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RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES

Part of: Diophantine equations

Published online by Cambridge University Press:  06 September 2011

ZHONGYAN SHEN  and
TIANXIN CAI
ZHONGYAN SHEN
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, PR China Department of Mathematics, Zhejiang International Study University, Hangzhou 310012, Zhejiang, PR China (email: [email protected])
TIANXIN CAI*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, PR China (email: [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, we obtain all rational points (x,y) on the superelliptic curves

Keywords

superelliptic curvesrational pointsDiophantine equations

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation 11D72: Equations in many variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 105 - 113
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The first author was supported by the Educational Commission of Zhejiang Province of China, Grant No. Y200803340, and Foundation of Zhejiang International Study University, Grant No. 09Z08. The second author was supported by the National Natural Science Foundation of China, Grant No. 10871169.

References

[1]Bennett, M. A., ‘Products of consecutive integers’, Bull. Lond. Math. Soc. 36 (2004), 683–694.CrossRefGoogle Scholar
[2]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
[3]Dickson, L. E., History of the Theory of Numbers: Diophantine Analysis, Vol. 2 (G. E. Stechert, New York, 1934).Google Scholar
[4]Erdős, P. and Selfridge, J. L., ‘The product of consecutive integers is never a power’, Illinois J. Math. 19 (1975), 292–301.Google Scholar
[5]Lakhal, M. and Sander, J. W., ‘Rational points on the superelliptic Erdős–Selfridge curve of fifth degree’, Mathematika 50 (2003), 113–124.Google Scholar
[6]Ribet, K. A., ‘On the equation a p+2αb p+c p=0’, Acta Arith. 79 (1997), 7–16.CrossRefGoogle Scholar
[7]Sander, J. W., ‘Rational points on a class of superelliptic curves’, J. Lond. Math. Soc. 59 (1999), 422–434.CrossRefGoogle Scholar
[8]Saradha, N. and Shorey, T. N., ‘Almost perfect powers in arithmetic progression’, Acta Arith. 99 (2001), 363–388.CrossRefGoogle Scholar
[9]Saradha, N. and Shorey, T. N., ‘Almost squares and factorisations in consecutive integers’, Compositio Math. 138 (2003), 113–124.Google Scholar
[10]Selmer, E., ‘The diophantine equation ax 3+by 3+cz 3=0’, Acta Math. 85 (1951), 203–362.Google Scholar