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1
Content available remote

On consecutive integers of the form ax², by² and cz²

100%
Bennett M.
Acta Arithmetica
|
1999
|
tom 88
|
nr 4
363-370
2
Content available remote

An ideal Waring problem with restricted summands

100%
Bennett M.
Acta Arithmetica
|
1994
|
tom 66
|
nr 2
125-132
3
Content available remote

Lucas' square pyramid problem revisited

88%
Bennett M.
Acta Arithmetica
|
2002
|
tom 105
|
nr 4
341-347
4
Content available remote

Effective results for restricted rational approximation to quadratic irrationals

64%
Bennett M., Bugeaud Y.
Acta Arithmetica
|
2012
|
tom 155
|
nr 3
259-269
5
Content available remote

On the equation a³ + b³ⁿ = c²

51%
Bennett M., Chen I., Dahmen S., Yazdani S.
Acta Arithmetica
|
2014
|
tom 163
|
nr 4
327-343
EN
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.
6
Content available remote

On the number of solutions of simultaneous Pell equations II

51%
Bennett M., Cipu M., Mignotte M., Okazaki R.
Acta Arithmetica
|
2006
|
tom 122
|
nr 4
407-417
7
Content available remote

Superelliptic equations arising from sums of consecutive powers

51%
Bennett M., Patel V., Siksek S.
Acta Arithmetica
|
2016
|
tom 172
|
nr 4
377-393
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.
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