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On the equation \(s^2+y^{2p} = \alpha^3\). (English) Zbl 1183.11030

Let \(p\) be a prime number. The author studies the diophantine equation \[ x^2+y^{2p}=z^3,(1) \] which is a special case of the generalized Fermat equation. A solution \((x,y,z)\in \mathbb Z^3\) of the equation (1) is said to be non-trivial if \(xy\neq 0\), and proper if \(\gcd(x,y,z)=1\). The author obtains a criterion which allows one often to prove that this equation has no non-trivial and proper solutions \((x,y,z)\in \mathbb Z^3\). He uses a modular method related to Galois representations and modular forms. His criterion involves an elliptic curve defined over \(\mathbb Q\) of conductor \(96\). He verifies computationally that the equation (1) has no non-trivial and proper solutions in case \(7<p<10^7\) and \(p\neq 31\).

MSC:

11G05 Elliptic curves over global fields
14G05 Rational points

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References:

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