The equation x 2 n + y 2 n = z 5

Michael A. Bennett[1]

  • [1] University of British Columbia 1984 Mathematics Road Vancouver, B.C. Canada

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 2, page 315-321
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Bennett, Michael A.. "The equation $x^{2n}+y^{2n}=z^5$." Journal de Théorie des Nombres de Bordeaux 18.2 (2006): 315-321. <http://eudml.org/doc/249644>.

@article{Bennett2006,
abstract = {We show that the Diophantine equation of the title has, for $n &gt; 1$, 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.},
affiliation = {University of British Columbia 1984 Mathematics Road Vancouver, B.C. Canada},
author = {Bennett, Michael A.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {higher degree Diophantine equations; Galois representations; elliptic curves},
language = {eng},
number = {2},
pages = {315-321},
publisher = {Université Bordeaux 1},
title = {The equation $x^\{2n\}+y^\{2n\}=z^5$},
url = {http://eudml.org/doc/249644},
volume = {18},
year = {2006},
}

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
PB - Université Bordeaux 1
VL - 18
IS - 2
SP - 315
EP - 321
AB - We show that the Diophantine equation of the title has, for $n &gt; 1$, 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.
LA - eng
KW - higher degree Diophantine equations; Galois representations; elliptic curves
UR - http://eudml.org/doc/249644
ER -

References

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  9. A. Kraus, Majorations effectives pour l’équation de Fermat généralisée. Canad. J. Math. 49 (1997), 1139–1161. Zbl0908.11017MR1611640
  10. A. Kraus, On the equation x p + y q = z r : a survey. Ramanujan J. 3 (1999), 315–333. Zbl0939.11016MR1714945
  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. Zbl0924.11022MR1488570
  12. L. Merel, Arithmetic of elliptic curves and Diophantine equations. J. Théor. Nombres Bordeaux 11 (1999), 173–200. Zbl0964.11028MR1730439
  13. K. Ribet, On modular representations of G a l ( ℚ ¯ / ℚ ) arising from modular forms. Invent. Math. 100 (1990), 431–476. Zbl0773.11039MR1047143
  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. Zbl0823.11029MR1333035

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