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Searching for simultaneous arithmetic progressions on elliptic curves

Published online by Cambridge University Press:  17 April 2009

Irene García-Selfa  and
José M. Tornero
Irene García-Selfa
Affiliation:
Deptartamento de Álgebra, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain, e-mail: igselfa@us.es, tornero@algebra.us.es
José M. Tornero
Affiliation:
Deptartamento de Álgebra, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain, e-mail: igselfa@us.es, tornero@algebra.us.es
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We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 3 , June 2005 , pp. 417 - 424
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bremner, A., ‘On arithmetic progressions on elliptic curves’, Experiment. Math. 8 (1999), 409–413.CrossRefGoogle Scholar
[2]Bremner, A., Silverman, J.H. and Tzanakis, N., ‘Integral points in arithmetic progression on y 2 = x(x 2 − n 2)’, J. Number Theory 80 (2000), 187–208.CrossRefGoogle Scholar
[3]Campbell, G., ‘A note on arithmetic progressions on elliptic curves’, J. Integer Seq. 6 (2003), 03.1.3.Google Scholar
[4]Lee, J.-B. and Vélez, W.Y., ‘Integral solutions in arithmetic progression for y 2 = x 3 + k’, Period. Math. Hungar. 25 (1992), 31–49.CrossRefGoogle Scholar
[5]Mohanty, S.P., ‘On consecutive integral solutions for y 2 = x 3 + k’, Proc. Amer. Math. Soc. 48 (1975), 261–265.Google Scholar
[6]Siegel, C.L., ‘Über einige Anwendungen diophantischer Approximationen (1929)’, in Collected works (Springer-Verlag, Berlin, Heidelberg, New York, 1966), pp. 209–266.Google Scholar
[7]Silverman, J., The arithmetic of elliptic curves (Springer-Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar