On The Elliptic Divisibility Sequences over Finite Fields

Authors: Osman Bizim

Abstract:

In this work we study elliptic divisibility sequences over finite fields. MorganWard in [11, 12] gave arithmetic theory of elliptic divisibility sequences. We study elliptic divisibility sequences, equivalence of these sequences and singular elliptic divisibility sequences over finite fields Fp, p > 3 is a prime.

Keywords: Elliptic divisibility sequences, equivalent sequences, singular sequences.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1329480

References:

[1] Chudnovsky D. V. and Chudnovsky G. V. Sequences of numbers generated by addition in formal groups and new primality factorization tests. Adv. in Appl. Math. 7 (1986), 385-434.
[2] Einsiedler M., Everest G., Ward T. Primes in elliptic divisibility sequences. LMS J. Comput. Math. 4 (2001), 1-13, electronic.
[3] Everest G., Van der Poorten A., Shparlinski I., Ward T. Recurrence Sequences, Mathematical Surveys and Monographs 104. AMS, Providence, RI, 2003.
[4] Everest G. and Ward T. Primes in divisibility sequences. Cubo Mat. Educ. 3(2001), 245-259.
[5] Shipsey R. Elliptic Divisibility Sequences. Dissertation, University of London, 2000.
[6] Silverman J.H. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986.
[7] Silverman J. H. and Stephens N. The sign of an elliptic divisibility sequences. Journal of Ramanujan Math. Soc. 21 (2006), 1-17.
[8] Silverman J. and Tate J. Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer, 1992.
[9] Swart, C.S. Elliptic Curves and Related Sequences. Dissertation, University of London, 2003.
[10] Tekcan A., Gezer B. and Bizim O. Some relations on Lucas numbers and their sums. Advanced Studies in Comtemporary Mathematics 15(2)(2007), 195-211.
[11] Ward M. The law of repetition of primes in an elliptic divisibility sequences. Duke Math. J. 15(1948), 941-946.
[12] Ward M. Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948), 31-74.