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This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Number theory Textbooks
 Resource Type:
 ebook

ebook
This version of YAINTT has a particular emphasis on connections to cryptology. The cryptologic material appears in Chapter 4 and §§5.5 and 5.6, arising naturally (I hope) out of the ambient number theory. The main cryptologic applications – being the RSA cryptosystem, DiffieHellman key exchange, and the ElGamal cryptosystem – come out so naturally from considerations of Euler’s Theorem, primitive roots, and indices that it renders quite ironic G.H. Hardy’s assertion [Har05] of the purity and eternal inapplicability of number theory. Note, however, that once we broach the subject of these cryptologic algorithms, we take the time to make careful definitions for many cryptological concepts and to develop some related ideas of cryptology which have much more tenuous connections to the topic of number theory. This material therefore has something of a different flavor from the rest of the text – as is true of all scholarly work in cryptology (indeed, perhaps in all of computer science), which is clearly a discipline with a different culture from that of “pure”mathematics. Obviously, these sections could be skipped by an uninterested reader, or remixed away by an instructor for her own particular class approach.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Number theory Textbooks
 Resource Type:
 ebook

ebook
All of the mathematics required beyond basic calculus is developed “from scratch.” Moreover, the book generally alternates between “theory” and “applications”: one or two chapters on a particular set of purely mathematical concepts are followed by one or two chapters on algorithms and applications; the mathematics provides the theoretical underpinnings for the applications, while the applications both motivate and illustrate the mathematics. Of course, this dichotomy between theory and applications is not perfectly maintained: the chapters that focus mainly on applications include the development of some of the mathematics that is specific to a particular application, and very occasionally, some of the chapters that focus mainly on mathematics include a discussion of related algorithmic ideas as well. The mathematical material covered includes the basics of number theory (including unique factorization, congruences, the distribution of primes, and quadratic reciprocity) and of abstract algebra (including groups, rings, fields, and vector spaces). It also includes an introduction to discrete probability theory—this material is needed to properly treat the topics of probabilistic algorithms and cryptographic applications. The treatment of all these topics is more or less standard, except that the text only deals with commutative structures (i.e., abelian groups and commutative rings with unity) — this is all that is really needed for the purposes of this text, and the theory of these structures is much simpler and more transparent than that of more general, noncommutative structures. There are a few sections that are marked with a “(∗),” indicating that the material covered in that section is a bit technical, and is not needed else where. There are many examples in the text, which form an integral part of the book, and should not be skipped. There are a number of exercises in the text that serve to reinforce, as well as to develop important applications and generalizations of, the material presented in the text. Some exercises are underlined. These develop important (but usually simple) facts, and should be viewed as an integral part of the book. It is highly recommended that the reader work these exercises, or at the very least, read and understand their statements. In solving exercises, the reader is free to use any previously stated results in the text, including those in previous exercises. However, except where otherwise noted, any result in a section marked with a “(∗),” or in §5.5, need not and should not be used outside the section in which it appears. There is a very brief “Preliminaries” chapter, which fixes a bit of notation and recalls a few standard facts. This should be skimmed over by the reader. There is an appendix that contains a few useful facts; where such a fact is used in the text, there is a reference such as “see §An,” which refers to the item labeled “An” in the appendix.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Number theory Algebra Textbooks Computer science  Mathematics
 Resource Type:
 ebook

Courseware
This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Number theory Algebraic number theory
 Resource Type:
 Courseware