The equation x 2n +y 2n =z 5
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 315-321.

Nous montrons que l’équation diophantienne ci-dessus n’admet pas de solutions entières x,y,z, telles que (x,y)=(y,z)=(x,z)=1 et xyz0. La démonstration utilise les courbes de Frey et des résultats liés à la modularité des représentations galoisiennes.

We show that the Diophantine equation of the title has, for n>1, no solution in coprime nonzero integers x,y and z. Our proof relies upon Frey curves and related results on the modularity of Galois representations.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.546
@article{JTNB_2006__18_2_315_0,
     author = {Michael A. Bennett},
     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},
     mrnumber = {2289426},
     zbl = {05135392},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.546/}
}
Michael A. Bennett. The equation $x^{2n}+y^{2n}=z^5$. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 315-321. doi : 10.5802/jtnb.546. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.546/

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