A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.

4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704

Offset

1,1

Comments

Apparently equal to the set of integers (A004709(k))^2, k>=2. [This is incorrect, as shown by the terms 256, 576, 1024, 1600, and 2304. - Jianing Song, Aug 25 2022]

From Jianing Song, Aug 25 2022: (Start)

Numbers which are perfect squares (A000290) but not perfect cubes (A000578). This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):

- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).

- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).

- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).

- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).

- In all the other cases, the torsion group is trivial. (End)

Links

Jianing Song, Table of n, a(n) for n = 1..10000

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Joseph H. Silverman, The Arithmetic of Elliptic Curves

Prog

(Sage) is_A179126 = lambda n: EllipticCurve([0, n]).torsion_order() == 3  # D. S. McNeil, Jan 04 2011
(PARI) isA179126(n) = my(k=ispower(n)); !(k%2) && (k%6) \\ Jianing Song, Aug 25 2022

Crossrefs

Cf. A179125, A002151, A002153, A002155, A002833, A031507, A228532.

Sequence in context: A080160 A110723 A084617 * A354180 A367802 A369567

Adjacent sequences: A179123 A179124 A179125 * A179127 A179128 A179129

Keyword

nonn,easy

Author

Artur Jasinski, Jun 30 2010

Status

approved