the nature of their work, we quote from their preface.

"This book has developed gradually from lectures delivered in a number of universities during the last ten years, and, like many books which have grown out of lectures, it has no very definite plan.

"It is not in any sense (as an expert can see by reading the table of contents) a systematic treatise on the theory of numbers. It does not even contain a fully reasoned account of any one side of that many-sided theory, but is an introduction, or a series of introductions, to almost all of these sides in turn. ... There is plenty of variety in our programme, but very little depth; it is impossible, in 400 pages, to treat any of these many topics at all profoundly."

Those who had the pleasure of hearing the senior author's lectures when he was in the United States ten years ago, will have pleasurable anticipations of what to expect; nor will they be disappointed. The book is like no other that was ever written on the theory of numbers, as an introduction or as a treatise; although Edouard Lucas might have written something like it had he been primarily interested in the analytic theory and were he living today. Some of the topics treated have been frequently discussed in the English and German journals of about the past decade. As might be anticipated from the authors' interests, analysis dominates much of the material. The treatment throughout, even of old things, is fresh and individual. ...

The foregoing sample from the two dozen chapters covering 400 pages may give some idea of the extraordinary richness of the material, and suggest the justice of the authors' own characterisation of their work as "a series of introductions" to a vast and many-sided theory. They have presented these introductions in a manner that should stimulate a reader to continue beyond some of them; and it seems safe to say a great deal more than what they themselves say, "we can hardly have failed completely, the subject-matter being so attractive that only extravagant incompetence could make it dull." The book is anything but dull; in fact it is as lively as the proverbial (not the English) cricket.

**1.3. Review by: George D Birkhoff.**

*Science, New Series*

**90**(2329) (1939), 158-159.

Mathematics undergoes continuous revitalisation through two main currents of thought: the one flowing from the inexhaustible supply of simple conundrums concerning the positive integers; the other flowing out of new conceptual ideas necessary for the comprehension of natural laws. For example, to quote Cajori, "in the study of the right triangle there arose questions of puzzling subtlety. Thus, given a number equal to the side of an isosceles right triangle, to find the number the hypotenuse is equal to .... The problem may have been attacked again and again. ... In some such manner arose the theory of irrational quantities" - as a by-product of the attempt to solve an arithmetic problem without rational solution. Similarly the imposing edifice of Euclidean geometry grew out of the physical concepts based on exact spatial measurement.

In the fascinating volume by G H Hardy and E M Wright under review here we find a remarkably happy and varied selection of important arithmetic problems treated with consummate clarity and distinction. The book will appeal to the many non-professional devotees of number theory and to professional mathematicians generally. As a basis for an attractive and profitable first course on the theory of numbers one could not find a better text. The title is the same as that of a well-known volume by our great American number-theorist, L E Dickson. But there is little overlapping, and no real confusion will be caused.

It will enhance the value of the work to mathematicians that one of the two distinguished authors (Hardy) has not only contributed greatly to the advancement of the subject on the side of the so-called "analytic theory of numbers" but also has long been one of the most inspiring mathematical figures in the world. We recall too his close association with the short-lived genius, Ramanujan of India, with the lamented Landau of Gottingen and with his remarkably gifted colleague Littlewood at Cambridge, all of whom have done so much for number theory.

Although the pages of the book treat a large variety of topics, nevertheless it is truly stated by the authors in the preface that "in the first eighteen chapters we assume nothing that is not commonly taught in schools, and any intelligent university student should find them comparatively easy reading. The last six are more difficult, and in them we presuppose a little more, but nothing beyond the content of the simpler university courses." The authors have discussed the distribution of primes, Farey series, the geometry of numbers of Minkowski, irrational numbers, the theory of congruences, Fermat's theorem and related topics, decimal representation of numbers, continued fractions, approximation of irrationals by rationals, algebraic integers, Diophantine equations, the familiar arithmetical functions, partitions, representation of numbers by two or four squares, Kronecker's theorem in one or more dimensions. In every case the discussion given is on an elementary level so far as technical knowledge is presupposed, but on a genuinely professional level as far as insight and thoroughness are concerned. ...

