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%I #39 Feb 17 2022 10:00:42
%S 1,6,19,657,21691,489489,9902523,1144421889,1683200989470,
%T 349043376293530,137006962414679910,13293998056584952174157235
%N a(n) is the smallest m such that the elliptic curve x^3 + y^3 = m has rank n, or -1 if no such m exists.
%C From Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: (Start)
%C I have verified that the first 5 entries are correct; the first two are basically trivial and the third is due to Selmer. I'm not sure who first discovered entries 4 and 5 and I expect that they had been previously proved to be the smallest values.
%C But I have rechecked that they are minimal for their respective rank using a combination of 3-descent, Magma and John Cremona's program mwrank.
%C There are new smaller values for ranks 6 and 7, namely k = 9902523 has rank 6 and k = 1144421889 has rank 7. 3-descent combined with Ian Connell's package apecs for Maple verifies that these are minimal subject to the Birch and Swinnerton-Dyer conjecture and the Generalized Riemann Hypothesis for L-functions associated to elliptic curves.
%C Finally, there are new entries for ranks 8 and 9: k = 1683200989470 has rank 8 and k = 148975046052222390 has rank 9. It seems somewhat likely that the rank 8 example is minimal. (End)
%C The sequence might be finite, even if it is redefined as smallest m such that x^3 + y^3 = m has rank >= n. - _Jonathan Sondow_, Oct 27 2013
%H Noam D. Elkies, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;c51b26cf.0310">Yet more rank records for x^3+y^3=k</a>, Posting to Number Theory List, Oct 19 2003, for a(9)
%H Noam D. Elkies and Nicholas F. Rogers, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;76505e0e.0307">Rank records for x^3+y^3=k, cont'd</a>, Posting to Number Theory List, Jul 18 2003, for a(8) and a(9).
%H Noam D. Elkies and Nicholas F. Rogers, <a href="https://doi.org/10.1007/978-3-540-24847-7_13">Elliptic curves x^3 + y^3 = k of high rank</a>, Algorithmic Number Theory, 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings, Springer, Berlin, Heidelberg, 2004, pp. 184-193. See also the <a href="https://arxiv.org/abs/math/0403116">arXiv version</a>arXiv:math/0403116 [math.NT], 2004.
%H Troy Kessler, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bc34aa4.0104">3 descent on elliptic curve</a>, Posting to Number Theory List, Apr 22, 2001.
%H Nick Rogers, <a href="http://projecteuclid.org/euclid.em/1045759524">Rank computations for the congruent number elliptic curves</a>. Experimental Mathematics 9 (2000), no. 4, 591-594.
%o (PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]<>n, k++); k} \\ _Seiichi Manyama_, Aug 24 2019
%Y Positions of records in A060838.
%Y Cf. A230564.
%K nonn,nice
%O 0,2
%A _N. J. A. Sloane_, Apr 23 2001
%E Definition clarified by _Jonathan Sondow_, Oct 27 2013
%E a(10)-a(11) from _Amiram Eldar_ were taken from the paper by Elkies & Rogers, Jul 27 2017.
%E Escape clause added by _N. J. A. Sloane_, Oct 26 2017
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