Anyone who has played with numbers has noticed the strange games they sometimes play. Consider, for example, the fact that 32 – 23 = 1: trivial, of course, but possessing a certain kind of pleasant symmetry. It might lead us, however, to ask a more general question: are there other consecutive powers? To be more formal, are there any other solutions to xm – yn = 1 with m and n both greater than one? In 1844, Catalan conjectured that the answer was no, that there are no other consecutive powers.
Catalan’s conjecture is quite a subtle question. It is easy to work out how the squares are distributed on the number line, how the cubes are distributed, and so on. But working out how the infinitely many lists of powers interact with each other is something else entirely, and the problem was open for a long time. There is a whole book by Paulo Ribenboim, Catalan’s Conjecture (Academic Press, 1994), that tells the story of the work done up to that point, the most important of which was a result of Tijdeman (1976). He showed that there were finitely many solutions to xm – yn = 1, and even gave a bound for their size. Unfortunately, the bound was huge; even after many improvements, the best estimate was that the exponents m and n would both need to be smaller than 8 × 1016. The exponents!
Then, in 2002, the news came out that Preda Mihailescu had proved the conjecture. It wasn’t really a complete surprise by that point, since Mihailescu had taken a huge step towards the proof three years before. The amazing thing, however, was that the proof was not too complicated. In particular, it did not require the kind of complicated machinery that allowed the proof of Fermat’s Last Theorem a few years earlier. Nor did the final form of the proof depend on Tijdeman’s result. No computer verifications either.
The neat little book under review gives a complete account of Mihailescu’s proof including most of the background material. The key ingredients are Runge’s method, explained here in terms of the theory of algebraic curves, and the theory of cyclotomic fields, including the theory of the Stickelberger ideal and a crucial theorem of Thaine. The latter is the hardest — and most recent — of the prerequisites, and is relegated to the final chapter.
Catalan’s Conjecture is very readable, direct, and full of insight. It manages to completely explain a significant recent result “from scratch” in a mere 120 pages, quite an achievement these days. It makes for delightful reading for any number theorist, but it would also be an excellent way to learn some algebraic number theory. One could easily build an undergraduate seminar around it; I’m sure students would enjoy it and learn a lot. It’s an excellent book.
Introduction.- The case "q = 2".- The case "p = 2".- The non-trivial solution.- Runge’s method.- Cassels' theorem.- An obstruction group.- Small p or q .- The Stickelberger ideal.- The double Wieferich criterion.- The minus argument.- The plus argument I.- Semi-simple group rings.- The plus argument II.- The density theorem.- Thaine’s theorem.- Appendix: Euler’s theorem.- Bibliography.- Index.