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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