On the Diophantine equation \((ax^m-4c)/(abx-4c)=by^2\) (Q2741978)
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scientific article; zbMATH DE number 1649629
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Diophantine equation \((ax^m-4c)/(abx-4c)=by^2\) |
scientific article; zbMATH DE number 1649629 |
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20 September 2001
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exponential Diophantine equations
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On the Diophantine equation \((ax^m-4c)/(abx-4c)=by^2\) (English)
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The author proves that the equation \((ax^m-4c)/(abx-4c)=by^2\), where \(a,b,x,y,m\in \mathbb{N}\), \(x>1\), \(y>1\), \(m>1\), \(2\nmid m\) and \(c\in \{-1,1\}\), has only the solutions \((a,b,c,x,y,m)=(1,1,1,5,11,3)\) and \((3^{(t-3)/2}+s,1,s,3,3^{(t-1)/2}+2s,t)\), where \(s\in \{-1,1\}\) and \(t\in \mathbb{N}\) with \(t>1\) and \(2\nmid t\). The proof uses some results on Pell's equation.
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