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 A031508 Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists. 5
 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]))<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019 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 EXTENSIONS Definition clarified by Jonathan Sondow, Oct 26 2013 STATUS approved

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Last modified September 15 22:10 EDT 2019. Contains 327088 sequences. (Running on oeis4.)