This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A031507 #91 Jul 06 2024 11:10:10
%S A031507 1,2,15,113,2089,66265,1358556
%N A031507 a(n) = smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.
%C A031507 See A031508 for the smallest negative k. - _Artur Jasinski_, Nov 21 2011
%C A031507 See A060950 for the rank of y^2 = x^3 + n. - _Jonathan Sondow_, Sep 10 2013
%C A031507 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 A031507 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
%C A031507 Womack (2003) gives further upper bounds: a(7) <= 47550317, a(8) <= 1632201497, a(9) <= 185418133372, a(10) <= 68513487607153. - _M. F. Hasler_, Jul 01 2024
%C A031507 The three questions for arbitrary k, positive k, and negative k are not very far from each other because the curves for k and -27k are related by a 3-isogeny and therefore have the same rank. It would be most natural to ask for the minimal |k| for k of either sign [see A373795]. - _Noam D. Elkies_, Jul 02 2024
%C A031507 a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - _N. J. A. Sloane_, Jul 05 2024
%D A031507 Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.
%H A031507 J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>
%H A031507 Noam D. Elkies and Zev Klagsbrun, <a href="https://doi.org/10.2140/obs.2020.4.233">New rank records for elliptic curves having rational torsion</a>, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.
%H A031507 J. Gebel, <a href="https://web.archive.org/web/20160825170558/http://tnt.math.se.tmu.ac.jp/simath/MORDELL/MORDELL+">Integer points on Mordell curves</a>, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. <a href="/A001014/a001014_2.txt">[Locally cached copy]</a>.
%H A031507 J. Gebel, A. Pethö and H. G. Zimmer, <a href="https://www.cambridge.org/core/journals/compositio-mathematica/article/on-mordells-equation/9BCC27AD0EA056233BFF224A5EEEDD7A">On Mordell's equation</a>, Compositio Math. 110 (1998), 335-367. (<a href="http://dx.doi.org/10.1023/A:1000281602647">doi:10.1023/A:1000281602647</a> not working as of July 2024.)
%H A031507 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 A031507 Tom Womack, <a href="https://homepages.warwick.ac.uk/~masgaj/theses/womack.pdf">Explicit Descent on Elliptic Curves</a>, PhD thesis, University of Nottingham, July 2003.
%H A031507 Tom Womack, <a href="https://web.archive.org/web/20170120143936/http://www.tom.womack.net/maths/mordellc.htm">Minimal-known positive and negative k for Mordell curves of given rank</a> (personal web page, latest available snapshot on web.archive.org from Jan. 2017), last modified Oct. 2002.
%F A031507 a(n) <= 27*A031508(n) and A031508(n) <= 27*a(n). - _Jonathan Sondow_, Sep 10 2013
%e A031507 a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack). - _Jonathan Sondow_, Sep 10 2013
%o A031507 (PARI) {A031507(n)=for(k=1, oo, ellrank(ellinit([0, k]))[1]==n && return(k))} \\ Use ellanalyticrank() for PARI version < 2.14. - _M. F. Hasler_, Jul 01 2024
%Y A031507 Cf. A031508, A373795.
%Y A031507 See also A060950, A002150-A002155, A102833, A179124, A031507, A060951, A081119, A179136, A179137.
%K A031507 nonn,nice,hard,more
%O A031507 0,2
%A A031507 _Noam D. Elkies_
%E A031507 Definition clarified by _Jonathan Sondow_, Oct 26 2013
%E A031507 Escape clause added to definition by _N. J. A. Sloane_, Jun 29 2024, because, as _John Cremona_ reminds me, it is not known if k always exists.