Abstract
We develop techniques first studied by Morgan Ward to characterize sequences which arise from elliptic curves and which contain a zero term. We first define elliptic divisibility sequences over finite fields by noting that they are not the sequences which arise by reduction from integer sequences. After that, we give general terms of these sequences over the finite fields Fp (p > 3 is a prime) and then we determine elliptic curves and singular curves associated with them.
Keywords
Elliptic divisibility sequences Singular sequences Elliptic curves Singular curves 2000 AMS Classification: 11 B 50
References
- 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, 385–434, 1986.
- Einsiedler, M., Everest, G. and Ward, T. Primes in elliptic divisibility sequences, LMS J. Comput. Math. 4, 1–13, 2001 (electronic).
- Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T. Recurrence Sequences (Math- ematical Surveys and Monographs 104, AMS, Providence, RI, 2003).
- Everest, G. and Ward, T. Primes in divisibility sequences, Cubo Mat. Educ. 3, 245–259, Shipsey, R. Elliptic divisibility sequences (Ph.D. Thesis, Goldsmith’s (University of London), ). Silverman, J. H. The Arithmetic of Elliptic Curves (Springer-Verlag, 1986).
- Silverman, J. H. and Stephens, N. The sign of an elliptic divisibility sequence, Journal of Ramanujan Math. Soc. 21, 1–17, 2006.
- Silverman, J. H. and Tate, J. Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics, Springer, 1992).
- Swart, C. S. Elliptic curves and related sequences (PhD. Thesis, Royal Holloway (University of London), 2003).
- Tekcan, A., Gezer, B. and Bizim, O. Some relations on Lucas numbers and their sums, Advanced Studies in Comtemporary Mathematics 15 (2), 195–211, 2007.
- Ward, M. The law of repetition of primes in an elliptic divisibility sequences, Duke Math. J. 15, 941–946, 1948.
- Ward, M. Memoir on elliptic divisibility sequences, Amer. J. Math. 70, 31–74, 1948.
Abstract
References
- 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, 385–434, 1986.
- Einsiedler, M., Everest, G. and Ward, T. Primes in elliptic divisibility sequences, LMS J. Comput. Math. 4, 1–13, 2001 (electronic).
- Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T. Recurrence Sequences (Math- ematical Surveys and Monographs 104, AMS, Providence, RI, 2003).
- Everest, G. and Ward, T. Primes in divisibility sequences, Cubo Mat. Educ. 3, 245–259, Shipsey, R. Elliptic divisibility sequences (Ph.D. Thesis, Goldsmith’s (University of London), ). Silverman, J. H. The Arithmetic of Elliptic Curves (Springer-Verlag, 1986).
- Silverman, J. H. and Stephens, N. The sign of an elliptic divisibility sequence, Journal of Ramanujan Math. Soc. 21, 1–17, 2006.
- Silverman, J. H. and Tate, J. Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics, Springer, 1992).
- Swart, C. S. Elliptic curves and related sequences (PhD. Thesis, Royal Holloway (University of London), 2003).
- Tekcan, A., Gezer, B. and Bizim, O. Some relations on Lucas numbers and their sums, Advanced Studies in Comtemporary Mathematics 15 (2), 195–211, 2007.
- Ward, M. The law of repetition of primes in an elliptic divisibility sequences, Duke Math. J. 15, 941–946, 1948.
- Ward, M. Memoir on elliptic divisibility sequences, Amer. J. Math. 70, 31–74, 1948.
Details
| Primary Language | English |
|---|---|
| Subjects | Statistics |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | February 1, 2009 |
| Published in Issue | Year 2009 Volume: 38 Issue: 2 |