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xxTable of Contents14.6.5 The Equation x 6 − y 4 = z 2 . . . . . . . . . . . . . . . . . . . . . . . . 109914.6.6 The Equation x 4 − y 6 = z 2 . . . . . . . . . . . . . . . . . . . . . . . . 109914.6.7 The Equation x 6 + y 4 = z 2 . . . . . . . . . . . . . . . . . . . . . . . . 110114.6.8 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110114.7 Applications of Mason’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 110314.7.1 Mason’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110314.7.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110414.8 Exercises for Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110515. The Modular Approach to Diophantine Equations . . . . . . . .110715.1 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110715.1.1 Introduction and Necessary Software Tools . . . . . . . . . . 110715.1.2 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110815.1.3 Rational Newforms and Elliptic Curves . . . . . . . . . . . . . 110915.2 Ribet’s Level-Lowering Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 111015.2.1 Definition of “Arises From” . . . . . . . . . . . . . . . . . . . . . . . . 111015.2.2 Ribet’s Level-Lowering Theorem . . . . . . . . . . . . . . . . . . . 111115.2.3 Absence of Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111315.2.4 How to use Ribet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 111515.3 Fermat’s Last Theorem and Similar Equations . . . . . . . . . . . . . 111515.3.1 A Generalization of FLT . . . . . . . . . . . . . . . . . . . . . . . . . . 111615.3.2 E Arises from a Curve with Complex Multiplication . . 111715.3.3 End of the Proof of Theorem 15.3.1 . . . . . . . . . . . . . . . . . 111815.3.4 The Equation x 2 = y p + 2 r z p for p 7 and r 2 . . . . . 111915.3.5 The Equation x 2 = y p + z p for p 7 . . . . . . . . . . . . . . . . 112115.4 An Occasional Bound for the Exponent . . . . . . . . . . . . . . . . . . . 112115.5 An Example of Serre–Mazur–Kraus .. . . . . . . . . . . . . . . . . . . . . . 112315.6 The Method of Kraus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112615.7 “Predicting Exponents of Constants” . . . . . . . . . . . . . . . . . . . . . 112915.7.1 The Diophantine Equation x 2 − 2 = y p . . . . . . . . . . . . . . 112915.7.2 Application to the SMK Equation . . . . . . . . . . . . . . . . . . 113315.8 Recipes for Some Ternary Diophantine Equations. . . . . . . . . . . 113415.8.1 Recipes for Signature (p, p, p) . . . . . . . . . . . . . . . . . . . . . . 113515.8.2 Recipes for Signature (p, p, 2) . . . . . . . . . . . . . . . . . . . . . . 113615.8.3 Recipes for Signature (p, p, 3) . . . . . . . . . . . . . . . . . . . . . . 113816. Catalan’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114116.1 Mihăilescu’s First Two Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 114116.1.1 The First Theorem: Double Wieferich Pairs . . . . . . . . . . 114216.1.2 The Equation (x p − 1)/(x − 1) = py q . . . . . . . . . . . . . . . 114416.1.3 Mihăilescu’s Second Theorem: p | h − q and q | h − p . . . . . . 114816.2 The + and − Subspaces and the Group S . . . . . . . . . . . . . . . . . 114916.2.1 The + and − Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 115016.2.2 The Group S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115216.3 Mihăilescu’s Third Theorem: p < 4q 2 and q < 4p 2 . . . . . . . . . . 1154
Table of Contentsxxi16.4 Mihăilescu’s Fourth Theorem: p ≡ 1 (mod q) or q ≡ 1 (mod p)115916.4.1 Preliminaries on Commutative Algebra . . . . . . . . . . . . . . 115916.4.2 Preliminaries on the Plus Part . . . . . . . . . . . . . . . . . . . . . 116116.4.3 Cyclotomic Units and Thaine’s Theorem . . . . . . . . . . . . 116416.4.4 Preliminaries on Power Series . . . . . . . . . . . . . . . . . . . . . . 116616.4.5 Proof of Mihăilescu’s Fourth Theorem . . . . . . . . . . . . . . 116916.4.6 Conclusion: Proof of Catalan’s Conjecture . . . . . . . . . . . 1172Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1173Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1182Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1182General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1189
Page 5 and 6: vKoblitz formula for Morita’s p-a
Page 8: viiiinstance) A(X) = A d X d +A d
Page 11 and 12: Table of Contentsxi3.6.1 Introducti
Page 13 and 14: Table of Contentsxiii6.3.1 The Gene
Page 15 and 16: Table of Contentsxv8.2.3 The Fundam
Page 17 and 18: Table of Contentsxvii10.2.2 The Use
Page 19: Table of Contentsxix12.8.3 A Genera

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