cp's OEIS Frontend

A031508 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.

%I A031508 #94 Feb 26 2025 11:05:05
%S A031508 1,2,11,174,2351,28279,975379
%N A031508 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 A031508 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
%C A031508 See A060951 for the rank of y^2 = x^3 - n. - _Jonathan Sondow_, Sep 10 2013
%C A031508 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 A031508 a(7) <= 56877643. a(8) <= 2520963512. a(9) <= 463066403167. a(10) <= 56736325657288. a(11) <= 46111487743732324. a(12) <= 6533891544658786928. See Table 3.3 in [Womack 2003]. - _Jose Aranda_, Jun 30 2024
%C A031508 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 A031508 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 A031508 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 A031508 J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>
%H A031508 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 A031508 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 A031508 J. Gebel, A. Pethö, and H. G. Zimmer, <a href="https://doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Mathematica 110 (1998), 335-367. <a href="http://www.ams.org/mathscinet-getitem?mr=1602064">MR1602064</a>.
%H A031508 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 A031508 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 10/2002.
%F A031508 a(n) = min { k >= 1 | A060951(k) == n }. - _M. F. Hasler_, Jul 01 2024
%e A031508 From _M. F. Hasler_, Jul 01 2024: (Start)
%e A031508 Sequence A060951 = (0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, ...) gives the analytic rank of the elliptic curve y^2 = x^3 - k for k = 1, 2, 3, ...
%e A031508 We can see that:
%e A031508   - the smallest k that gives rank 0 is k = 1 = a(0);
%e A031508   - the smallest k that gives rank 1 is k = 2 = a(1);
%e A031508   - the smallest k that gives rank 2 is k = 11 = a(2); etc. (End)
%o A031508 (PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ _Seiichi Manyama_, Aug 24 2019
%o A031508 (PARI) {A031508(n)=for(k=1,oo, ellrank(ellinit([0, -k]))[1]==n && return(k))} \\ _M. F. Hasler_, Jul 01 2024
%Y A031508 Cf. A031507, A373795.
%Y A031508 See also A060950, A002150-A002155, A102833, A179124, A031507, A060951, A081119, A179136, A179137.
%K A031508 nonn,nice,hard,more
%O A031508 0,2
%A A031508 _Noam D. Elkies_
%E A031508 Definition clarified by _Jonathan Sondow_, Oct 26 2013.
%E A031508 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.