Currently displaying 1 – 9 of 9

Showing per page

Order by Relevance | Title | Year of publication

The equation x 2 n + y 2 n = z 5

Michael A. Bennett — 2006

Journal de Théorie des Nombres de Bordeaux

We show that the Diophantine equation of the title has, for n > 1 , no solution in coprime nonzero integers x , y and z . Our proof relies upon Frey curves and related results on the modularity of Galois representations.

Superelliptic equations arising from sums of consecutive powers

Michael A. BennettVandita PatelSamir Siksek — 2016

Acta Arithmetica

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...

On the equation a³ + b³ⁿ = c²

Michael A. BennettImin ChenSander R. DahmenSoroosh Yazdani — 2014

Acta Arithmetica

We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.

Page 1

Download Results (CSV)