Many new discoveries, both theoretical and numerical, are introduced. Coverage includes four Mersenne primes, numerous new world records, and the latest evidence supporting open conjectures. Recent theoretical discoveries are described, including the Tao-Green theorem about arbitrarily long arithmetic progressions of primes. New biographies of Terence Tao, Etienne Bezout, Norman MacLeod Ferrers, Clifford Cocks, and Waclaw Sierpinski supplement the already rich collection of biographies in the book. This edition also includes historical information about secret British cryptographic discoveries that predate the work of Rivest, Shamir, and Adelman. Expanded treatment of both resolved and open conjectures about prime numbers is provided. Combinatorial number theory -partitions are covered in a new section of the book. This provides an introduction to combinatorial number theory, which was not covered in previous editions.

This new section covers many aspects of this topics including Ferrers diagrams, restricted partition identities, generating functions, and the famous Ramanujan congruences. Partition identities are proved using both generating functions and bijections. Congruent numbers and elliptic curves -a new section is devoted to the famous congruent number problem, which asks which positive integers are the area of a right triangle with rational side lengths. This section shows that the congruent number problem is equivalent to finding rational points on certain elliptic curves and introduces some basic properties of elliptic curves. The use of geometric reasoning in the solution of diophantine problems has been added to the new edition. In particular, finding rational points on the unit circle is shown to be equivalent to finding Pythgaorean triples. Finding rational triangles with a given integer as area is shown to be equivalent to finding rational points on an associated elliptic curve. Greatest common divisors are now defined in Chapter 1.

The terminology on Bezout coefficients is now introduced in Chapter 3, where properties of greatest common divisors are developed. An expanded discussion on the usefulness of the Jacobi symbol in evaluating Legendre symbols is now provided. Extensive revisions to the already-strong exercise sets include several hundred new exercises, ranging from routine to challenging. In particular, there are many new and revised computational exercises. The Companion Website for this edition (www .pearsonhighered.com/rosen) has been considerably expanded. Among the new features are an expanded collection of applets, a manual for using computational engines to explore number theory, and a Web page devoted to number theory news.