An Introduction to the Theory of Numbers

I'm not a mathematician, but I was interested enough to make my way through the first few chapters and found them relatively accessible. Though I imagine most university courses use more contemporary texts, this is still a decent introduction to the topic for advanced undergraduates/graduate students. Besides, it's G.H. Hardy.

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 show more 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. show less