Proofs in Number Theory: Sums and Products

Begin Your Journey in Number Theory How do you learn to write proofs in number theory? This book teaches how to write direct proofs in number theory. It uses finite sums and finite products to introduce the essential concepts. Being a book about proofs, it contains over 300 detailed proofs. Occasionally, two distinct proofs are given for a particular theorem. Corollaries are proven as a warm-up to the main theorem. Thus, the reader is always prepared to understand the ideas used in the proof of a theorem. Number theory is important in the modern world. It has many applications in technology and computing. For example, it enables the creation of algorithms that provide a safe internet. This book is an introductory text for number theory. It was written to make number theory accessible. Accordingly, the prerequisite is merely secondary school math. The target audience is high schoolers. The text contains the few topics in number theory that can be learned without a deep understanding of logic. A minimalist approach is employed: each concept introduced is either essential or useful for explaining some essential concept. As a reader, you may not always know the purpose or relevance of each section. Thus, each section begins with a paragraph that explains why the section is important and what it contains. There are over 1000 exercises in the book, spread across each chapter. They test comprehension, develop the reader’s problem-solving skills, and offer an opportunity to practice number theory. Some exercises contain hints to their solution. Method of Proof Traditionally, the first method of proof number theory students learn is proof by induction. This method of proof is quite powerful. However, it should not be the first method of proof a student learns. Direct proof is the simplest type of proof. It is easier for a beginner to understand. This book replaces inductive proofs with direct proofs to make number theory accessible. All theorems in the book are proven using direct proofs. This book proves the validity of various formulas of sums and products in number theory: sum of positive integers sum of cubes product of triangular numbers product of odd numbers product of predecessors of squares telescoping sums alternating sums telescoping products —- ISBN: 9798838984449 | ISBN10: B0B5KXGS6J | ISBN-13: 979-8838984449