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An Introduction to the Theory of Numbers
Ivan M. Niven, Herbert S. Zuckerman, and Hugh L. Montgomery
Publisher: John Wiley (1991)
Details: 544 pages, Hardcover
Topics: Elementary Number Theory
This book is in the MAA's basic library list.
MAA Review[Reviewed by Allen Stenger, on 12/23/2008]
This undergraduate textbook is a comprehensive survey of everything that might be considered elementary number theory. The approach emphasizes breadth rather than depth, but some deep results are covered as multi-part exercises. It is a modern look at number theory and (despite being published in 1991) is very much up-to-date; the only recent developments that might be added today would be the solutions of Fermat's Last Theorem and of Catalan's conjecture. There's a moderate amount of numeric work, including some modern factorization methods such as Pollard's rho and elliptic curve factorization.
The authors often work up to a difficult theorem by proving a simpler version and explaining the strategy being followed. For example they start working on Schnirelmann's and Mann's theorems on sums of sets of integers by first proving that every integer > 1 is the sum of two square-free integers. This is a neat result in itself and surprisingly simple to prove. They quote but do not prove Dirichlet's theorem that there are an infinity of primes in an arithmetic progression, but they do prove it for common difference 4, and the proof uses all the ingredients of the full proof so you get a good understanding of how it works.
The exercises are especially good, and approach the subject from several viewpoints. There are simple numeric exercises to verify proved theorems and proof problems of various difficulties. There are even sketches of some quite advanced theorems, for example, I. M. Vinogradov's theorem that the error term in the Dirichlet divisor problem is O(x1/3(ln x)2) (the text proves O(x1/2)). There's a topological proof that there are an infinity of primes. Be sure to read the exercises; some of the most interesting results are there!
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
BLL*** — The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.