The Higher Arithmetic: An Introduction to the Theory of Numbers

The higher arithmetic is more commonly known as number theory and is one of the most enjoyable and complex areas of mathematics. Simultaneously simple and hard, the problems are generally easy to understand yet can be horrendously difficult to solve. Furthermore, the initial areas of number theory are easy to comprehend; in general it only takes a basic knowledge of algebra to manage the main points. In this book, Davenport takes you through the basics of number theory, starting with prime factorization and going through some simple Diophantine equations. The chapter titles are: *) Factorization and the primes *) Congruences *) Quadratic residues *) Continued fractions *) Sums of squares *) Quadratic forms *) Some Diophantine equations This book is a solid introduction to number theory and can be understood by the advanced high school student. The primary drawback for the modern reader is that there is no coverage of the use of number theory in modern encryption techniques.

Well, this is definitely a very good introduction to number theory. The author provides clear, readable proofs of all the most basic theorems on topics such as congruences, sums of squares, etc. He explains things quite well. However, despite costing almost 2.5 times as much, I would recommend Hardy and Wright's book An Introduction to the Theory of Numbers more highly than Davenport's book. Seriously, although it may seem good that Davenport doesn't require a knowledge of calculus as a prerequisite for his book (which Hardy DOES require), one probably shouldn't learn number theory until one has a good backrground on topics ranging from improper integrals to infinite series. Because Davenport does not require calculus as a prerequisite, he neglects HUGE aspects of what could actually be considered BASIC number theory: namely, the basic analytic aspects (such as Tchebycheff's results on the Prime Number Theorem) and the additive theory (i.e. partitions and such, as well as the basics of the generalized theory surrounding Waring's problem for high powers of integers). So, my recommendation is, wait until you know integral calculus and the theory of infinite series BEFORE buying a book on number theory, and then buy Hardy and Wright's book rather than this one.

This book is an AMAZING introduction to the Theory of Numbers. It assumes no previous exposure to the subject, or any technical mathematical knowledge for that matter. Its prose is lucid and the style appealing. Davenport chose NOT to write a lemma-theorem-proof kind of book, and the result is a marvelous, eminently readable introduction to the subject. Its wonderful to read a book where good prose is used to appropiately substitute a massive collection of uninviting symbols. I've also been reading other books on Number Theory, such as Hardy & Wright, but none are as clear as this one.I found the chapter on quadratic residues (which includes the reciprocity law) to be especially well written. The section on computers and number theory is excelent as well. A concise and coherent discussion of crytography and the RSA system is included here. The organization of the book's chapters is fantastic. Each chapter builds up on results proven in the previous ones, showing well the connections between the different aspects of Number Theory. The exercises of the book range from simple to challenging, but are all accesible to someone willing to put effort into them.This would be an excelent source for learning number theory for mathematical competition purposes, such as the ASHME, AIME, USAMO, and even for the International Mathematical Olympiad. The book contains much more than what is needed for these competitions, but the olympiad/contest reader will benefit greatly from a study of Davenport's work.The book can certainly be used for an undergraduate course in Number Theory, though it might need supplementary materials, to cover a semester's worth of work. I know the book has been used in the past in previous editions as the main text for Math 124: Number Theory at Harvard University.I would also recommend this book to anyone interested in acquanting themselves with Number Theory.Awesome! There is simply no other word that describes The Higher Arithmetic.