Superelliptic equations arising from sums of consecutive powers

• Autorzy: Michael A. Bennett , Vandita Patel , Samir Siksek
• Języki publikacji: 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.

Kategorie tematyczne

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 172
• Numer: 4
• Strony: 377-393

Daty

wydano

2016

Twórcy

autor

Michael A. Bennett

• Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2 Canada

autor

Vandita Patel

• Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

autor

Samir Siksek

• Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Identyfikatory

DOI

10.4064/aa8305-12-2015