Superelliptic equations arising from sums of consecutive powers (Q2804248)
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scientific article; zbMATH DE number 6574968
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Superelliptic equations arising from sums of consecutive powers |
scientific article; zbMATH DE number 6574968 |
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Superelliptic equations arising from sums of consecutive powers (English)
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28 April 2016
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exponential equation
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Galois representation
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Frey-Hellegouarch curve
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modularity
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level lowering
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multi-Frey-Hellegouarch
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The authors consider the following Diophantine equation NEWLINE\[CARRIAGE_RETURNNEWLINE(x-1)^k+x^k+(x+1)^k=z^n,\qquad x,z,k,n\in \mathbb{Z},\quad k,n\geq2.CARRIAGE_RETURNNEWLINE\]NEWLINE Using multi-Frey-Hellegouarch curves, the authors show that when \(k=5\), the only solutions of the latter equation are given by \(x=z=0\).NEWLINENEWLINEFurthermore, the authors develop a version of the method of bounding exponents to show that when \(k=6\), the equation has no solutions.
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