Wyniki wyszukiwania - Biblioteka Nauki

Nowa wersja platformy, zawierająca wyłącznie zasoby pełnotekstowe, jest już dostępna.
Przejdź na https://bibliotekanauki.pl

Ograniczanie wyników

3 Acta Arithmetica

Znaleziono wyników: 3

Liczba wyników na stronie

Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

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

100%

Siksek S. , Cremona J.

Acta Arithmetica

2003

tom 109

nr 2

143-149

Perfect powers expressible as sums of two fifth or seventh powers

100%

Dahmen S. , Siksek S.

Acta Arithmetica

2014

tom 164

nr 1

65-100

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.

Superelliptic equations arising from sums of consecutive powers

80%

Bennett M. , Patel V. , Siksek S.

tom 172

nr 4

377-393

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.