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| Preface | ||
| Reader's Guide | ||
| Introduction | 1 | |
| History | 5 | |
| Pt. P | Preliminaries | 9 |
| 1 | Binomials and Cyclotomic Polynomials | 9 |
| 2 | The Cyclotomic Field | 27 |
| 3 | The Pythagorean Equation, Special Cases of Fermat's Last Theorem and Related Equations | 31 |
| 4 | Continued Fractions | 55 |
| 5 | The Equations EX[superscript 2] - DY[superscript 2] = [actual symbol not reproducible]C | 60 |
| Pt. A | Special Cases | 67 |
| 1 | Preliminary Lemmas | 67 |
| 2 | The Sequence of Squares or Cubes | 69 |
| 3 | The Equation X[superscript m] - Y[superscript 2] = 1 | 78 |
| 4 | The Result of Stormer on Fermat's Equation | 80 |
| 5 | The Attempts to Solve X[superscript 2] - Y[superscript n] = 1 | 89 |
| 6 | The Equation X[superscript 2] - Y[superscript n] = 1, n [actual symbol not reproducible] 3 | 92 |
| 7 | The Equations X[superscript 3] - Y[superscript n] = 1 and X[superscript m] - Y[superscript 3] = 1, with m, n [actual symbol not reproducible] 3 | 96 |
| 8 | The Equation [actual symbol not reproducible] | 110 |
| 9 | The Sequence of Powers of 2 or 3 | 124 |
| 10 | Interlude | 127 |
| 11 | The Equation 2X[superscript n] - 1 = Z[superscript 2] | 129 |
| 12 | [pi] and Grave's Problem | 132 |
| 13 | A Problem of Fermat on Pythagorean Triangles and the Equation 2X[superscript 4] - Y[superscript 4] = Z[superscript 2] | 144 |
| 14 | The Equations [actual symbol not reproducible] and [actual symbol not reproducible] | 164 |
| 15 | Representation of Integers by Binary Cubic Forms | 177 |
| 16 | Some Quartic Equations | 192 |
| Pt. B | Divisibility Conditions | 201 |
| 1 | Getting the Consecutive Powers 8 and 9 | 201 |
| 2 | The Theorem of Cassels and First Consequences | 204 |
| 3 | Prime Factors of Solutions of Catalan's Equation | 214 |
| 4 | The Theorem of Hyyro | 216 |
| 5 | The Theorems of Inkeri | 219 |
| Pt. C | Analytical Methods | 241 |
| 1 | Some General Theorems for Diophantine Equations | 241 |
| I | The Equation X[superscript m] - Y[superscript n] = 1 | 247 |
| 2 | Upper Bounds for the Number and Size of Solutions | 247 |
| 3 | Lower Bounds for Solutions | 256 |
| 4 | Algorithm to Determine the Eventual Solutions | 266 |
| II | The Equation a[superscript U] - b[superscript V] = 1 | 269 |
| 5 | What Will Be Discussed | 269 |
| 6 | Finiteness of the Number of Solutions | 271 |
| 7 | Algorithm to Determine the Eventual Solutions | 282 |
| 8 | The Largest Prime Factor of Values of Quadratic Polynomials | 283 |
| 9 | Effective Results | 288 |
| III | The Equation X[superscript U] - Y[superscript V] = 1 | 298 |
| 10 | The Theorem of Tijdeman | 298 |
| 11 | A Density Result | 308 |
| Appendix 1. Catalan's Equation in Other Domains | 313 | |
| (A) | Catalan's Equation over Number Fields | 313 |
| (B) | Catalan's Equation over Fields K(t) and Domains K[t] | 314 |
| (C) | Catalan's Equation Over Function Fields of Projective Varieties | 318 |
| Appendix 2. Powerful Numbers | 319 | |
| (A) | Distribution of Powerful Numbers | 320 |
| (B) | Additive Problems | 322 |
| (C) | Difference Problems | 323 |
| Bibliography | 331 | |
| Index of Names | 359 | |
| Subject Index | 363 |