A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic (Undergraduate Texts in Mathematics)

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. Preface......Page 3 Contents......Page 8 Notation......Page 11 1.1 The Pythagorean Theorem......Page 13 1.2 Pythagorean triples......Page 16 1.3 The questions......Page 18 2.1 Natural numbers, mathematical induction, and the Well-ordering Principle......Page 23 2.2 Divisibility and prime factorization......Page 24 2.3 The Chinese Remainder Theorem......Page 32 2.4 Euler's Theorem......Page 34 2.5 Polynomials modulo a prime......Page 40 2.6 Digit expansions......Page 42 2.7 Digit expansions of rational numbers......Page 49 2.8 Primitive roots......Page 51 3.1 Solutions......Page 69 3.2 Geometric method to find solutions......Page 71 3.3 Geometric method to find solutions: Non-Pythagorean examples......Page 75 3.4 Application: X4 + Y4 = Z4......Page 80 4.1 Congruent numbers......Page 90 4.2 Small numbers......Page 92 4.3 Connection to cubic equations......Page 93 5.1 The theorem......Page 99 5.2 Gaussian integers......Page 101 5.3 The proof of Theorem 5.2......Page 103 5.4 Irreducible elements in mathbbZ[i]......Page 106 5.5 Proof of Theorem 5.1......Page 107 6.1 Euclid's theorem on the infinitude of primes......Page 113 6.2 Quadratic residues......Page 115 6.3 An application of the Law of Quadratic Reciprocity......Page 120 7.1 Gauss sums and Quadratic Reciprocity......Page 127 7.2 The Jacobi Symbol......Page 132 8.1 The Pythagorean Equation modulo a prime number p......Page 139 8.2 Solutions modulo n for a natural number n......Page 144 9.1 The case of two squares......Page 156 9.2 More than two squares......Page 160 9.3 Integral points on arcs......Page 161 10.1 Lattices in mathbbRn......Page 170 10.2 Minkowski's Theorem......Page 173 10.3 Sums of two squares......Page 177 10.4 Sums of four squares......Page 178 10.5 Sums of three squares......Page 181 11.1 Quaternions......Page 191 11.2 Matrix representation......Page 193 11.3 Four squares......Page 194 12.1 Quadratic forms with integral coefficients......Page 199 12.2 Binary forms......Page 204 12.3 Ternary forms......Page 207 12.4 Three squares......Page 210 13.1 The asymptotic formula......Page 215 13.2 The computation of C2......Page 221 14.1 The real line......Page 231 14.2 The unit circle......Page 244 A.1 Sine, cosine, and exponentials......Page 251 A.2 The Binomial Theorem......Page 252 A.3 The Pigeon-Hole Principle......Page 253 Algebraic Integers......Page 258 SageMath......Page 263 C.1 Basic operations......Page 264 C.2 Basic number theory......Page 265 C.3 Polynomial operations......Page 269 Refs......Page 273 Index......Page 278