Introduction to Number Theory: (Textbooks in Mathematics)

One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics. This classroom-tested, student-friendly text covers a wide range of subjects, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments that include cryptography, the theory of elliptic curves, and the negative solution of Hilbert’s tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Euler’s theorem in RSA encryption, and quadratic residues in the construction of tournaments. The book interweaves the theoretical development of the material with Mathematica® and Maple™ calculations while giving brief tutorials on the software in the appendices. Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory.

Table of Contents:
Core Topics Introduction What is number theory? The natural numbers Mathematical induction Divisibility and Primes Basic definitions and properties The division algorithm Greatest common divisor The Euclidean algorithm Linear Diophantine equations Primes and the fundamental theorem of arithmetic Congruences Residue classes Linear congruences Application: Check digits and the ISBN system Fermat’s theorem and Euler’s theorem The Chinese remainder theorem Wilson’s theorem Order of an element mod n Existence of primitive roots Application: Construction of the regular 17-gon Cryptography Monoalphabetic substitution ciphers The Pohlig–Hellman cipher The Massey–Omura exchange The RSA algorithm Quadratic Residues Quadratic congruences Quadratic residues and nonresidues Quadratic reciprocity The Jacobi symbol Application: Construction of tournaments Consecutive quadratic residues and nonresidues Application: Hadamard matrices Further Topics Arithmetic Functions Perfect numbers The group of arithmetic functions Möbius inversion Application: Cyclotomic polynomials Partitions of an integer Large Primes Prime listing, primality testing, and prime factorization Fermat numbers Mersenne numbers Prime certificates Finding large primes Continued Fractions Finite continued fractions Infinite continued fractions Rational approximation of real numbers Periodic continued fractions Continued fraction factorization Diophantine Equations Linear equations Pythagorean triples Gaussian integers Sums of squares The case n = 4 in Fermat’s last theorem Pell’s equation Continued fraction solution of Pell’s equation The abc conjecture Advanced Topics Analytic Number Theory Sum of reciprocals of primes Orders of growth of functions Chebyshev’s theorem Bertrand’s postulate The prime number theorem The zeta function and the Riemann hypothesis Dirichlet’s theorem Elliptic Curves Cubic curves Intersections of lines and curves The group law and addition formulas Sums of two cubes Elliptic curves mod p Encryption via elliptic curves Elliptic curve method of factorization Fermat’s last theorem Logic and Number Theory Solvable and unsolvable equations Diophantine equations and Diophantine sets Positive values of polynomials Logic background The negative solution of Hilbert’s tenth problem Diophantine representation of the set of primes APPENDIX A: Mathematica Basics APPENDIX B: Maple Basics APPENDIX C: Web Resources APPENDIX D: Notation References Index Notes appear at the end of each chapter.