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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 (list; graph; refs; listen; history; text; internal format)
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 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. (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, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings, Springer, Berlin, Heidelberg, 2004, pp. 184-193. [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, 591-594.

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

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Last modified January 20 16:46 EST 2019. Contains 319335 sequences. (Running on oeis4.)