On the Diophantine equation \(x^2= y^p+2^kz^p\) (Q558171)
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scientific article; zbMATH DE number 2184627
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Diophantine equation \(x^2= y^p+2^kz^p\) |
scientific article; zbMATH DE number 2184627 |
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On the Diophantine equation \(x^2= y^p+2^kz^p\) (English)
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30 June 2005
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The equation of the title is attacked using a Frey curve, Ribet's level-lowering theorem and a method due to Darmon and Merel. All the solutions in pairwise coprime integers \(x, y, z\) \((p \geq 7, k \geq 2)\) are determined. In particular, previous results of various authors are combined to obtain a complete solution for the equation \(x^2 + 2^k = y^n\) for \(n \geq 3\).
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exponential Diophantine equations
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elliptic curves
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