There are a number of major changes in the Fourth Edition. Many new exercises appear throughout the text. A flowchart giving chapter dependencies is included to help instructors choose the most appropriate mix of topics for their students. Content Updates There is a new chapter on mathematical induction (Chapter 26). Some material on proof by contradiction has been moved forward to Chapter 8. It is used in the proof that a polynomial of degree d has at most d roots modulo p. This fact is then used in place of primitive roots as a tool to prove Euler’s quadratic residue formula in Chapter 21. (In earlier editions, primitive roots were used for this proof.) The chapters on primitive roots (Chapters 28–29) have been moved to follow the chapters on quadratic reciprocity and sums of squares (Chapters 20–25). The rationale for this change is the author’s experience that students find the Primitive Root Theorem to be among the most difficult in the book. The new order allows the instructor to cover quadratic reciprocity first, and to omit primitive roots entirely if desired. Chapter 22 now includes a proof of part of quadratic reciprocity for Jacobi symbols, with the remaining parts included as exercises. Quadratic reciprocity is now proved in full. The proofs for (-1/p) and (2/p) remain as before in Chapter 21, and there is a new chapter (Chapter 23) that gives Eisenstein’s proof for (p/q)(q/p). Chapter 23 is significantly more difficult than the chapters that precede it, and it may be omitted without affecting the subsequent chapters. As an application of primitive roots, Chapter 28 discusses the construction of Costas arrays. Chapter 39 includes a proof that the period of the Fibonacci sequence modulo p divides p - 1 when p is congruent to 1 or 4 modulo 5. Number theory is a vast and sprawling subject, and over the years this book has acquired many new chapters. In order to keep the length of this edition to a reasonable size, Chapters 47, 48, 49, and 50 have been removed from the printed version of the book. These omitted chapters are freely available online at http://math.brown.edu/~jhs/frint.html or www.pearsonhighered.com/silverman. The online chapters are included in the index.