login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A031507 Smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, if k exists. 11
1, 2, 15, 113, 2089, 66265, 1358556 (list; graph; refs; listen; history; text; internal format)
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, Pethö, & 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 out---see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
LINKS
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. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sc. Paris I, 305 (1987), 215-218.
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
Sequence in context: A052874 A360432 A074622 * A207998 A246570 A052861
KEYWORD
nonn,nice,hard,more
AUTHOR
EXTENSIONS
Definition clarified by Jonathan Sondow, Oct 26 2013
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)