On the Diophantine equation \(a\frac{x^n-1}{x-1}=y^q\). (Q2710112)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Diophantine equation \(a\frac{x^n-1}{x-1}=y^q\). |
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19 April 2001
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On the Diophantine equation \(a\frac{x^n-1}{x-1}=y^q\). (English)
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The author considers the equation NEWLINE\[NEWLINEa \frac{x^n-1}{x-1}=y^qNEWLINE\]NEWLINE in integers \(n \geq 3, x \geq 2, 1\leq a <x, y \geq 2, q \geq 2\). This equation asks for perfect powers whose digits are all equal to \(a\) to the base \(x\). The author extends a result of Inkeri and solves completely the above equation for \(x \leq 100 \) and \( x=1000.\) Besides some elementary results and computations, the proof depends on a crucial result due to the author and Mignotte that the above equation with \(a=1,\;x=z^t, 2 \leq z \leq 10^4\) and \(t \geq 1\) has no solution.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].
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