Wyniki wyszukiwania

1

Perfect powers expressible as sums of two fifth or seventh powers

100%

Dahmen S., Siksek S.

Acta Arithmetica

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2014

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tom 164

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nr 1

65-100

EN

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.

2

On the Diophantine equation $x^2 + 7 = y^m$

100%

Siksek S., Cremona J.

Acta Arithmetica

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2003

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tom 109

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nr 2

143-149

3

Superelliptic equations arising from sums of consecutive powers

81%

Bennett M., Patel V., Siksek S.

Acta Arithmetica

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2016

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tom 172

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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.

Ograniczanie wyników

3 Acta Arithmetica

3 Siksek S.

1 Bennett M.

1 Cremona J.

1 Dahmen S.

1 Patel V.

1 2016

1 2014

1 2003

Perfect powers expressible as sums of two fifth or seventh powers

On the Diophantine equation $x^2 + 7 = y^m$

Superelliptic equations arising from sums of consecutive powers