Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-22T08:46:40.355Z Has data issue: false hasContentIssue false
  • English
  • Français

On the integers represented by x4 − y4

Published online by Cambridge University Press:  17 April 2009

Andrzej Dąbrowski
Andrzej Dąbrowski
Affiliation:
Institute of Mathematics, University of Szczecin, ul. Wielkpolska 15, 70–451 Szczecin, Poland, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

Let p be a prime number ≥ 5, and n a positive integer > 1. This note is concerned with the diophantine equation x4 − y4 = nzp. We prove that, under certain conditions on n, this equation has no non-trivial solution in Z if p ≥ C(n), where C(n) is an effective constant.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 1 , August 2007 , pp. 133 - 136
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Darmon, H., ‘The equation x 4 − y 4 = z p’, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 286–290.Google Scholar
[2]Diamond, F., ‘On deformation rings and Hecke rings’, Ann. of Math. 144 (1996), 137–166.CrossRefGoogle Scholar
[3]Ivorra, W., ‘Courbes elliptiques sur Q, ayant un point d'ordre 2 rationnel sur Q, de conducteur 2Np’, Dissertationes Math. 429 (2004), 55pp.CrossRefGoogle Scholar
[4]Kraus, A., ‘Majorations effectives pour l'équation de Fermat généralisée’, Canad. J. Math. 49 (1997), 1139–1161.CrossRefGoogle Scholar
[5]Kraus, A., ‘Détermination du poids et du conducteur associé aux représentations des points de p-torsion d'une courbe elliptique’, Dissertationes Math. 364 (1997), 39pp.Google Scholar
[6]Mazur, B., ‘Rational isogenies of prime degree’, Invent. Math. 44 (1978), 129–162.CrossRefGoogle Scholar
[7]Merel, L., ‘Arithmetic of elliptic curves and diophantine equations’, J. Théor. Nombres Bordeaux 11 (1999), 173–200.CrossRefGoogle Scholar
[8]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
[9]Ribet, K., ‘On modular representations of Gal(Q/Q) arising from modular forms’, Invent. Math. 100 (1990), 431–476.CrossRefGoogle Scholar
[10]Serre, J.-P., ‘Sur les représentations modulaires de degré 2 de Gal(Q/Q)’, Duke Math. J. 54 (1987), 179–230.CrossRefGoogle Scholar
[11]Wiles, A., ‘Modular elliptic curves and Fermat's Last Theorem’, Ann. of Math. 141 (1995), 443–551.CrossRefGoogle Scholar