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On the equations \(z^ m = F(x,y)\) and \(Ax^ p + By^ q = Cz^ r\). (English) Zbl 0838.11023
The first result of this paper concerns the diophantine equation \(F(x,y) = z^m\), where \(F\) is a homogeneous polynomial defined over an algebraic number field \(K\). It is shown, using Falting’s theorem, that there are only finitely many primitive \(K\)-integral solutions, except when \(F\) takes certain special forms. In the latter case it is shown that there can be infinitely many solutions.
The second result states that if \(A,B,C\) are natural numbers then the equation \(Ax^p + By^q = Cz^r\) has only finitely many primitive solutions if \(p^{-1} + q^{-1} + r^{-1} < 1\). Again the proof uses Falting’s theorem. (In both results a ‘primitive’ solution is one in which \(x,y,z\) have no nontrivial common factor.) The paper investigates the generalized Fermat equation above in considerable detail, giving a full discussion of the cases \(p^{-1} + q^{-1} + r^{-1} = 1\) and \(> 1\), too.

MSC:
11D41 Higher degree equations; Fermat’s equation
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