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• # Artykuł - szczegóły

## Acta Arithmetica

2016 | 172 | 4 | 377-393

## Superelliptic equations arising from sums of consecutive powers

EN

### Abstrakty

EN
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.

377-393

wydano
2016

### Twórcy

autor
• Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 Canada
autor
• Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
autor
• Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom