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Primitive Divisors on Twists of Fermat's Cubic

Published online by Cambridge University Press:  01 February 2010

Graham Everest ,
Patrick Ingram  and
Shaun Stevens
Graham Everest
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK, g.everest@uea.ac.uk
Patrick Ingram
Affiliation:
Department of Mathematics, University of Toronto, Canada, M5S 2E4, pingram@math.utoronto.ca
Shaun Stevens
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK, shaun.stevens@uea.ac.uk

Abstract

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We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 + v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 54 - 81
Copyright
Copyright © London Mathematical Society 2009

References

1.Ayad, M., ‘Points S-entiers des courbes elliptiques’, Manuscripta Math. 76 (1992) 305–324.CrossRefGoogle Scholar
2.Bang, A. S., ‘Taltheoretiske Undersøgelser’, Tidskrift f. Math. 5 (1886) 70–80, 130–137.Google Scholar
3.Bilu, Y., Hanrot, G. and Voutier, P. M., ‘Existence of primitive divisors of Lucas and Lehmer numbers’. With an appendix by M. Mignotte, J. reine angew. Math. 539 (2001) 75–122.Google Scholar
4.Bremner, A., Silverman, J. H. and Tzanakis, N., ‘Integral points in arithmetic progression on y 2 = x(x 2 – n 2)’, J. Number Theory 80 (2000) 187–208.CrossRefGoogle Scholar
5.Carmichael, R. D., ‘On the numerical factors of the arithmetic forms αn ± bn’, Ann. of Math. (2) 15 (1914) 49–70.CrossRefGoogle Scholar
6.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) 385–434.CrossRefGoogle Scholar
7.Cremona, J. E., ‘Elliptic Curve Data up-dated 05–01–2007’, www.Warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.htmGoogle Scholar
8.Elkies, N., ‘Rational points with low canonical height’, http://www.math.harvard.edu/~elkies/low_height.htmlGoogle Scholar
9.Erdős, P., ‘Arithmetical properties of polynomials’, J. London Math. Soc. 28 (1953) 416–425.CrossRefGoogle Scholar
10.Everest, G., Miller, V. and Stephens, N., ‘Primes generated by elliptic curves’, Proc. Amer. Math. Soc. 132 (2004) 955–963.CrossRefGoogle Scholar
11.Everest, G. and King, H., ‘Prime powers in elliptic divisibility sequences’, Math. Comp. 74 (2005) 2061–2071.CrossRefGoogle Scholar
12.Everest, G., McLaren, G. and Ward, T., ‘Primitive divisors of elliptic divisibility sequences’, J. Number Theory 118 (2006) 71–89.CrossRefGoogle Scholar
13.Everest, G. R., Stevens, S., Tamsett, D. and Ward, T., ‘Primes generated by recurrence sequences’, American Mathematical Monthly, May 2007 (American Mathematical Society, Providence, RI).Google Scholar
14.Ingram, P., ‘Elliptic divisibility sequences over certain curves’, J. Number Theory 123 (2007) 473–486.CrossRefGoogle Scholar
15.Ingram, P., ‘Multiples of integral points on elliptic curves’, arXiv:math.NT/0802.2651v2.Google Scholar
16.Ingram, P. and Silverman, J. H., ‘Uniform estimates for primitive divisors in elliptic divisibility sequences’, to appear in a forthcoming memorial volume for Serge Lang (Springer).Google Scholar
17.Jedrzejak, T., ‘Height estimates on cubic twists of the Fermat elliptic curve’, Bull. Austral. Math. Soc. 72 (2005) 177–186.CrossRefGoogle Scholar
18.MAGMA, version V2.13, University of Sydney, http://magma.maths.usyd.edu.au/magma/index.html.Google Scholar
19.PARI/GP, version 2.3.0, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/.Google Scholar
20.Praeger, C., ‘Primitive prime divisor elements in finite classical groups’, Groups St. Andrews 1997 in Bath II (Cambridge University Press, 1999) 605–623.CrossRefGoogle Scholar
21.Schinzel, A., ‘Primitive divisors of the expression An – Bn in algebraic number fields’, J. reine angew. Math. 268/269 (1974) 27–33.Google Scholar
22.Shorey, T. and Tijdeman, R., Exponential diophantine equations, Cambridge Tracts in Mathematics 87 (Cambridge University Press, 1986).CrossRefGoogle Scholar
23.Silverman, J. H., ‘Integer points and the rank of Thue elliptic curves’, Invent. Math. 66 (1982) 395–404.CrossRefGoogle Scholar
24.Silverman, J. H., ‘The arithmetic of elliptic curves’, Graduate Texts in Mathematics 106 (Springer, New York, 1986).CrossRefGoogle Scholar
25.Silverman, J. H., ‘Common divisors of elliptic divisibility sequences over function fields’, Manuscripta Mathematica 114 (2004) 431–446.CrossRefGoogle Scholar
26.Silverman, J. H. and Stephens, N., ‘The sign of an elliptic divisibility sequence’, J. Ramanujan Math. Soc. 21 (2006) 1–17.Google Scholar
27.Silverman, J. H., ‘p-adic properties of division polynomials and elliptic divisibility sequences’, Math. Ann. 332 (2005) 443–471.CrossRefGoogle Scholar
28.Silverman, J. H., ‘Wieferich's criterion and the abc-conjecture’, J. Number Theory 30 (1988) 226–237.CrossRefGoogle Scholar
29.Ward, M., ‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948) 31–74.CrossRefGoogle Scholar
30.Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. 3 (1892) 265–284.CrossRefGoogle Scholar