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On the equation $ s^2+y^{2p} = \alpha^3$


Author: Imin Chen
Journal: Math. Comp. 77 (2008), 1223-1227
MSC (2000): Primary 11G05; Secondary 14G05
DOI: https://doi.org/10.1090/S0025-5718-07-02083-2
Published electronically: October 23, 2007
MathSciNet review: 2373199
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Abstract | References | Similar Articles | Additional Information

Abstract: We describe a criterion for showing that the equation $ s^2+y^{2p} = \alpha^3$ has no non-trivial proper integer solutions for specific primes $ p > 7$. This equation is a special case of the generalized Fermat equation $ x^p + y^q + z^r = 0$. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific $ p$ in the final stage of the method (1998). The criterion can be computationally verified for primes $ 7<p < 10^7$ and $ p \not= 31$.


References [Enhancements On Off] (What's this?)

  • 1. M. Bennett and C. Skinner.
    Ternary diophantine equations via galois representations and modular forms.
    Canadian Journal of Mathematics, 56(1):23-54, 2004. MR 2031121 (2005c:11035)
  • 2. C. Breuil, B. Conrad, F. Diamond, and R. Taylor.
    Modularity of elliptic curves over $ Q$: wild $ 3$-adic exercises.
    Journal of the American Mathematical Society, 14(4):843-939, 2001. MR 1839918 (2002d:11058)
  • 3. N. Bruin.
    Chabauty methods and covering techniques applied to generalised Fermat equations.
    Ph.D. thesis, University of Leiden, 1999.
  • 4. J.E. Cremona.
    Algorithms for modular elliptic curves.
    Cambridge University Press, second edition, 1997. MR 1628193 (99e:11068)
  • 5. H. Darmon.
    Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation.
    C.R. Math. Rep. Acad. Sci. Canada, 19(1):3-14, 1997. MR 1479291 (98h:11034a)
  • 6. A. Granville and H. Darmon.
    On the equations $ x^p+y^q=z^r$ and $ z^m=f(x,y)$.
    Bulletin of the London Math. Society, 27(129):513-544, 1995. MR 1348707 (96e:11042)
  • 7. A. Kraus.
    Sur l'équation $ a^3+b^3=c^p$.
    Experiment. Math., 7:1-13, 1998. MR 1618290 (99f:11040)
  • 8. A. Kraus.
    On the equation $ x^p+y^q=z^r$: A survey.
    The Ramanujan Journal, 3:315-333, 1999. MR 1714945 (2001f:11046)
  • 9. B. Mazur.
    Rational isogenies of prime degree.
    Inventiones Mathematicae, 44:129-162, 1978. MR 482230 (80h:14022)
  • 10. B. Mazur and J. Vélu.
    Courbes de Weil de conducteur $ 26$.
    C. R. Acad. Sci. Paris Sér. A-B, 275:A743-A745, 1972. MR 0320010 (47:8551)
  • 11. K. Ribet.
    On modular representations of Gal$ (\overline{\mathbb{Q}}\mid \mathbb{Q})$ arising from modular forms.
    Inventiones Mathematicae, 100:431-476, 1990. MR 1047143 (91g:11066)

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Additional Information

Imin Chen
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
Email: ichen@math.sfu.ca

DOI: https://doi.org/10.1090/S0025-5718-07-02083-2
Received by editor(s): October 13, 2004
Received by editor(s) in revised form: January 20, 2005
Published electronically: October 23, 2007
Additional Notes: This research was supported by NSERC
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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