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Prime powers in elliptic divisibility sequences

Authors: Graham Everest and Helen King
Journal: Math. Comp. 74 (2005), 2061-2071
MSC (2000): Primary 11G05, 11A41
Published electronically: March 1, 2005
MathSciNet review: 2164113
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Abstract: Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.

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  • 1. J. W. S. Cassels,
    Lectures on Elliptic Curves,
    London Math. Soc. Student Texts 24, Cambridge Univ. Press, 1991. MR 1144763 (92k:11058)
  • 2. J. E. Cremona, Elliptic Curve Data up-dated 14-1-02, personal/jec/ftp/data/INDEX.html
  • 3. D. V. Chudnovsky and G. V. Chudnovsky,
    Sequences of numbers generated by addition in formal groups and new primality and factorization tests,
    Adv. in Appl. Math. 7 (1986), 385-434. MR 0866702 (88h:11094)
  • 4. Sinnou David,
    Minorations de formes linéaires de logarithmes elliptiques,
    Mém. Soc. Math. France (N.S.) (1995), no. 62, iv+143. MR 1385175 (98f:11078)
  • 5. Manfred Einsiedler, Graham Everest and Thomas Ward,
    Primes in elliptic divisibility sequences,
    LMS J. Comp. Math. 4 (2001), 1-13. MR 1815962 (2002e:11181)
  • 6. Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward,
    Recurrence Sequences, Mathematical Surveys and Monographs 104, Amer. Math. Soc., 2003. MR 1990179 (2004c:11015)
  • 7. Graham Everest, Victor Miller and Nelson Stephens,
    Primes generated by elliptic curves,
    Proc. Amer. Math. Soc. 32 (2004), 955-963. MR 2045409
  • 8. PARI-GP,
  • 9. Rachel Shipsey,
    Elliptic divisibility sequences,
    Ph.D. thesis, Univ. of London, 2000.
  • 10. Joseph H. Silverman,
    The Arithmetic of Elliptic Curves,
    Springer-Verlag, New York, 1986. MR 0817210 (87g:11070)
  • 11. Joseph H. Silverman,
    Common divisors of elliptic divisibility sequences over function fields,
    Manuscripta Math. 114 (2004), 431-446.
  • 12. R. J. Stroeker and N. Tzanakis,
    Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms,
    Acta. Arith., 67 (1994), 177-196. MR 1291875 (95m:11056)
  • 13. Christine Swart,
    Elliptic divisibility sequences,
    Ph.D. thesis, Univ. of London, 2003.
  • 14. J. Velu,
    Isogénies entre courbes elliptiques,
    C. R. Acad. Sci. Paris, 273 (1971), 238-241. MR 0294345 (45:3414)
  • 15. Morgan Ward,
    Memoir on elliptic divisibility sequences,
    Amer. J. Math., 7 (1948), 31-74. MR 0023275 (9:332j)

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Additional Information

Graham Everest
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Helen King
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

Keywords: Elliptic curve, isogeny, prime, elliptic divisibility sequence
Received by editor(s): March 30, 2004
Received by editor(s) in revised form: April 26, 2004
Published electronically: March 1, 2005
Additional Notes: The second author was supported by an EPSRC Doctoral Training Award. Both authors thank the referee for several comments leading to improvements in the text.
Article copyright: © Copyright 2005 American Mathematical Society

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