It is much to be hoped that other mathematical works having the appeal of the book by Hardy and Wright will soon be written; and that a much wider public than at present will come to realise how through such works the highest artistic and intellectual enjoyment may be obtained, only to be compared with that to be derived from literature, art and music.

**1.4. Review by: Harald Bohr.**

*Matematisk Tidsskrift. B*(1939), 22-24.

The present book has been prepared on the basis of lectures by Prof Hardy over a long period of time has delivered at various universities. Therefore, as the authors themselves point out, "like many other books that have grown out of lectures, it has no specific plan". That the book thus has no completely systematic structure is, however, greatly offset by the life which abounds in it; in many places, Prof Hardy personally feels easy to speak again. And it is far from bearing the mark of chance, on the contrary, one feels everywhere, what an exceedingly great work the authors have created in designing and final polishing, and in so doing they have succeeded in making the work appear as an artistic whole.

The book deals with a great diversity of different topics of theoretical number theory. As the authors write, "there has been no shortage of variation in their programme"; however, when they add "but only a little depth", this only means that they have not been able to find space within the framework of the book for evidence of the real "severe" proofs; however, such proofs are mentioned in numerous places and often accompanied by brief hints of the method of proof.

**1.5. Review by: Louis J Mordell.**

*The Mathematical Gazette*

**23**(257) (1939), 482-486.

Not so many years ago a book with such a title would have suggested an account of number theory limited for the main part to the classical elements and so based chiefly upon the work of Gauss and Dirichlet. The subject has, however, continued to spread in many directions; its ramifications are so extensive, important and full of life that there has been a recent tendency to enlarge considerably the scope of an introductory book. The authors of this book have taken full advantage of this trend and have given a wide interpretation of the title.

The table of contents, as well as the authors' preface, suggests that a more appropriate title might have been "An introduction to some aspects of the theory of numbers". Thus the book contains some account of the following topics:

(1) The prime numbers.

(2) The more familiar arithmetical functions, including the partition function; their generating functions, i.e. the Dirichlet series or power series associated with them; and their orders of magnitude.

(3) Diophantine approximation, including Farey series and theorems of Minkowski, Kronecker and Hurwitz.

(4) Congruences, and of course the law of quadratic reciprocity.

(5) Irrational numbers, including decimals, continued fractions, transcendental numbers and the transcendence of $e$ and $\pi$.

(6) Simple arithmetical fields, e.g. $k(\xi)$, where $\xi = i, r, √2, √5$; and quaternions.

(7) Diophantine equations and also the representation of numbers by sums and differences of squares, cubes and fourth powers.

These subjects are distributed among the twenty-four chapters without great regard to order and to strictly logical development. It would have added greater unity to the book if the matter had been so arranged that chapters and parts of chapters containing connected subjects followed consecutively. The third chapter on primes - namely chapter 22 of the book - might well have been nearer the first two chapters on primes, namely chapters 1 and 2. Similarly, chapter 24 on some more theorems of Minkowski might be nearer chapter 3 containing one of his fundamental theorems. The account of quaternions in the middle of chapter 20 might also have been given in a separate chapter nearer the quadratic fields in chapter 12.

The authors have made their point of view and object quite clear in the preface. It is to produce a book giving an account of material, not too hackneyed, which they consider interesting, entertaining or important, or about which they felt they had something to say. There is, as they state, no question about a systematic development of a general theory, and results are sometimes used long before they are proved, as for example the existence of primitive roots. There is now and then a tendency to give ad hoc proofs, which, while striking, are not nearly as significant or far-reaching as the general theory, and are apt to be concerned more with details. There is no suggestion of any really axiomatic development of arithmetic operations. Thus the definition of the addition of two integers is taken for granted, but not that of the division.

