Introduction to Diophantine Equations (eBook)

Preface 6
Contents 10
Part I Diophantine Equations 14
I.1 Elementary Methods for Solving Diophantine Equations 15
1.1 The Factoring Method 15
1.2 Solving Diophantine Equations Using Inequalities 25
1.3 The Parametric Method 32
1.4 The Modular Arithmetic Method 41
1.5 The Method of Mathematical Induction 48
1.6 Fermat’s Method of Infinite Descent (FMID) 59
1.7 Miscellaneous Diophantine Equations 70
I.2 Some Classical Diophantine Equations 78
2.1 Linear Diophantine Equations 78
2.2 Pythagorean Triples and Related Problems 87
2.3 Other Remarkable Equations 99
2.3.1. Some Quadratic Diophantine Equations and Related Problems 99
2.3.2. Some Higher-Degree Diophantine Equations 111
I.3 Pell-Type Equations 128
3.1 Pell’s Equation: History and Motivation 129
3.2 Solving Pell’s Equation 132
3.3 The Equation ax2 by2 = 1 146
3.4 The Negative Pell’s Equation 151
I.4 Some Advanced Methods for Solving Diophantine Equations 157
4.1 The Ring Z[i] of Gaussian Integers 161
4.2 The Ring of Integers of Q[vd] 172
4.3 Quadratic Reciprocity and Diophantine Equations 188
4.4 Divisors of Certain Forms 191
4.4.1 Divisors of a2 +b2 192
4.4.2 Divisors of a2 +2b2 196
4.4.3 Divisors of a2 2b2 198
Part II Solutions to Exercises and Problems 201
II.1 Solutions to Elementary Methods for Solving Diophantine Equations 202
1.1 The Factoring Method 202
1.2 Solving Diophantine Equations Using Inequalities 211
1.3 The Parametric Method 222
1.4 The Modular Arithmetic Method 228
1.5 The Method of Mathematical Induction 238
1.6 Fermat’s Method of Infinite Descent (FMID) 248
1.7 Miscellaneous Diophantine Equations 262
II.2 Solutions to Some Classical Diophantine Equations 273
2.1 Linear Diophantine Equations 273
2.2 Pythagorean Triples and Related Problems 281
2.3 Other Remarkable Equations 286
II.3 Solutions to Pell-Type Equations 296
3.1 Solving Pell’s Equation by Elementary Methods 296
3.2 The Equation ax2 by2 = 1 305
II.4 Solutions to Some Advanced Methods in Solving Diophantine Equations 315
4.1 The Ring Z[i] of Gaussian Integers 315
4.2 The Ring of Integers of Q[vd] 320
4.3 Quadratic Reciprocity and Diophantine Equations 328
4.4 Divisors of Certain Forms 330
References 333
Glossary 337
Index 346