A060748 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.

1, 6, 19, 657, 21691, 489489, 9902523, 1144421889, 1683200989470, 349043376293530, 137006962414679910, 13293998056584952174157235

Offset: 0

Author

N. J. A. Sloane, Apr 23 2001

Keywords

• nonn
• nice

Comments

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

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.

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.

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.

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)

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

Links

• Noam D. Elkies, Yet more rank records for x^3+y^3=k, Posting to Number Theory List, Oct 19 2003, for a(9)
• Noam D. Elkies and Nicholas F. Rogers, Rank records for x^3+y^3=k, cont'd, Posting to Number Theory List, Jul 18 2003, for a(8) and a(9).
• Noam D. Elkies and Nicholas F. Rogers, Elliptic curves x^3 + y^3 = k of high rank, 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 arXiv versionarXiv:math/0403116 [math.NT], 2004.
• Troy Kessler, 3 descent on elliptic curve, Posting to Number Theory List, Apr 22, 2001.
• Nick Rogers, Rank computations for the congruent number elliptic curves. Experimental Mathematics 9 (2000), no. 4, 591-594.

Crossrefs

Positions of records in A060838.

Cf. A230564.

Programs

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

Extensions

Definition clarified by Jonathan Sondow, Oct 27 2013

a(10)-a(11) from Amiram Eldar were taken from the paper by Elkies & Rogers, Jul 27 2017.

Escape clause added by N. J. A. Sloane, Oct 26 2017