The equation $x^{2n}+y^{2n}=z^5$

Bennett, Michael A.

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 $xyz\ne 0$. 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.

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

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

[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 | Zbl

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

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

[6] H. Darmon, A. Granville, On the equations $z^m=F(x,y)$ and $A{x}^p+B{y}^q=C{z}^r$. Bull. London Math. Soc. 27 (1995), 513–543. | MR | Zbl

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

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

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

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

[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 | Zbl

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

[13] K. Ribet, On modular representations of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ arising from modular forms. Invent. Math. 100 (1990), 431–476. | MR | Zbl

[14] W. Stein, Modular forms database. http://modular.fas.harvard.edu/Tables/

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