Diophantine Equations (eBook)

Front Cover 1
Diophantine Equations 4
Copyright Page 5
Contents 10
Preface 6
Chapter 1. Introduction 14
Chapter 2. Equations Proved Impossible by Congruence Considerations 16
Chapter 3. Equations Involving Sums of Squares 25
Chapter 4. Quartic Equations with only Trivial Solutions 29
Chapter 5. Some Linear Equations 43
Chapter 6. Properties of Congruences 47
Chapter 7. Homogeneous Equations of the Second Degree 55
Chapter 8. Pell's Equation 66
Chapter 9. Rational Solutions Derived from Given Ones 79
Chapter 10. Rational Points on Some Cubic Curves 89
Chapter 11. Rational Points on Cubic Surfaces 95
Chapter 12. Rational and Integer Points on Quartic Surfaces 103
Chapter 13. Integer Solutions of Some Cubic Equations in Three Variables 113
Chapter 14. Simple Algebraic Considerations 124
Chapter 15. Applications of Algebraic Number Theory 134
Chapter 16. Finite Basis Theorem for the Rational Points on a Cubic Curve f(x, y , z) = 0 of Genus one 151
Chapter 17. Rational Points on Curves of Genus g = 0 or 1 and g > 1
Chapter 18. Representation of Numbers by Homogeneous Forms in Two Variables 170
Chapter 19. Representation of Numbers by Special Binary Quadratic and Quaternary Quadratic Forms 177
Chapter 20. Representation of Numbers by Homogeneous Forms in Several Variables 187
Chapter 21. Representation of Numbers by Polynomials 194
Chapter 22. Thue's Theorem on the Integer Solutions of f(x, y ) = m 199
Chapter 23. Local Methods or p-Adic Applications 213
Chapter 24. Binary Cubic Forms 226
Chapter 25. Binary Quartic Forms 245
Chapter 26. The Equation y2 = x3 + k 251
Chapter 27. The equation y2 = ax3 + bx2 + cx + d 268
Chapter 28. Some Equations of Degree > 3
Chapter 29. Fermat's Last Theorem 293
Chapter 30. Miscellaneous Results 300
Bibiography 319
List of Equations and Congruences 320
Pure and Applied Mathematics 326
In preparation 327