

A060748


a(n) = smallest m such that the elliptic curve x^3+y^3=m has rank n, or 1 if no such m exists.


9



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

0,2


COMMENTS

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 3descent, 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. 3descent combined with Ian Connell's package apecs for Maple verifies that these are minimal subject to the Birch and SwinnertonDyer conjecture and the Generalized Riemann Hypothesis for Lfunctions 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. (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

Table of n, a(n) for n=0..11.
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, ANTSVI, Burlington, VT, USA, June 1318, 2004, Proceedings, Springer, Berlin, Heidelberg, 2004, pp. 184193. [arXiv: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, 591594.


CROSSREFS

Positions of records in A060838.
Cf. A230564.
Sequence in context: A118411 A091876 A041066 * A176559 A241715 A224919
Adjacent sequences: A060745 A060746 A060747 * A060749 A060750 A060751


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Apr 23 2001


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


STATUS

approved



