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

%I A179126 #25 Aug 25 2022 09:54:30
%S A179126 4,9,16,25,36,49,81,100,121,144,169,196,225,256,289,324,361,400,441,
%T A179126 484,529,576,625,676,784,841,900,961,1024,1089,1156,1225,1296,1369,
%U A179126 1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,2304,2401,2500,2601,2704
%N A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.
%C A179126 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]
%C A179126 From _Jianing Song_, Aug 25 2022: (Start)
%C A179126 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):
%C A179126 - 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).
%C A179126 - If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
%C A179126 - If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
%C A179126 - If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
%C A179126 - In all the other cases, the torsion group is trivial. (End)
%H A179126 Jianing Song, <a href="/A179126/b179126.txt">Table of n, a(n) for n = 1..10000</a>
%H A179126 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H A179126 Joseph H. Silverman, <a href="https://www.math.ens.psl.eu/~benoist/refs/Silverman.pdf">The Arithmetic of Elliptic Curves</a>
%o A179126 (Sage) is_A179126 = lambda n: EllipticCurve([0,n]).torsion_order() == 3  # _D. S. McNeil_, Jan 04 2011
%o A179126 (PARI) isA179126(n) = my(k=ispower(n)); !(k%2) && (k%6) \\ _Jianing Song_, Aug 25 2022
%Y A179126 Cf. A179125, A002151, A002153, A002155, A002833, A031507, A228532.
%K A179126 nonn,easy
%O A179126 1,1
%A A179126 _Artur Jasinski_, Jun 30 2010