cp's OEIS Frontend

A201268 Distances d=x^3-y^2 for primary extremal points {x,y} of Mordell elliptic curves with quadratic extensions over rationals.

%I A201268 #21 Jan 21 2013 04:34:36
%S A201268 52488,15336,-20088,219375,-293625,-474552,1367631,-297,100872,
%T A201268 -105624,6021000,-6615000,40608000,-45360000,-423360000,69641775,
%U A201268 -72560097,110160000,-114912000,-1216512,1418946687,-1507379625,1450230912,-1533752064,2143550952,4566375
%N A201268 Distances d=x^3-y^2 for primary extremal points {x,y} of Mordell elliptic curves with quadratic extensions over rationals.
%C A201268 For successive x coordinates see A201047.
%C A201268 For successive y coordinates see A201269.
%C A201268 One elliptic curve with particular d can contain a finite number of extremal points.
%C A201268 Theorem (*Artur Jasinski*):
%C A201268 One elliptic curve cannot contain more than 1 extremal primary point with quadratic extension over rationals.
%C A201268 Consequence of this theorem is that any number in this sequence can't appear more than 1 time.
%C A201268 Conjecture (*Artur Jasinski*):
%C A201268 One elliptic curve cannot contain more than 1 point with quadratic extension over rationals.
%C A201268 Mordell elliptic curves contained points with extensions which are roots of polynomials : 2 degree (with Galois 2T1), 4 degree (with Galois 4T3) and 6 degree (with not soluble Galois PGL(2,5) <most of points {x,y} belonging here and rest are only rare exceptions>). Order of minimal polynomial of any extension have to divided number 12. Theoretically points can exist which are roots of polynomial of 3 degree but any such point isn't known yet.
%C A201268 Particular elliptic curves x^3-y^2=d can contain more than one extremal point e.g. curve x^3-y^2=-297=a(8) contained 3 of such points with coordinates x={48, 1362, 93844}={A134105(7),A134105(8),A134105(9)}.
%C A201268 Conjecture (*Artur Jasinski*): Extremal points are k-th successive points with maximal coordinates x.
%F A201268 a(n) = (A201047(n))^3-(A201269(n))^2.
%Y A201268 Cf. A200218, A201047, A201269.
%K A201268 sign
%O A201268 1,1
%A A201268 _Artur Jasinski_, Nov 29 2011