A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic

Description

Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed. Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath. By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters. Front Matter ....Pages i-xviii Front Matter ....Pages 1-1 Introduction (Ramin Takloo-Bighash)....Pages 3-12 Basic number theory (Ramin Takloo-Bighash)....Pages 13-58 Integral solutions to the Pythagorean Equation (Ramin Takloo-Bighash)....Pages 59-79 What integers are areas of right triangles? (Ramin Takloo-Bighash)....Pages 81-89 What numbers are the edges of a right triangle? (Ramin Takloo-Bighash)....Pages 91-104 Primes of the form \(4k+1\) (Ramin Takloo-Bighash)....Pages 105-118 Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol (Ramin Takloo-Bighash)....Pages 119-130 Front Matter ....Pages 131-131 Counting Pythagorean triples modulo an integer (Ramin Takloo-Bighash)....Pages 133-149 How many lattice points are there on a circle or a sphere? (Ramin Takloo-Bighash)....Pages 151-164 What about geometry? (Ramin Takloo-Bighash)....Pages 165-185 Another proof of the four squares theorem (Ramin Takloo-Bighash)....Pages 187-194 Quadratic forms and sums of squares (Ramin Takloo-Bighash)....Pages 195-210 How many Pythagorean triples are there? (Ramin Takloo-Bighash)....Pages 211-226 How are rational points distributed, really? (Ramin Takloo-Bighash)....Pages 227-246 Back Matter ....Pages 247-279