Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett, Vandita Patel, and Samir Siksek. "Superelliptic equations arising from sums of consecutive powers." Acta Arithmetica 172.4 (2016): 377-393. <http://eudml.org/doc/279748>.

@article{MichaelA2016,
abstract = {Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization $(x-1)^k+x^k+(x+1)^k = z^n$ (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.},
author = {Michael A. Bennett, Vandita Patel, Samir Siksek},
journal = {Acta Arithmetica},
keywords = {exponential equation; Galois representation; Frey-Hellegouarch curve; modularity; level lowering; multi-Frey-Hellegouarch},
language = {eng},
number = {4},
pages = {377-393},
title = {Superelliptic equations arising from sums of consecutive powers},
url = {http://eudml.org/doc/279748},
volume = {172},
year = {2016},
}

TY - JOUR
AU - Michael A. Bennett
AU - Vandita Patel
AU - Samir Siksek
TI - Superelliptic equations arising from sums of consecutive powers
JO - Acta Arithmetica
PY - 2016
VL - 172
IS - 4
SP - 377
EP - 393
AB - Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization $(x-1)^k+x^k+(x+1)^k = z^n$ (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.
LA - eng
KW - exponential equation; Galois representation; Frey-Hellegouarch curve; modularity; level lowering; multi-Frey-Hellegouarch
UR - http://eudml.org/doc/279748
ER -