On The Elliptic Divisibility Sequences over Finite Fields

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.

Authors:

• Osman Bizim

Keywords:

• Elliptic divisibility sequences
• equivalent sequences
• singular sequences.

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.