cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

1, 5, 34, 1254, 29274, 48272239, 6611719866
Offset: 0

Views

Author

Keywords

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.

Crossrefs

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

A309060 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.

Original entry on oeis.org

1, 3, 17, 627, 14637
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2019

Keywords

Comments

From Jose Aranda, Jun 30 2024: (Start)
A319510(n even) = A309061(n/2), A319510(n odd) = A309061(2*n) (Empirical).
A194687(5) = 48272239 which implies a(5) <= 96544478 (Checked).
A194687(6) = 6611719866 which implies a(6) <= 3305859933 (Checked).
A194687(7) <= 797507543735 which implies a(7) <= 1595015087470 (Checked). (End)

Examples

			A309061(1) = 0.
A309061(3) = 1.
A309061(17) = 2.
		

Crossrefs

Programs

  • PARI
    {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, k^2, 0]))[1]<>n, k++); k}

Formula

A309061(a(n)) = n.

A309029 Smallest k>0 such that the elliptic curve y^2 = x^3 - k*x has rank n, if k exists.

Original entry on oeis.org

1, 2, 17, 82, 5037, 49042
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2019

Keywords

Comments

See A309028 for the smallest positive k.

Crossrefs

Extensions

a(5) from Vaclav Kotesovec, Jul 09 2019

A309190 Numbers k for which rank of the elliptic curve y^2 = x^3 + k*x is 5.

Original entry on oeis.org

195843, 196168, 233864
Offset: 1

Views

Author

Vaclav Kotesovec, Jul 16 2019

Keywords

Crossrefs

Showing 1-4 of 4 results.