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%I #44 Aug 24 2022 09:02:20
%S 1,2,15,113,2089,66265,1358556
%N Smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, if k exists.
%C 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
%C For bounds on later terms see the Gebel link. - _N. J. A. Sloane_, Jul 05 2010
%C See A031508 for the smallest negative k. - _Artur Jasinski_, Nov 21 2011
%C See A060950 for the rank of y^2 = x^3 + n. - _Jonathan Sondow_, Sep 10 2013
%C 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
%C 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
%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H J. Gebel, A. Pethö and H. G. Zimmer, <a href="http://dx.doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Math. 110 (1998), 335-367.
%H J. Quer, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k54952139/f15.image">Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12</a>, C. R. Acad. Sc. Paris I, 305 (1987), 215-218.
%H Tom Womack, <a href="http://www.tom.womack.net/maths/mordellc.htm">Minimal-known positive and negative k for Mordell curves of given rank</a>.
%F a(n) <= 27*A031508(n) and A031508(n) <= 27*a(n). - _Jonathan Sondow_, Sep 10 2013
%e a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack). - _Jonathan Sondow_, Sep 10 2013
%Y Cf. A060950, A002151, A002153, A002155, A102833, A179124, A031508.
%K nonn,nice,hard,more
%O 0,2
%A _Noam D. Elkies_
%E Definition clarified by _Jonathan Sondow_, Oct 26 2013
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