Introduction To The Theory Of Numbers (Dover Books on Mathematics) by Harold N. Shapiro (PDF)

Description

Geared toward advanced undergraduates and graduate students, this text starts with the fundamentals of number theory and advances to an intermediate level.It explores evolutions from the notion of congruence, examines a variety of applications related to counting problems, and develops the roots of number theory. 1983 edition.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Andrew Wile’s said that the first text he reaches for when first getting acquainted with specific area of number theory is Davenport’s “Higher Arithmetic.” I think many feel similarly and that book is coincidentally the text that I first used.However, as I learned more number theory, beyond Davenport’s work, it has always struck me how unfortunate it was that Davenport didn’t discuss things like Bertrand’s postulate or any of the elementary results using calculus of a real variable. As a first go at these topics, Shapiro’s work is unparalleled in my opinion – I simply can’t recommend it highly enough. As I’ve pointed out in other reviews, no amount of mathematical rigor can substitute for clarity of thought and clarity of prose. Though H. N. Shapiro seems to have moved away from number theory in recent years (in terms of publications), it’s clear that he was, minimally, a dedicated student of the subject and uncommonly able see all the ideas with 20/20 vision, through what is often encumbered formalisms. Many proofs and theoretical explanations I have not seen written about in such a clear manner anywhere else, undoubtedly.In addition to the quality of writing, I also want to point out the excellent and quite rare selection of topics. There are discussions of many things other books ignore at an elementary level, such as the divisor function for polynomials and Dirichlet’s divisor problem for arithmetic progressions. Also, in one place are the three most notable proofs of Bertrand’s postulate with an interesting discussion on how they relate. There are also some nice and surprisingly broad discussions of algebraic topics.Again, this is a fantastic book that seems to suffer from undue neglect.

⭐

⭐Die Zahlen…yes, maybe it was God himself who created them, as Kronecker once declared. I always felt a kind of religious fear of (interger) numbers. But in the last years, I have realised that there are a handful of affordable introductions to number theory, number theoy for dummies like me (for example Pettofrezzo-Byrkit’s

⭐

⭐, or LeVeque’s

⭐

⭐). This book is not exactly that kind, but I must declare that it is one that has much to admire and to enjoy, even at a distance. The author is a wellknown researcher and scholar, his Thesis advisor was Emil Artin. But he has avoided to cite his own work (for example, there is no mention in the book to his celebrated Tauberian theorem, which appears in Apostol’s

⭐

⭐) section 4.6). Erudition, elegance, freshnesh and an incredible amount of originality are offered here in spades. Starting from such simple things as GCD and unique factorization, the book leads us to arithmetic functions and shows that they form a valuated integral domain, where convolution is the product. In this context, the properties of some conspicuous arithmetic functions (like Euler’s phi or Moebius’ mu) seem no more extravagant oddities, but just pearls on a crown, on a ring. But there is no overkill. The algebraic hardcore is always under control, and the link with the glorious history of number theory is permanent. Of course, a little group theory is used for better understanding of congruences and primes in arithmetic progression. The chinese remainders theorem is offered in full but crystal clear generality. Simultaneous polynomial congruences are then introduced. A chapter on quadratic congruences includes two proofs of the reciprocity law. The rest of the book gives an elmentary proof of the prime number theorem and the Dirichlet theorem on primes in an arithmeic progression, which is proved in two different ways. Another jewel is the proof of a Theorem by I. Shur on the existence of primes in “integer intervals of arbitrry lenght”.

⭐

⭐Although I bought this gem years ago at a Chicago Borders (oh, Borders…), I plan to buy a backup or two through Amazon in the near future — even while my copy, though well-used, is in pristine shape. That’s just how I treat my most treasured books, and this is indeed among them. The other reviews here splendidly articulate all of this work’s virtues, so I needn’t rehearse them again; let me just reinforce the other reviewers’ comments by stating that this work does an excellent job of carrying its student along from simple fundamentals, through all of the elements of number theory, into an amazingly thorough and comprehensible rendering of the Prime Number Theorem. Every theorem is deduced with perfect clarity, and every example is useful. I cannot recommend this book highly enough: if you are interested in studying number theory beyond the basics (well-covered in works by LeVeque, Dudley and Andrews), but do not plan on postdoctoral work in mathematics (i.e. if mathematics is recreational and aesthetic), this is for you.

⭐

Keywords

Free Download Introduction To The Theory Of Numbers (Dover Books on Mathematics) in PDF format
Introduction To The Theory Of Numbers (Dover Books on Mathematics) PDF Free Download
Download Introduction To The Theory Of Numbers (Dover Books on Mathematics) 2008 PDF Free
Introduction To The Theory Of Numbers (Dover Books on Mathematics) 2008 PDF Free Download
Download Introduction To The Theory Of Numbers (Dover Books on Mathematics) PDF
Free Download Ebook Introduction To The Theory Of Numbers (Dover Books on Mathematics)