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

Author(s): Imin Chen.
Journal: Math. Comp. 77 (2008), 1223-1227.
MSC (2000): Primary 11G05; Secondary 14G05
Posted: October 23, 2007
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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$.

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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: 10.1090/S0025-5718-07-02083-2
PII: S 0025-5718(07)02083-2
Received by editor(s): October 13, 2004
Received by editor(s) in revised form: January 20, 2005
Posted: October 23, 2007
Additional Notes: This research was supported by NSERC
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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