On the equation 𝑠²+𝑦^{2𝑝}=𝛼³

Chen, Imin

doi:10.1090/S0025-5718-07-02083-2

On the equation $s^2+y^{2p} = \alpha ^3$
HTML articles powered by AMS MathViewer

by Imin Chen;

Math. Comp. 77 (2008), 1223-1227

DOI: https://doi.org/10.1090/S0025-5718-07-02083-2

Published electronically: October 23, 2007

PDF | Request permission

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

Michael A. Bennett and Chris M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 (2004), no. 1, 23–54. MR 2031121, DOI 10.4153/CJM-2004-002-2
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
N. Bruin. Chabauty methods and covering techniques applied to generalised Fermat equations. Ph.D. thesis, University of Leiden, 1999.
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
H. Darmon, Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation, C. R. Math. Rep. Acad. Sci. Canada 19 (1997), no. 1, 3–14. MR 1479291
Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), no. 6, 513–543. MR 1348707, DOI 10.1112/blms/27.6.513
Alain Kraus, Sur l’équation $a^3+b^3=c^p$, Experiment. Math. 7 (1998), no. 1, 1–13 (French, with English and French summaries). MR 1618290
Alain Kraus, On the equation $x^p+y^q=z^r$: a survey, Ramanujan J. 3 (1999), no. 3, 315–333. MR 1714945, DOI 10.1023/A:1009835521324
B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
Barry Mazur and Jacques Vélu, Courbes de Weil de conducteur $26$, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A743–A745 (French). MR 320010
K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195