Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T22:48:11.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational points on the superelliptic Erdös Selfridge curve of fifth degree

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

M. Lakhal  and
J. W. Sander
M. Lakhal
Affiliation:
Institut für Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.
J. W. Sander
Affiliation:
Institut für Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.
Get access

Extract

§1. Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equation

with integers k≥2 and m≥2, has only the trivial solutions. x = −j(j = i, …, m), y = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 113 - 124
Copyright
Copyright © University College London 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Darmon, H. and Mcrel, L.. Winding quotients and some variants of Fermat's Last Theorem, j. reine angew. Math. 490 (1997). 81–100.Google Scholar
2.Dickson, L. E.. History of the Theory of Numbers, volume II. Carnegie Institute of Washington, (1919).CrossRefGoogle Scholar
3.Erdös, P. and Sclfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. 19 (1975). 292–301.CrossRefGoogle Scholar
4.Faltings, G., Endlichkeitssätze fur abelsche Varietäten iiber Zahlkörpern. Invent, math. 73 (1983), 349–366.CrossRefGoogle Scholar
5.Poonen, B., Some diophantine equations of the form xn + yn = zm, Ada Arith. 86 (1998). 193–205.CrossRefGoogle Scholar
6.Ribet, K. A., On the equation ap + 2αbp + cp = 0, Ada Arith. 79 (1997). 7–16.CrossRefGoogle Scholar
7.Sander, J. W., Rational points on a class of superelliptic curves, J. London Math. Soc (2) 59 (1999), 422–434.CrossRefGoogle Scholar
8.Serre, J. P., Sur les representations modulaires de degre de Gal ***. Duke Math. J. 54 (1987). 179–230.CrossRefGoogle Scholar
9.Wiles, A., Modular elliptic curves and Fermat's Last Theorem, Ann. Math. 141 (1995). 443–551.CrossRefGoogle Scholar