1
$\begingroup$

Is the rank of the elliptic curve x^3 + y^3 = a(n) over the rationals, where a(n) is the n-th cubefree taxicab number A080642(n) in the OEIS, known?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ For partial results towards this see Silverman's paper maa.org/sites/default/files/images/upload_library/22/Ford/… . In particular it says that the rank of the elliptic curve is bounded below in terms of the number of solutions to the taxicab equation (see page 339). I am not aware of a more precise solution. $\endgroup$
    – Lucia
    Commented Oct 22, 2013 at 22:26
  • $\begingroup$ In Unsolved Problems in Number Theory, 3rd edition, section D1, 2004, R. K. Guy says, "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively." But Taxicab(n) equals A011541(n), not the cubefree taxicab number A080642(n). @Lucia $\endgroup$
    – Jonathan Sondow
    Commented Oct 25, 2013 at 20:07
  • $\begingroup$ I think the answer to this question is simply: no. $\endgroup$
    – Kimball
    Commented Feb 10, 2016 at 11:25

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .