A228532 -id:A228532 - cp's OEIS frontend

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

Artur Jasinski, Jun 30 2010

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)

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

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

isA179126(n) = my(k=ispower(n)); !(k%2) && (k%6) \\ Jianing Song, Aug 25 2022

is_A179126 = lambda n: EllipticCurve([0,n]).torsion_order() == 3  # D. S. McNeil, Jan 04 2011

cp's OEIS Frontend

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

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