A062693 -id:A062693 - cp's OEIS frontend

A062695 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 2.

34, 41, 65, 137, 138, 145, 154, 161, 194, 210, 219, 226, 257, 265, 291, 299, 313, 323, 330, 353, 371, 386, 395, 410, 426, 434, 442, 457, 465, 505, 514, 546, 561, 602, 609, 651, 658, 674, 689, 721, 723, 731, 761, 777, 793, 866, 889, 890, 905, 915, 985, 987, 995

Offset: 1

Noam D. Elkies, Jul 04 2001

nonn

These n are precisely the primitive congruent numbers (A006991) with n==1, n==2, or n==3 (mod 8). - T. D. Noe, Aug 02 2006

Jinyuan Wang, Table of n, a(n) for n = 1..453
A. Dujella, A. S. Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340.
G. Kramarz, All congruent numbers less than 2000, Math. Annalen, 273 (1986), 337-340. [Annotated, corrected, scanned copy]
Kazunari Noda and Hideo Wada, All congruent numbers less than 10000, Proc. Japan Acad. Ser. A Math. Sci., Volume 69, Number 6 (1993), 175-178.

Cf. A003273, A006991, A062693, A062694, A259680-A259687.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
for(n=1,1e3,if(issquarefree(n)&&r(n)==2,print1(n", "))) \\ Charles R Greathouse IV, Sep 01 2011; corrected by Frank M Jackson, Aug 04 2016

More terms from Jinyuan Wang, Dec 12 2020

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

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

A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Sep 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A003273, A060952, A062693, A062695, A194687, A273929.

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

a(n) = A060952(n^2).
a(A003273(n)) > 0.
a(A194687(n)) = n.
Empirical: a(n) = a(4*n). - Jose Aranda, Jul 02 2024

A062694 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3 and nontrivial SHA[2].

Original entry on oeis.org

42486, 68839, 80189, 82205, 83845, 88502, 92045, 112326, 116645, 125749, 142222, 182005, 199805, 202742, 270805, 275286, 282613, 287246, 295222, 342205, 372742, 392502, 440453, 450079, 473263, 477581, 487302, 488047

Offset: 0

Noam D. Elkies, Jul 04 2001

nonn

Conjectural, as detailed in the pages from which it is extracted (see the first few links at the web site mentioned for details), but the conjecture is supported by much numerical and theoretical evidence.

A. Dujella, A. S.Janfeda, S. Salami, A Search for High Rank Congruent Number Elliptic Curves, JIS 12 (2009) 09.5.8
N. D. Elkies, Algorithmic (a.k.a. Computational) Number Theory: Tables, Links, etc.

Cf. A062693, A062695.

cp's OEIS Frontend

A062695 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 2.

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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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A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

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A062694 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3 and nontrivial SHA[2].

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