ABSTRACT
I. Introduction 92II. General Number Theory 92A. General texts 92B. Major journals and article sources 98C. Supplements: reference tools, study guides, forums, etc. 99III. Algebraic Number Theory 99A. Global theory of algebraic numbers 100B. Local theory of algebraic numbers 101C. Advanced algebraic number theory 102IV. Arithmetic Algebraic Geometry 103A. General sources for arithmetic on curves 103B. Background on algebraic curves 104C. Elliptic functions and elliptic curves 105D. Equations over finite fields 106V. Analytic Number Theory 107A. General sources 107B. Distribution of prime numbers 108C. Advanced topics: zeta functions, modularand automorphic forms 109VI. Selected Topics 110A. Irrational and transcendental numbers andDiophantine approximation 110B. Quadratic forms 111 91
C. Geometry of numbers 112D. Partitions of a number and additive number theory 113E. Diophantine equations 113F. Prime numbers 114VII. Probabilistic Number Theory 114VIII. Applied and Algorithmic Number Theoryand Cryptography 114IX. History and Texts with a Historical Perspective 115X. Historically Important Works 117
Number theory is the study of those properties of the integers and rational numbers that go beyond the ordinary manipulations of everyday arithmetic. It is the oldest branch of mathematics, tracing back to the Babylonians, and it is also one of the most active areas of current research. It has always been appealing to mathematical laymen because of the simple-sounding nature of many of its deepest theorems and conjectures. For example, the statement of Fermat’s Last Theorem can be understood by a high school student, but Wiles’s proof is as deep and difficult as they come. Another great appeal of the subject is that it is possible to study it at any level, from high school student up to research mathematician. There are important theorems at every level.The resources discussed here were chosen to cover the broad range of topics within number theory at a variety of levels. Our organization of topics has a certain arbitrariness since a strong aspect of modern number theory is the interrelation of the topics. Recommendations and qualitative comments represent the views of the chapter authors; some descriptive comments are drawn from other sources, mainly the books themselves. II. GENERAL NUMBER THEORY
The following six recommended books, listed approximately in order of increasing depth and difficulty, together provide an orientation to the whole field. Some alternative treatments of the same topics are given later.
