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

Author

Charles R Greathouse IV, Sep 01 2011

Keywords

• nonn
• hard
• more

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

• 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

Crossrefs

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

Programs

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)))

Extensions

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