On the solvability of a family of Diophantine equations (Q2721674)
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scientific article; zbMATH DE number 1616385
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the solvability of a family of Diophantine equations |
scientific article; zbMATH DE number 1616385 |
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4 November 2002
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Pellian equation
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triangular numbers
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On the solvability of a family of Diophantine equations (English)
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The authors prove that for any nonsquare integer \(d>1\) and \(m\in \mathbb{Z}\setminus \{0\}\), the equation NEWLINE\[NEWLINEx(x+m)= dy(y+m) \quad\text{in }x,y\in \mathbb{Z}NEWLINE\]NEWLINE has infinitely many solutions. When \(d=3\), the solutions are explicitly given in a parametric form. A connection between the Pellian equation \(X^2- 2Y^2 =-1\) and the triangular numbers is also established.
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