The equation x 2n +y 2n =z 5
Michael A. Bennett
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 xyz≠0. 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 : 2004-10-07
Publié le : 2008-12-02
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 = {jtnb.centre-mersenne.org/item/JTNB_2006__18_2_315_0/}
}
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/item/JTNB_2006__18_2_315_0/

[1] A. Battaglia, Impossibilità dell’equazione indeterminata x 2n +y 2n =z 5 . Archimede 20 (1968), 300–305. | MR 240051 | Zbl 0185.10803

[2] M.A. Bennett, C. Skinner, Ternary Diophantine equations via Galois representations and modular forms. Canad. J. Math. 56 (2004), 23–54. | MR 2031121 | Zbl 1053.11025

[3] N. Bruin, On powers as sums of two cubes. Algorithmic number theory (Leiden, 2000), 169–184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000. | MR 1850605 | Zbl 0986.11021

[4] J. Cremona, Algorithms for Modular Elliptic Curves. Cambridge University Press, 1992. | MR 1201151 | Zbl 0758.14042

[5] H. Darmon, Rigid local systems, Hilbert modular forms, and Fermat’s last theorem. Duke. Math. J. 102 (2000), 413–449. | MR 1756104 | Zbl 1008.11023

[6] H. Darmon, A. Granville, On the equations z m =F(x,y) and Ax p +By q =Cz r . Bull. London Math. Soc. 27 (1995), 513–543. | MR 1348707 | Zbl 0838.11023

[7] H. Darmon, L. Merel, Winding quotients and some variants of Fermat’s Last Theorem. J. Reine Angew Math. 490 (1997), 81–100. | MR 1468926 | Zbl 0976.11017

[8] J. S. Ellenberg, Galois representations attached to ℚ-curves and the generalized Fermat equation A 4 +B 2 =C p . Amer. J. Math. 126 (2004), 763–787. | MR 2075481 | Zbl 1059.11041

[9] A. Kraus, Majorations effectives pour l’équation de Fermat généralisée. Canad. J. Math. 49 (1997), 1139–1161. | MR 1611640 | Zbl 0908.11017

[10] A. Kraus, On the equation x p +y q =z r : a survey. Ramanujan J. 3 (1999), 315–333. | MR 1714945 | Zbl 0939.11016

[11] R.D. Mauldin, A generalization of Fermat’s last theorem: the Beal conjecture and prize problem. Notices Amer. Math. Soc. 44 (1997), 1436–1437. | MR 1488570 | Zbl 0924.11022

[12] L. Merel, Arithmetic of elliptic curves and Diophantine equations. J. Théor. Nombres Bordeaux 11 (1999), 173–200. | Numdam | MR 1730439 | Zbl 0964.11028

[13] K. Ribet, On modular representations of Gal(ℚ ¯/ℚ) arising from modular forms. Invent. Math. 100 (1990), 431–476. | MR 1047143 | Zbl 0773.11039

[14] W. Stein, Modular forms database.

[15] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), 443–551. | MR 1333035 | Zbl 0823.11029