

A031508


Smallest k>0 such that the elliptic curve y^2 = x^3  k has rank n, if k exists.


5




OFFSET

0,2


COMMENTS

The sequence might be finite, even if it is redefined as smallest k>0 such that the elliptic curve y^2 = x^3  k has rank >= n.  Jonathan Sondow, Sep 26 2013
For bounds on later terms see the Gebel link.  N. J. A. Sloane, Jul 05 2010
See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n.  Jonathan Sondow, Sep 06 2013
See A060951 for the rank of y^2 = x^3  n.  Jonathan Sondow, Sep 10 2013
Gebel, PethÃ¶, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log k/log log k^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol.  Charles R Greathouse IV, Sep 10 2013


LINKS

Table of n, a(n) for n=0..6.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
J. Gebel, A. PethÃ¶, H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335367. MR1602064.
Tom Womack, Minimalknown positive and negative k for Mordell curves of given rank.


PROG

(PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, k]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019


CROSSREFS

Cf. A002150, A002152, A002154, A031507, A060951, A179136, A179137.
Sequence in context: A122527 A039747 A049531 * A202140 A011806 A012953
Adjacent sequences: A031505 A031506 A031507 * A031509 A031510 A031511


KEYWORD

nonn,nice,hard,more


AUTHOR

Noam D. Elkies


EXTENSIONS

Definition clarified by Jonathan Sondow, Oct 26 2013


STATUS

approved



