A note on the equation \(1^ k+2^ k+\cdots+(x-1)^ k=y^ m\) (Q1359390)
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scientific article; zbMATH DE number 1029290
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A note on the equation \(1^ k+2^ k+\cdots+(x-1)^ k=y^ m\) |
scientific article; zbMATH DE number 1029290 |
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A note on the equation \(1^ k+2^ k+\cdots+(x-1)^ k=y^ m\) (English)
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26 November 1997
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The purpose of this paper is to give a sharp explicit estimate for \(m\) in the equation of the title; namely: All the solutions \(x,y,m\) to this equation with \(x>10^3 (k/2)^{k+ (5/2)} =c(k)\), \(y>1\) and \(m\geq 2\) satisfy \[ m<c_1 \cdot k\log 2k, \] where \(c_1\) is an effective absolute constant.
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exponential equations
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linear form in logarithms
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