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

Author

Noam D. Elkies, Jul 04 2001

Keywords

• nonn

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.

Crossrefs

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

Programs

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

Extensions

More terms from Jinyuan Wang, Dec 12 2020