There is no doubt, however, that the authors have produced a refreshing and attractive volume, written in a pleasant and readable style. It contains a wealth of interesting and often unexpected material which has not yet found its way into other so easily accessible books. The authors have spread their nets far and wide in gathering interesting material and their haul is a very nice one indeed. It is really surprising what an immense storehouse they have filled. There is sufficient variety in it to satisfy the most catholic taste and to cater for the reader in all his moods. He may go through the book from cover to cover, or study a chapter here and there, or dip in now and then for a pleasant morsel. In lighter moments he may turn to the theory of the game of Nim, while on more austere occasions he may study the question of Euclidean algorithms in algebraic fields, or the Rogers-Ramanujan identities in the theory of partitions.

The book, which is well printed, nicely set out and easy to read, is sure to extend the circle of those interested in number-theory. It will act as a stimulus to further study and research, since it contains much recent material still occupying the attention of investigators. The notes at the ends of the chapters are particularly valuable, not only from a historical point of view, but also in supplementing the text and in putting the subject matter in its proper perspective. The book is sure to have a long and successful life.

#### 2. An Introduction to the Theory of Numbers (Second Edition) (1945), by G H Hardy and E M Wright.

**2.1. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**30**(288) (1946), 55-56.

We must be grateful to the Oxford Press for giving us a second edition of this fascinating book (in spite of the difficulties under which publishers still work). It can hardly be necessary to describe it in detail, but it may be said that it gives a full, vivid and exciting account of its subject, as far as this can be done without using too much advanced theory, such as, for instance, the theory of functions of a complex variable. In fact, all the equipment the reader requires is a firm grasp of algebra, of the type that we must call "old-fashioned". Granted this, he will learn a great deal about: prime numbers, congruences, irrational numbers, the simpler number fields, Diophantine equations, Diophantine approximation and the "geometry of numbers ". The absence of function-theory does not mean that the book is elementary or devoid of "outlook value", since it brings us well up to the present bounds of mathematical knowledge; Chapter XXIV, "Geometry of Numbers", for example, is concerned with matters which are very much in activity at the moment, as a reference to recent papers by Davenport, Mordell and others in the publications of the London Mathematical Society will show.

#### 3. An Introduction to the Theory of Numbers (Third Edition) (1954), by G H Hardy and E M Wright.

**3.1. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**39**(328) (1955), 174.

"Hardy and Wright" is an established classic, and general recommendation of this third edition would be superfluous. We need only note that the most substantial change is the extension of the third of the chapters on primes to include Selberg's proof of the prime number theorem. This proof is "elementary" in the technical sense, since it makes no appeal to the theory of the Riemann zeta-function. Following the first proofs, by Hadamard and de la Vallee Poussin, steady sapping had reduced the appeal to function theory to what was believed to be the irreducible minimum, namely that $\zeta(s)$ has no zeros on the line $a = 1$, so that the Selberg-Erdos proof was something of a sensation. But it must be remembered that proofs which are " elementary " in this technical sense are often more difficult to find and to grasp than the non-elementary arguments.

Professor Wright is surely too modest when in his preface he attributes to himself only the faults of the book; if this were true, the determination of his own contributions would be extremely difficult.

#### 4. An introduction to the theory of numbers (Fourth Edition) (1979), by G H Hardy and E M Wright.

**4.1. Review by: John Hunter.**

*Proceedings of the Edinburgh Mathematical Society*

**12**(3) (1961), 161.

The fourth edition of this famous book differs only slightly from the third edition. Some of the notes and several sections have been altered to include information on recent results and in particular to note arithmetical results obtained by the use of electronic computers. In the latter category there is in Section 2.5 the latest information on the largest known prime (the corresponding note in the third edition appeared in the Notes to Chapter I) and in the Notes to Chapter XIII there is the latest statement on Fermat's Last Theorem and other Diophantine equations.

There are new or revised proofs of several theorems, including some simplification in the proofs of certain identities in Sections 19.8 and 19.9 in Chapter XIX, the chapter on partitions. There is also in Section 19.12 of this chapter a note on the recent work on partitions by Atkin and Swinnerton-Dyer.

