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On the equation of Catalan. (English) Zbl 0286.10013

Acta Arith. 29, No. 2, 197-209 (1976).
The following conjecture was first made by E. Catalan in [J. Reine Angew. Math. 27, 192 (1844; Zbl 02750995)] but has never been proved. The only solution in integers \(p > 1\), \(q > 1\), \(x > 1\), \(y > 1\) of the equation
\[ x^p - y^q =1 \tag{*} \]
is \(p=y=2\), \(q=x=3\). In the present paper it is proved that (*) has only finitely many solutions and that effective bounds for the solutions \(p, q, x, y\) can be given. The proof depends heavily on the method of Gel-fond-Baker. By a multiple application of a refinement of a theorem of A. Baker [Acta Arith. 21, 117–129 (1972; Zbl 0244.10031)] it is shown that there exist effective upper bounds for \(p\) and \(q\). The full assertion then is an immediate consequence of another result of A. Baker [Proc. Camb. Philos. Soc. 65, 439–444 (1969; Zbl 0174.33803)].

MSC:

11D61 Exponential Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
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