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Superelliptic equations arising from sums of consecutive powers. (English) Zbl 1401.11076

The authors consider the following Diophantine equation \[ (x-1)^k+x^k+(x+1)^k=z^n,\qquad x,z,k,n\in \mathbb{Z},\quad k,n\geq2. \] Using multi-Frey-Hellegouarch curves, the authors show that when \(k=5\), the only solutions of the latter equation are given by \(x=z=0\).
Furthermore, the authors develop a version of the method of bounding exponents to show that when \(k=6\), the equation has no solutions.

MSC:

11D61 Exponential Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11F80 Galois representations
11F11 Holomorphic modular forms of integral weight

Software:

Magma
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Full Text: DOI arXiv

References:

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