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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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