Perfect powers expressible as sums of two fifth or seventh powers

Sander R. Dahmen, and Samir Siksek. "Perfect powers expressible as sums of two fifth or seventh powers." Acta Arithmetica 164.1 (2014): 65-100. <http://eudml.org/doc/279348>.

@article{SanderR2014,
abstract = {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.},
author = {Sander R. Dahmen, Samir Siksek},
journal = {Acta Arithmetica},
keywords = {generalized Fermat; curves; elliptic curves; Galois representation; Jacobian; modular forms; Fermat-Catalan},
language = {eng},
number = {1},
pages = {65-100},
title = {Perfect powers expressible as sums of two fifth or seventh powers},
url = {http://eudml.org/doc/279348},
volume = {164},
year = {2014},
}

TY - JOUR
AU - Sander R. Dahmen
AU - Samir Siksek
TI - Perfect powers expressible as sums of two fifth or seventh powers
JO - Acta Arithmetica
PY - 2014
VL - 164
IS - 1
SP - 65
EP - 100
AB - 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.
LA - eng
KW - generalized Fermat; curves; elliptic curves; Galois representation; Jacobian; modular forms; Fermat-Catalan
UR - http://eudml.org/doc/279348
ER -