An Introduction to the Theory of Numbers

by G.H. Hardy

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory, not usually combined in a single volume; and, although it is written for mathematicians, the range of mathematical knowledge presupposed is not greater than that of an intelligent first-year student. In this edition the main changes are in the notes at the end of each chapter; Sir Edward Wright seeks 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.

Genres: Mathematics, Number Theory, Science, Nonfiction, Textbooks, Reference, Education

456 pages, Paperback

First published April 3, 1980

About the author

Godfrey Harold Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.

Non-mathematicians usually know him for A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman.

His relationship as mentor, from 1914 onwards, of the Indian mathematician Srinivasa Ramanujan has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. He called their collaboration "the one romantic incident in my life."

Reviews

[Joey Comeau · Rating 4 out of 5]

I read a two sentence review of this once that has really stuck with me. It went along the lines of

"If I could bring only one book with me to a desert island, it would be [some other book] if I thought I would be rescued. It would be G.H. Hardy's Theory of Numbers if I knew I was never coming back."

Sold.

[Cyrus · Rating 5 out of 5]

As a perfectionist, Hardy tries in this book to find the easiest way to teach us the theory of numbers without downgrading its beautiful and complex structure to a dummbed-down level. Rumer has it that Hardy didn’t look at mirror even for shaving. Perhaps he believed no phenomenon in the world is as perfect as it supposed to be, even his handsome face. The only exception is probably Math, and particularly the way he put together the building blocks of such a traditional structure, Number Theory: in a perfect, Hardian way!

[William Schram · Rating 5 out of 5]

Introduction to the Theory of Numbers by Godfrey Harold Hardy is more sturdy than the other book by him that I had read recently. It is also significantly longer. While E. M. Wright also went and wrote some things for this book, he wasn’t included on the spine of the book, so I forgot about him. The opening section of the book states that it arose out of series of lectures given at Oxford, Cambridge, and other Universities. Given that, it is not a systematic treatment of the subject, though it does attempt to touch upon all of the facets of Number Theory.

This book starts out by discussing the terminology and symbols used. It is divided into twenty-four chapters starting out with Prime Numbers. The chapters go like this:

(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(1), k(i), and k(ρ)
(13. Some Diophantine Equations
(14. Quadratic Fields (1)
(15. Quadratic Fields (2)
(16. The Arithmetical Functions ϕ(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. Some More Theorems of Minkowski

Most of the chapters are self-explanatory. Some of them are rather opaque at first glance. Take chapter 19 for example, it is called Partitions. What exactly is a partition? Looking into it tells you that a partition is a way to show a number using any number of positive integral parts.

I especially liked the chapters on modular arithmetic since that is something I never really learned in school for some reason. This particular version was written in 1938 and I don't know what edition it is.

[Vaibhav · Rating 5 out of 5]

Only a few books this good have been written.

[Sindhuja · Rating 5 out of 5]

This is the best of reference materials available on number theory, if anyone is interested. I read it out of interest, and hence at my own pace. You cannot do random chapters unless you already have significant knowledge on the subject. I initially felt that the lack of problems to work out might be an issue. Sooner, you find out that this book is more of a building block on a step by step basis. The book perse solves a lot of problem in the form of theorems and proofs. My familiarity with number theory might have rusted a bit, while picking up this book. So I did spend a little bit of time, proving the theorems or their intermediate steps myself. This is more of a beginner literature in advanced number theory, which also makes it a necessary but not sufficient book to have. In some vague sense, it was a sneak peek into the minds of the likes of Fermat, Euler and Ramanujan. An extra star for that.

[Mikesokolov · Rating 4 out of 5]

I got a lot out of it, but ultimately didn't finish.. can't say why really; maybe the same reason I didn't get a graduate math degree

[Ryan Kirkish · Rating 4 out of 5]

Brain pushups. It goes proof by proof. if you want to know what a mathematical proof is and have no desire to become a mathematician read only the first 2 or three Chapters.

[Alexa Daskalakis · Rating 5 out of 5]

Numbers are not human inventions. They are the raw machinery of existence, the silent structure behind all that can be quantified, reasoned, or imagined. G.H. Hardy and E.M. Wright’s An Introduction to the Theory of Numbers does not simply explain these objects—it reveals their deepest secrets, the rules they follow, and the patterns they obey across the infinite.

At its core, number theory is not just mathematics—it is the closest thing we have to an encounter with absolute, unchangeable truth. Every prime number, every modular residue, every congruence relation is not just an abstract fact—it is a fundamental feature of the mathematical universe, something that would exist even if no conscious mind had ever discovered it.

Hardy and Wright lead the reader down a corridor of mathematical revelation:

• Prime numbers—random and irregular, yet bound by inescapable laws.
• Diophantine equations—questions so simple a child could ask them, yet so deep they have resisted the greatest minds for centuries.
• Quadratic reciprocity—the hidden symmetry between numbers that should, by all intuition, be unrelated.
• The Riemann Hypothesis—an unsolved problem whose truth would shake the entire foundation of mathematics.

This is not a book that hands you answers. It places you on the precipice of discovery, staring into the vastness of the mathematical unknown. Every theorem proved here, every function explored, is a glimpse into an eternal structure that exists independent of time, space, or human cognition.

The greatest power of number theory is that it is both terrifyingly rigid and endlessly mysterious. The objects it studies—integers, primes, modular forms—are the most basic elements of mathematics, yet they behave in ways that defy prediction, appearing random on the surface but ultimately governed by deep, hidden order.

Hardy famously believed in the beauty of pure mathematics, untouched by application. Yet, from these pages emerged the mathematics that would one day define cryptography, quantum mechanics, and theoretical physics. The unknowable became known.

This is a book that does not just teach mathematics—it forces you to confront the fact that numbers are not human creations. They were always there. They were simply waiting to be found.

[Mike · Rating 3 out of 5]

Hardy’s Introduction to The Theory of Numbers, God’s gift to Mathematicians or merely a steppingstone to better books. First off, I do consider this one of Hardy’s better books. I read it so I could tell myself and others I read it. I would probably never use it as a reference, unlike the masterful book Introduction to Algebraic Number theory by Alaca and Williams, which I consult frequently. If I really wanted to learn a topic covered in this book like continued fractions, I’d probably read an exhaustive treaty on the subject like C. D. Olds book titled Continued Fractions. If I had to choose other intro level books that covered similar topics in number theory I would of course read A Classical Introduction to Modern Number Theory by Ireland and Rosen and the second book Introduction to Number Theory by Hua Long Keng. I feel that I'd simply learn the material faster with these books. But keep in mind these books are written for people with at least a Bachelor’s degree understanding in Mathematics. I read the 4th edition, hardcover. There is no index, but I believe the 5th and 6th editions do have it.

[powei · Rating 5 out of 5]

classic. hardy died before providing a proof for God's inexistence. so this book remained his most popular.

[Samira]

Just recieved the book,so happy,It seems it has to open up my mind

[Tim · Rating 4 out of 5]

encyclopedic, fairly elementary proof of prime number theorem

[sqrt2 · Rating 5 out of 5]

didn't finish , theres no button that says "stopped reading"