

A031507


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


8




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 A031508 for the smallest negative k.  Artur Jasinski, Nov 21 2011
See A060950 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
The curves for k and 27*k are isogenous (as Noam Elkies points outsee Womack), so they have the same rank.  Jonathan Sondow, 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¶ and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335367.
J. Quer, Corps quadratiques de 3rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sc. Paris I, 305 (1987), 215218.
Tom Womack, Minimalknown positive and negative k for Mordell curves of given rank.


FORMULA

a(n) <= 27*A031508(n) and A031508(n) <= 27*a(n).  Jonathan Sondow, Sep 10 2013


EXAMPLE

a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack).  Jonathan Sondow, Sep 10 2013


CROSSREFS

Cf. A002150, A002152, A002154, A031508, A060950, A179136, A179137.
Sequence in context: A026113 A052874 A074622 * A207998 A246570 A052861
Adjacent sequences: A031504 A031505 A031506 * A031508 A031509 A031510


KEYWORD

nonn,nice,hard,more


AUTHOR

Noam D. Elkies


EXTENSIONS

Definition clarified by Jonathan Sondow, Oct 26 2013


STATUS

approved



