A194687 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x is n, or -1 if no such k exists.

1, 5, 34, 1254, 29274, 48272239, 6611719866

Offset

0,2

Comments

Fermat found a(0), Biling found a(1), and Wiman found a(2)-a(4). Rogers found upper bounds on a(5) and a(6) equal to their true value; Rathbun and an unknown author verified them as a(5) and a(6), respectively.

a(7) <= 797507543735, see Rogers 2004.

References

G. Billing, "Beiträge zur arithmetischen theorie der ebenen kubischen kurven geschlechteeins", Nova Acta Reg. Soc. Sc. Upsaliensis (4) 11 (1938), Nr. 1. Diss. 165 S.

N. Rogers, "Elliptic curves x^3 + y^2 = k with high rank", PhD Thesis in Mathematics, Harvard University (2004).

A. Wiman, "Über rationale Punkte auf Kurven y^2 = x(x^2-c^2)", Acta Math. 77 (1945), pp. 281-320.

Links

Table of n, a(n) for n=0..6.

Andrej Dujella, Ali S. Janfada, and Sajad Salami, A search for high rank congruent number elliptic curves, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.8.

Randall L. Rathbun, Posting to NMBRTHRY, Aug 25 2011

N. F. Rogers, Rank computations for the congruent number elliptic curves, Exper. Math. 9:4 (2000), pp. 591-594.

K. Rubin and A. Silverberg, Ranks of elliptic curves, p.464, Table 2.

Mark Watkins, On elliptic curves and random matrix theory, Journal de Theorie des Nombres de Bordeaux

Author?, LfunctionsAndModularFormsII / CentralValues / Rank4

Prog

(PARI) r(n)=ellanalyticrank(ellinit([0, 0, 0, -n^2, 0]))[1]
rec=0; for(n=1, 1e4, t=r(n); if(t>rec, rec=t; print("r("n") = "t)))

Crossrefs

Cf. A062693, A062695, A003273, A309028, A309029, A319510.

Sequence in context: A295545 A309534 A348375 * A208098 A216759 A144995

Adjacent sequences: A194684 A194685 A194686 * A194688 A194689 A194690

Keyword

nonn,hard,more

Author

Charles R Greathouse IV, Sep 01 2011

Extensions

Escape clause added to definition by N. J. A. Sloane, Jul 01 2024

Status

approved