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Primes in Elliptic Divisibility Sequences

Published online by Cambridge University Press:  01 February 2010

Manfred Einsiedler
Affiliation:
Mathematical Institute, University of Vienna, Strudlhofgasse 4, A-1090 Wien, Austria, Manfred.Einsiedler@univie.ac.at
Graham Everest
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, G.Everest@uea.ac.uk
Thomas Ward
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, T.Ward@uea.ac.uk

Abstract

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Morgan Ward pursued the study of elliptic divisibility sequences, originally initiated by Lucas, and Chudnovsky and Chudnovsky subsequently suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here, from both a theoretical and a practical viewpoint. We show calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2001

References

1.Bézivin, Jean Paul, Pethö, Attila and van der Poorten, Alfred J., ‘A full characterization of divisibility sequences’, Amer. J. Math. 112 (1990) 9851001.CrossRefGoogle Scholar
2.Chahal, J. S., Topics in number theory (Plenum Press, New York, 1988).CrossRefGoogle Scholar
3.Chudnovsky, D. V. and Chudnovsky, G. V., ‘Sequences of numbers generated by addition in formal groups and new primality and factorization tests’, Adv. in Appl. Math. 7 (1986) 385434.CrossRefGoogle Scholar
4.Chudnovsky, D. V. and Chudnovsky, G. V., ‘Computer assisted number theory with applications’, Number theory (New York, 19841985) (Springer, Berlin, 1987) 168.Google Scholar
5.Dubner, Harvey and Keller, Wilfrid, New Fibonacci and Lucas primes, Math. Comp. 68 (1999) 417427, S1-S12.CrossRefGoogle Scholar
6.Einsiedler, M., Everest, G. and Ward, T., ‘Morphic heights and periodic points’, New York Number Th. Sem., to appear (2001).Google Scholar
7.Einsiedler, M., Everest, G. and Ward, T., ‘Entropy and the canonical height.’ J. Number Theory, to appear (2001).CrossRefGoogle Scholar
8.Einsiedler, M., Everest, G. and Ward, T., ‘Primes in sequences associated to polynomials (after Lehmer)’, LMS J. Comput. Math. 3 (2000) 125139–; http://www.lms.ac.uk/jcm/3/lms2000-004/.CrossRefGoogle Scholar
9.Everest, G. and Ward, T., ‘The canonical height of an algebraic point on an elliptic curve’, New York J. Math. 6 (2000).Google Scholar
10.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 5th edn (Clarendon Press, Oxford, 1979).Google Scholar
11.Lehmer, D. H., ‘Factorization of certain cyclotomic functions’, Ann. of Math. 34 (1933) 461479.CrossRefGoogle Scholar
13.Pierce, T. A., ‘Numerical factors of the arithmetic forms πni=1 (1±αmi)’, Ann. of Math. 18 (1917) 5364.CrossRefGoogle Scholar
14.Ribenboim, Paulo, ‘The Fibonacci numbers and the Arctic ocean’, Proceedings of the 2nd Gauss Symposium. Conference A: Mathematics and Theoretical Physics (Munich, 1993) (de Gruyter, Berlin, 1995) 4183.Google Scholar
15.Shipsey, R. 'Elliptic divisibility sequences', PhD Thesis, Goldsmith's College (University of London), 2000.Google Scholar
16.Silverman, J. H., The arithmetic of elliptic curves (Springer, New York, 1986).CrossRefGoogle Scholar
17.Silverman, J. H., Advanced topics in the arithmetic of elliptic curves (Springer, New York, 1994).Google Scholar
18.Wagstaff, S. S., ‘Divisors of Mersenne numbers’, Math. Comp. 40 (1983) 385397.CrossRefGoogle Scholar
19.Ward, M., ‘The law of repetition of primes in an elliptic divisibility sequence’, Duke Math. J. 15 (1948) 941946.CrossRefGoogle Scholar
20.Ward, M., ‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948) 3174.Google Scholar