In 1844, Catalan conjectured that 8 and 9 were the only consecutive integral powers. The problem of consecutive powers is more easily grasped than Fermats last theorem, so recently vanquished. In this book, Paulo Ribenboim brings together for the first time the diverse approaches to proving Catalans conjecture. These range from the elementary methods that apply in special cases (Part A) to powerful and general results in diophantine approximation (Part C) which yieldthe strongest results to date.
In 1976, Tijdeman found a computable bound above which there are no consecutive powers. Langerin reduced this to an exponential: exp (exp (exp (exp (730)))). The enormous gap between such bonds-and the only known consecutive powers, 8 and 9-has been narrowed, but its immensity suggests the difficulty of such problems, as well as the interesting and fruitful methods developed for attacking them.
Ribenboims book will appeal to anyone interested in how number theorists attack difficult problems. Its concrete focus on an easy to state problem is a relief from such generalities as the related A, B, C conjecture, and will prove to be a good point ofdeparture for seminars in number theory. The text includes many beautiful results of classical number theory not found in any other book. The treatment is fully accessible and self-contained, making this book as attractive as the authors
13 Lectures on Fermats Last Theorem, and his book,
Book of Prime Records.