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Superelliptic equations arising from sums of consecutive powers

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Bennett M., Patel V., Siksek S.

Acta Arithmetica

2016

tom 172

nr 4

377-393

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Bennett M.

1 Patel V.

1 Siksek S.

1 2016

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Superelliptic equations arising from sums of consecutive powers