Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T05:56:20.984Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A CLASS OF GENERALIZED FERMAT EQUATIONS

Part of: Arithmetic algebraic geometry Diophantine equations Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  18 June 2010

ANDRZEJ DĄBROWSKI
ANDRZEJ DĄBROWSKI*
Affiliation:
Institute of Mathematics, University of Szczecin, ul. Wielkopolska 15, 70-451 Szczecin, Poland (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation xn+yn=2αpz2’, C. R. Math. Acad. Sci. Soc. R. Can.28 (2006), 6–11] concerning the solubility of the diophantine equation xn+yn=2αpz2. We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3,p) or (4,2,p) can be reduced, in certain cases, to studying several equations of the type (p,p,2).

Keywords

generalized Fermat equationelliptic curvemodular formGalois representation

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11F80: Galois representations 11G05: Elliptic curves over global fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 505 - 510
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bennett, M. A. and Mulholland, J., ‘On the diophantine equation x n+y n=2αpz 2’, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), 6–11.Google Scholar
[2]Bennett, M. A. and Skinner, C. M., ‘Ternary diophantine equations via Galois representations and modular forms’, Canad. J. Math. 56 (2004), 23–54.CrossRefGoogle Scholar
[3]Bennett, M. A., Vatsal, V. and Yazdani, S., ‘Ternary diophantine equations of signature (p,p,3)’, Compositio Math. 140 (2004), 1399–1416.CrossRefGoogle Scholar
[4]Billerey, N., ‘Formes homogènes de degré 3 et puissances p-ièmes’, J. Number Theory 128 (2008), 1272–1294.CrossRefGoogle Scholar
[5]Breuil, C., Conrad, B., Diamond, F. and Taylor, R., ‘On the modularity of elliptic curves over ℚ: wild 3-adic exercises’, J. Amer. Math. Soc. 14 (2001), 843–939.CrossRefGoogle Scholar
[6]Dąbrowski, A., ‘On the integers represented by x 4−y 4’, Bull. Aust. Math. Soc. 76 (2007), 133–136.CrossRefGoogle Scholar
[7]Darmon, H. and Merel, L., ‘Winding quotients and some variants of Fermat’s last theorem’, J. reine angew. Math. 490 (1997), 81–100.Google Scholar
[8]Ivorra, W., ‘Courbes elliptiques sur ℚ, ayant un point d’ordre 2 rationnel sur ℚ, de conducteur 2Np’, Dissert. Math. 429 (2004), 55pp.CrossRefGoogle Scholar
[9]Ivorra, W. and Kraus, A., ‘Quelques résultats sur les équations ax p+by p=cz 2’, Canad. J. Math. 58 (2006), 115–153.CrossRefGoogle Scholar
[10]Kraus, A., ‘Majorations effectives pour l’équation de Fermat généralisée’, Canad. J. Math. 49 (1997), 1139–1161.CrossRefGoogle Scholar
[11]Kraus, A., ‘Sur l’équation a 3+b 3=c p’, Experiment Math. 7 (1998), 1–13.CrossRefGoogle Scholar
[12]Papadopoulos, I., ‘Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle 2 et 3’, J. Number Theory 44 (1993), 119–152.CrossRefGoogle Scholar
[13]Ribet, K., ‘On modular representations of arising from modular forms’, Invent. Math. 100 (1990), 431–476.CrossRefGoogle Scholar
[14]Serre, J.-P., ‘Sur les représentations modulaires de degré 2 de ’, Duke Math. J. 54 (1987), 179–230.CrossRefGoogle Scholar
[15]Wiles, A., ‘Modular elliptic curves and Fermat’s last theorem’, Ann. of Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar