Catalans Conjecture: Are 8 and 9 the Only Consecutive Powers?

Pt. P. Preliminaries. 1. Binomials And Cyclotomic Polynomials. 2. The Cyclotomic Field. 3. The Pythagorean Equation, Special Cases Of Fermat's Last Theorem And Related Equations. 4. Continued Fractions. 5. The Equations Ex[superscript 2] -- Dy[superscript 2] = [actual Symbol Not Reproducible]c -- Pt. A. Special Cases. 1. Preliminary Lemmas. 2. The Sequence Of Squares Or Cubes. 3. The Equation X[superscript M] -- Y[superscript 2] = 1. 4. The Result Of Stormer On Fermat's Equation. 5. The Attempts To Solve X[superscript 2] -- Y[superscript N] = 1. 6. The Equation X[superscript 2] -- Y[superscript N] = 1, N [actual Symbol Not Reproducible] 3. 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. 8. The Equation [actual Symbol Not Reproducible]. 9. The Sequence Of Powers Of 2 Or 3. 10. Interlude. 11. The Equation 2x[superscript N] -- 1 = Z[superscript 2]. 12. [pi] And Grave's Problem. 13. A Problem Of Fermat On Pythagorean Triangles And The Equation 2x[superscript 4] -- Y[superscript 4] = Z[superscript 2]. 14. The Equations [actual Symbol Not Reproducible] And [actual Symbol Not Reproducible]. 15. Representation Of Integers By Binary Cubic Forms. 16. Some Quartic Equations -- Pt. B. Divisibility Conditions. 1. Getting The Consecutive Powers 8 And 9. 2. The Theorem Of Cassels And First Consequences. 3. Prime Factors Of Solutions Of Catalan's Equation. 4. The Theorem Of Hyyro. 5. The Theorems Of Inkeri -- Pt. C. Analytical Methods. 1. Some General Theorems For Diophantine Equations. I. The Equation X[superscript M] -- Y[superscript N] = 1. 2. Upper Bounds For The Number And Size Of Solutions. 3. Lower Bounds For Solutions. 4. Algorithm To Determine The Eventual Solutions. Ii. The Equation A[superscript U] -- B[superscript V] = 1. 5. What Will Be Discussed. 6. Finiteness Of The Number Of Solutions. 7. Algorithm To Determine The Eventual Solutions. 8. The Largest Prime Factor Of Values Of Quadratic Polynomials. 9. Effective Results. Iii. The Equation X[superscript U] -- Y[superscript V] = 1. 10. The Theorem Of Tijdeman. 11. A Density Result -- Appendix 1. Catalan's Equation In Other Domains. (a). Catalan's Equation Over Number Fields. (b). Catalan's Equation Over Fields K(t) And Domains K[t]. (c). Catalan's Equation Over Function Fields Of Projective Varieties -- Appendix 2. Powerful Numbers. (a). Distribution Of Powerful Numbers. (b). Additive Problems. (c). Difference Problems. Paulo Ribenboim. Includes Bibliographical References And Indexes.