A062694 -id:A062694 - 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

A062693 Squarefree n such that the elliptic curve n*y^2 = x^3 - x arising in the "congruent number" problem has rank 3.

Original entry on oeis.org

1254, 2605, 2774, 3502, 4199, 4669, 4895, 6286, 6671, 7230, 7766, 8005, 9015, 9430, 9654, 10199, 10549, 11005, 11029, 12166, 12270, 12534, 12935, 13317, 14965, 15655, 16151, 16206, 16887, 17958, 18221, 19046, 19726, 20005, 20366

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.
Fidel Ronquillo Nemenzo, All congruent numbers less than 40000, Proc. Japan Acad. Ser. A Math. Sci., Volume 74, Number 1 (1998), 29-31. See Table IV p. 31.

Cf. A062694, A062695.

r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
for(n=1,1e4,if(r(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Sep 01 2011

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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