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Noam D. Elkies and Nicholas F. Rogers, <a href="httphttps://link.springerdoi.com/chapterorg/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</a> [math.NT], 2004.]

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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.

but But I have rechecked that they are minimal for their respective rank using a combination of 3-descent, MAGMA Magma and John Cremona's program mwrank.

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

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From Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: (Start)

Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: 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, (cont.)

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. (cont.)

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. (cont.)

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. (endEnd)

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