We show that the Diophantine equation of the title has, for , no solution in coprime nonzero integers and . Our proof relies upon Frey curves and related results on the modularity of Galois representations.
Nous montrons que l’équation diophantienne ci-dessus n’admet pas de solutions entières , telles que et . La démonstration utilise les courbes de Frey et des résultats liés à la modularité des représentations galoisiennes.
@article{JTNB_2006__18_2_315_0, author = {Bennett, Michael A.}, title = {The equation $x^{2n}+y^{2n}=z^5$}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {315--321}, publisher = {Universit\'e Bordeaux 1}, volume = {18}, number = {2}, year = {2006}, doi = {10.5802/jtnb.546}, zbl = {05135392}, mrnumber = {2289426}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.546/} }
TY - JOUR AU - Bennett, Michael A. TI - The equation $x^{2n}+y^{2n}=z^5$ JO - Journal de théorie des nombres de Bordeaux PY - 2006 SP - 315 EP - 321 VL - 18 IS - 2 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.546/ DO - 10.5802/jtnb.546 LA - en ID - JTNB_2006__18_2_315_0 ER -
Bennett, Michael A. The equation $x^{2n}+y^{2n}=z^5$. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 315-321. doi : 10.5802/jtnb.546. http://www.numdam.org/articles/10.5802/jtnb.546/
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