Catalan’s conjecture. Are 8 and 9 the only consecutive powers? (English) Zbl 0824.11010

Ribenboim, Paulo

London: Academic Press. xiv, 364 p. (1994).

With the solution of Fermat’s last theorem, the second most notorious unsolved problem in diophantine equations is Catalan’s conjecture: Does the equation \(x^ p- y^ q =1\) have any solutions in integers \(x\), \(y\) and primes \(p\), \(q\) other than \((3,2,2,3)\)?
Since Catalan proposed this in 1844 many famous mathematicians have worked on the problem. The problem may be attacked from two directions. The first is to find an upper bound for \(p\) and \(q\). In 1976, R. Tijdeman and M. Langevin [Acta Arith. 29, 197-209 (1976; Zbl 0316.10008), respectively Unpublished Manuscript, 1976], in a celebrated paper, established explicit upper bounds for \(x\), \(y\), \(p\), \(q\) by using Baker’s theory of linear forms in logarithms of algebraic numbers. Since then successive improvements in Baker’s constants have yielded successive improvements in these bounds. Currently, the best known bounds for \(p\) and \(q\) to date are \(10^{18}\) and \(10^{13}\) respectively. These were found by Tim O’Neil, a PhD student in mathematics at BGSU (submitted), who improved some results of Bennet et al. on linear forms in logarithms of three algebraic numbers (submitted) to obtain them.
The problem is also being attacked from below. Here one must resort to algebraic number theory to make any headway. The use of ideals and class numbers of the relevant fields is essential in this regard. Using this approach K. Inkeri [Acta Arith. 9, 285-290 (1964; Zbl 0127.271); J. Number Theory 34, 142-152 (1990; Zbl 0699.10029)] was able to rule out many \(p\) and \(q\) for which Catalan’s equation admits a nontrivial solution. He also gave criteria for those \(p\) and \(q\) for which a solution can exist. However, these lead to questions about the class numbers of cyclotomic fields, which can be very large and difficult to compute. M. Mignotte (to appear) improved the latter criterion by showing that one only needs to compute the class number of the maximal 2 subfield of the cyclotomic field. Again, this leads to a difficult criterion: The class numbers of these fields may be split into two factors, the first factor and the relative class number. The first factor is, in general, very difficult to compute. However, W. Schwarz (to appear) circumvented this problem by showing that it suffices just to compute the relative class number of the relevant fields.
Though it has been possible to compute these relative class numbers for the last 50 years, all known results involved obtaining two large numbers and cancelling out their common factors. This made implementation of the algorithm on a computer quite difficult. The theory was developed by Borevich and Shafarevich in Russia and Hasse in Germany. Recently, in a fine Masters thesis at Bowling Green State University, Rob Clother was able to merge their ideas into a single formula, thus taking advantage of both viewpoints. Further, he used some ideas of Kummer from the 1850’s to eliminate the computation of all large numbers in his formula. This led to an algorithm for computation of class numbers of Mignotte’s fields which was much faster than any achieved to date. Finally, M. Mignotte (to appear) has given methods for dealing with those cases of Catalan’s problem for which all these criteria fail. In view of all this, it is safe to say that the problem will (hopefully) be completely solved quite soon.
Ribenboim’s book is not only a book about Catalan’s problem, but also an excellent one on diophantine equations and related topics. Thus, after some preliminary sections on basic facts needed in the proofs, we find others devoted to the Pythagorean equation, continued fractions, results of Størmer on Pell’s equation, representation of integers by binary cubic forms, J. H. E. Cohn’s [J. Lond. Math. Soc. 42, 750-752 (1967; Zbl 0154.29701)] results on binary quartic diophantine equations, powerful numbers and many other topics, some of which are still open. However, we also find a complete treatment of all classical results on Catalan’s equation and their proofs.
In the last part of the book the author proves an effective version of Tijdeman’s result on an upper bound for \(p\) and \(q\). As pointed out above, this is not the best upper bound known, but it does give the essential ideas of Tijdeman’s work. We also find a derivation of S. Hyyrö’s [Ann. Univ. Turku, Ser. A I 79 (1964; Zbl 0127.01904)] algorithm to determine all eventual solutions \((x, y)\). Unfortunately, the known upper bounds on \(x\) and \(y\) are too large for this algorithm to be useful. It would require a major improvement in known lower bounds for linear forms in two logarithms for this.
All in all, this is a very fine book on an exciting research topic – indeed, it is the first to be devoted to this topic. What makes it particularly noteworthy is that many useful results on diophantine equations from many journals are gathered together in one place for the first time.
It is certainly worthy of translation into many languages.

Reviewer: R.P.Steiner (Bowling Green)

MSC:

11Dxx	Diophantine equations
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11D41	Higher degree equations; Fermat’s equation
11D61	Exponential Diophantine equations
11D09	Quadratic and bilinear Diophantine equations
11D25	Cubic and quartic Diophantine equations

Keywords:

Catalan’s conjecture; upper bound; Pythagorean equation; Pell’s equation; representation of integers by binary cubic forms; binary quartic diophantine equations; powerful numbers

Citations:

Zbl 0316.10008; Zbl 0127.271; Zbl 0699.10029; Zbl 0154.29701; Zbl 0127.01904