A319510 -id:A319510 - cp's OEIS frontend

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

Charles R Greathouse IV, Sep 01 2011

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.

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.

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

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

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)))

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

A349511 a(n) = Sum_{k=n^2..3n^2-3n+1} binomial(n^3, k).

Original entry on oeis.org

1, 1, 162, 129426405, 16891063036609237658, 18250180714636047151855346313907038815, 1291091703201646062849529792547495285890126156377393082996087554, 15934719293558661243731879701489946881532638280926268234547722632676376681552065231576737967805230

Offset: 0

Stefano Spezia, Nov 20 2021

nonn

a(n) is an upper bound of the number of vertices of the polytope of the n X n X n stochastic tensors, or equivalently, of the number of Latin squares of order n, or equivalently, of the number of n X n X n line-stochastic (0,1)-tensors (see Zhang et al.).

Fuzhen Zhang and Xiao-Dong Zhang, Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach, Optimization, 69:4, 729-741, (2020). arXiv:2008.04655 [math.CO], 2020. See p. 6.
Fuzhen Zhang and Xiao-Dong Zhang, Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors, arXiv:2110.12337 [math.CO], 2021. See p. 4.

Cf. A000290, A000578.

Cf. A349506, A349507, A349508, A349509, A349510, A349512.

a[n_]:=Sum[Binomial[n^3,k],{k,n^2,3n^2-3n+1}]; Array[a,8,0]

A349508(n)/A349509(n) <= A319510(n) < a(n) < A349512(n) (see Corollary 7 in Zhang et al., 2021).
a(n) = binomial(n^3, n^2)*2F1([1, n^2-n^3], [1+n^2], -1) - binomial(n^3, 2-3*n+3*n^2)*2F1([1, 2-3*n+3*n^2-n^3], [3(1-n+n^2)], -1), where 2F1 is the hypergeometric function.
a(n) ~ exp(3*n^2 - 9*n/2 + 3) * n^(3*n*(n-1)) / (sqrt(2*Pi) * 3^(3*n^2 - 3*n + 3/2)). - Vaclav Kotesovec, Dec 05 2021

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

Seiichi Manyama, Jul 09 2019

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)

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

Cf. A194687, A309028, A309061, A319510.

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

A309061(a(n)) = n.

A309061 Rank of elliptic curve y^2 = x^3 + n^2 * x.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2

Offset: 1

Seiichi Manyama, Jul 09 2019

nonn

Cf. A060953, A309060, A319510.

{a(n) = ellanalyticrank(ellinit([0, 0, 0, n^2, 0]))[1]}

a(n) = A060953(n^2).

cp's OEIS Frontend

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.

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A349511 a(n) = Sum_{k=n^2..3n^2-3n+1} binomial(n^3, k).

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

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A309061 Rank of elliptic curve y^2 = x^3 + n^2 * x.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Extensions

A349511 a(n) = Sum_{k=n^2..3*n^2-3*n+1} binomial(n^3, k).

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Author

Keywords

Comments

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Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A309061 Rank of elliptic curve y^2 = x^3 + n^2 * x.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A349511 a(n) = Sum_{k=n^2..3n^2-3n+1} binomial(n^3, k).