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!)
A031508 Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists. 6
1, 2, 11, 174, 2351, 28279, 975379 (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 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, 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
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, H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064.
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
Sequence in context: A122527 A039747 A049531 * A202140 A011806 A012953
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 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)