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A class number free criterion for Catalan’s conjecture. (English) Zbl 1049.11036
Catalan’s conjecture [E. Catalan, J. Reine Angew. Math. 27, 192 (1844; ERAM 027.0790cj)] predicts that 8 and 9 are the only consecutive integers which are both perfect powers. This conjecture was recently proved by the present author. This paper contains the first part of his contribution to the final solution of this problem.
Catalan’s conjecture corresponds to the Diophantine equation \(x^p-y^q=1\) where \(p\) and \(q\) are prime numbers. Several arithmetical criteria were obtained by K. Inkeri [J. Number Theory 34, 142–152 (1990; Zbl 0699.10029)] and after M. Mignotte [C. R. Math. Acad. Sci., Soc. R. Can. 15, 199–200 (1993; Zbl 0802.11010)] and W. Schwarz [Acta Arith. 72, 277–279 (1995; Zbl 0837.11014)], but all of them implied some condition on certain class numbers. In this paper the author is able to get rid of these conditions and proves the remarkable fact that if the above equation has a nontrivial solution in rational integers then
\[ p^{q-1} \equiv 1 \pmod {q^2} \quad \text{ and} \quad q^{p-1} \equiv 1 \pmod {p^2}. \]
The proof follows Inkeri’s proof, except for a very ingenious use of Stickelberger’s theorem.

MSC:
11D61 Exponential Diophantine equations
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