A bound for the solutions of the Diophantine equation \(D_1x^2+D_2^m=4y^n\) (Q1565799)
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scientific article; zbMATH DE number 1918573
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A bound for the solutions of the Diophantine equation \(D_1x^2+D_2^m=4y^n\) |
scientific article; zbMATH DE number 1918573 |
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A bound for the solutions of the Diophantine equation \(D_1x^2+D_2^m=4y^n\) (English)
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26 January 2004
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The main result of this note is: every solution \((D_1,D_2,x,m,n)\) of the Diophantine equation \(D_1x^2+D_2^m=4y^n\) with \(n\) prime and not dividing the class number of \(\mathbb Q(\sqrt{-D_1 D_2})\) satisfies \(n\leq 5351\). This result improves on results of \textit{M. H. Le} [Monatsh. Math. 120, 121-125 (1995; Zbl 0877.11020)] and \textit{M. Mignotte} [Port. Math. 54, 457-460 (1997; Zbl 1028.11019)]. The proof uses a sharp result of Voutier for linear forms in two logarithms (complex case).
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exponential Diophantine equations
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generalized Ramanujan-Nagell equation
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linear forms in logarithms
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0.8603352308273315
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0.8600202202796936
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