The postscript on prime-pairs is now incorporated in the text as Section 22.20. Also an index of the names occurring throughout the book has been added. The reviewer feels that it would have been better to have produced instead, or, in addition, some form of general index. The aim of the list of books given on pages 414 and 415 might also be altered so that some of the more recent books could be included. These are however very minor criticisms. The book remains unique among books on Number Theory, with its wealth of material and its erudite and interesting comments on a great variety of topics.

#### 5. An introduction to the theory of numbers (Fifth Edition) (1979), by G H Hardy and E M Wright.

**5.1. From the Preface.**

The main changes to this edition are in the Notes at the end of each chapter. I [E M Wright] have sought to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present, both in the Notes and in the text, a reasonably accurate account of the present state of knowledge. For this I have been dependent on the relevant sections of those invaluable publications, the

*Zentralblatt*and the

*Mathematical Reviews*. But I was also greatly helped by several correspondents who suggested amendments or answered queries. I am especially grateful to Professors J W S Cassels and H Halberstam, each of whom supplied me at my request with a long and most valuable list of suggestions and references.

There is a new, more transparent proof of Theorem 445 and an account of my changed opinion about Theodorus' method in irrationals. To facilitate the use of this edition for reference purposes, I have, so far as possible, kept the page numbers unchanged. For this reason, I have added a short appendix on recent progress in some aspects of the theory of prime numbers, rather than insert the material in the appropriate places in the text.

#### 6. An introduction to the theory of numbers (Sixth Edition) (2008), by G H Hardy and E M Wright.

**6.1. From the Preface (by D R Heath-Brown and J H Silverman).**

This sixth edition contains a considerable expansion of the end-of-chapter notes. There have been many exciting developments since these were last revised, which are now described in the notes. It is hoped that these will provide an avenue leading the interested reader towards current research areas. The notes for some chapters were written with the generous help of other authorities. Professor D Masser updated the material on Chapters 4 and 11, while Professor G E Andrews did the same for Chapter 19. A substantial amount of new material was added to the notes for Chapter 21 by Professor T D Wooley, and a similar review of the notes for Chapter 24 was undertaken by Professor R Hans-Gill. We are naturally very grateful to all of them for their assistance.

In addition, we have added a substantial new chapter, dealing with elliptic curves. This subject, which was not mentioned in earlier editions, has come to be such a central topic in the theory of numbers that it was felt to deserve a full treatment. The material is naturally connected with the original chapter on Diophantine Equations.

Finally, we have corrected a significant number of misprints in the fifth edition. A large number of correspondents reported typographical or mathematical errors, and we thank everyone who contributed in this way.

The proposal to produce this new edition originally came from Professors John Maitland Wright and John Coates. We are very grateful for their enthusiastic support.

**6.2. Review by: Ian Anderson.**

*The Mathematical Gazette*

**94**(529) (2010), 184.

This is the sixth edition of the well-established text that first appeared in 1938. Since the fifth edition (revised by E M Wright, and briefly reviewed in the October 1980

*Gazette*) appeared in 1979, there have been many important developments in number theory, such as Andrew Wiles' work on Fermat's Last Theorem, so a new edition is most welcome. The revisions have been made by the distinguished number theorist Roger Heath-Brown, with a foreword by Andrew Wiles and a completely new chapter on elliptic curves by J H Silverman.

The revision strategy has been to keep the text of the fifth edition unchanged, and to add footnotes where necessary. Thus, for example, the statement "It is not known whether there is an infinity of Carmichael numbers" is retained, but a footnote states "This has now been settled, see the end of chapter notes." ...

Overall, though, this new edition is to be welcomed warmly; it will continue to play its part as one of the leading textbooks in the theory of numbers, very readable and clear in its exposition. And, finally, it is extremely pleasant to find that this new edition has something its predecessors greatly lacked - a general index, which makes the book so much more useful and user friendly.

Last Updated July 2020