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A031507 Smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, if k exists. 8
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

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. 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.

Tom Womack, Minimal-known 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

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Last modified November 18 05:25 EST 2017. Contains 294853 sequences.