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

1,1

Comments

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

Links

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.

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A337695 A045044 A108303 * A107730 A320703 A095419

Adjacent sequences: A062692 A062693 A062694 * A062696 A062697 A062698

Keyword

nonn

Author

Noam D. Elkies, Jul 04 2001

Extensions

More terms from Jinyuan Wang, Dec 12 2020

Status

approved