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Rational periodic sequences for the Lyness recurrence
1. | Dept. de Matemµatiques, Universitat Autónoma de Barcelona, Edifici C, 08193-Bellaterra, Barcelona, Spain |
2. | Dept. de Matemàtica Aplicada III (MA3), Control, Dynamics and Applications Group (CoDALab), Universitat Politècnica de Catalunya (UPC), Colom 1, 08222 Terrassa, Spain |
3. | Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain |
References:
[1] |
A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399-405.
doi: 10.1090/S0025-5718-1993-1140645-1. |
[2] |
E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291-306.
doi: 10.1080/10236199508808028. |
[3] |
G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$, J. Difference Equations Appl., 10 (2004), 977-1003.
doi: 10.1080/10236190410001728104. |
[4] |
A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières, C. R. Acad. Sci. Paris, Série I Math., 294 (1982), 657-660. |
[5] |
F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191-200.
doi: 10.1006/jdeq.1997.3359. |
[6] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[7] |
H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations," Graduate Texts in Mathematics, 239, Springer, New York, 2007. |
[8] |
J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().
|
[9] |
A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().
|
[10] |
N. Elkies, Rational points near curves and small nonzero $|x^3 - y^2|$ via lattice reduction, in "Algorithmic Number Theory" (Leiden, 2000), 33-63. Lecutre Notes in Comput. Sci., 1838, Springer, Berlin, 2000. |
[11] |
J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., 25 (2001), 477-502.
doi: 10.1007/s004540010075. |
[12] |
D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, Springer-Verlag, New York, 1987. |
[13] |
D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[14] |
I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'' Fifth edition, John Wiley & Sons, Inc., New York, 1991. |
[15] |
F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'' Thèse de doctorat, Université de Caen, 2008. |
[16] |
W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().
|
[17] |
J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'' Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994. |
[18] |
J. Silverman, "The Arithmetic of Elliptic Curves,'' Second edition, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009. |
[19] |
J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'' Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. |
[20] |
J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507-508.
doi: 10.2307/2304540. |
[21] |
E. C. Zeeman, Geometric unfolding of a difference equation, Hertford College, Oxford, (1996), Unpublished paper, Reprinted as a Preprint of the Warwick Mathematics Institute, 2008. A video of the distinguished lecture, with the same title, at PIMS on March 21, 2000, can be downloaded from: http://www.pims.math.ca/resources/multimedia/video. The slides can be obtained at: http://zakuski.utsa.edu/~gokhman/ecz/gu.html. |
show all references
References:
[1] |
A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399-405.
doi: 10.1090/S0025-5718-1993-1140645-1. |
[2] |
E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equations Appl., 1 (1995), 291-306.
doi: 10.1080/10236199508808028. |
[3] |
G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$, J. Difference Equations Appl., 10 (2004), 977-1003.
doi: 10.1080/10236190410001728104. |
[4] |
A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulières, C. R. Acad. Sci. Paris, Série I Math., 294 (1982), 657-660. |
[5] |
F. Beukers and R. Cushman, Zeeman's monotonicity conjecture, J. Differential Equations, 143 (1998), 191-200.
doi: 10.1006/jdeq.1997.3359. |
[6] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[7] |
H. Cohen, "Number Theory. Volume I: Tools and Diophantine Equations," Graduate Texts in Mathematics, 239, Springer, New York, 2007. |
[8] |
J. E. Cremona, "Elliptic Curve Data," Web page maintained by W. Stein, University of Warwick., Available from: \url{http://www.warwick.ac.uk/staff/J.E.Cremona//ftp/data/}., ().
|
[9] |
A. Dujella, "Elliptic Curve Tables," Web page maintained by author, University of Zagreb., Available from: \url{http://web.math.hr/~duje/}., ().
|
[10] |
N. Elkies, Rational points near curves and small nonzero $|x^3 - y^2|$ via lattice reduction, in "Algorithmic Number Theory" (Leiden, 2000), 33-63. Lecutre Notes in Comput. Sci., 1838, Springer, Berlin, 2000. |
[11] |
J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., 25 (2001), 477-502.
doi: 10.1007/s004540010075. |
[12] |
D. Husemoller, "Elliptic Curves," With an appendix by Ruth Lawrence, Graduate Texts in Mathematics, 111, Springer-Verlag, New York, 1987. |
[13] |
D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[14] |
I. Niven, H. S. Zukerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,'' Fifth edition, John Wiley & Sons, Inc., New York, 1991. |
[15] |
F. P. Rabarison, "Torsion et Rang des Courbes Elliptiques Définies sur les Corps de Nombres Algébriques,'' Thèse de doctorat, Université de Caen, 2008. |
[16] |
W. Stein, et al., Sage: Open Source Mathematical Software (Version 4.0), The Sage Group, 2009., Available from: \url{http://www.sagemath.org/}., ().
|
[17] |
J. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves,'' Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994. |
[18] |
J. Silverman, "The Arithmetic of Elliptic Curves,'' Second edition, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009. |
[19] |
J. Silverman and J. Tate., "Rational Points on Elliptic Curves,'' Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. |
[20] |
J. M. H. Olmsted, Rational values of trigonometric functions, Amer. Math. Monthly, 52 (1945), 507-508.
doi: 10.2307/2304540. |
[21] |
E. C. Zeeman, Geometric unfolding of a difference equation, Hertford College, Oxford, (1996), Unpublished paper, Reprinted as a Preprint of the Warwick Mathematics Institute, 2008. A video of the distinguished lecture, with the same title, at PIMS on March 21, 2000, can be downloaded from: http://www.pims.math.ca/resources/multimedia/video. The slides can be obtained at: http://zakuski.utsa.edu/~gokhman/ecz/gu.html. |
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