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A031508 Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists. 3
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

REFERENCES

J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

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]

Tom Womack, Minimal-known positive and negative k for Mordell curves of given rank.

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

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.