Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu). (English) Zbl 1080.11030

The first published proof [cf. the reviewer, J. Reine Angew. Math. 572, 167–195 (2004; Zbl 1067.11017)] of Catalan’s conjecture reduces essentially to showing that \[ x^p-y^q=1 \tag{1} \] has no integer solutions with odd primes \(p, q\) unless \(p \equiv 1 \mod q^2\). This is done by using a relatively simple apparatus of cyclotomic fields. However, the case resulting from the condition \(p \equiv 1 \mod q^2\) – which shall be denoted as non semisimple case, since the condition gives rise to non semisimple group rings – is discarded in the Crelle paper (cited above) (and also in Yu. F. Bilu’s Bourbaki presentation of the same proof [cf. Astérisque 294, 1–26, Exp. No. 909 (2004; Zbl 1094.11014)]) by using reminiscent results from older theorems based on the Baker theory.
In the present paper, dedicated to Rob Tijdeman, Yuri Bilu gives an own deduction of some simpler results which lead to the elimination of the non semisimple case. The condition \(q | h_p^-\) for \(p < q\) was proved by Y. Bugeaud and G. Hanrot [Mathematika 47, No. 1–2, 63–73 (2000; Zbl 1008.11011)] and was subsequently symmetrized (thus \(p, q\) verify no additional inequality) by Mihailescu. It results that odd primes for which (1) has a solution verify \(p | h_q^-\) and \(q | h_p^-\). This condition helps dealing with small values of \(p\) and \(q\); for the remaining pairs, Bilu gives a demonstration of the fact that there are no solutions if \(q > 3(p-1)^2\). This is a theorem of Mihailescu and leads to the conclusion that the non semisimple case has no solution. It thus yields a proof of Catalan’s conjecture “without logarithmic forms”, as stated in the title.
The paper is very well written and self-contained, being an exposition of the results of Bugeaud, Hanrot and Mihailescu, as explained by the author.


11D61 Exponential Diophantine equations
11R18 Cyclotomic extensions
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