A201268 - OEIS

%I #21 Jan 21 2013 04:34:36

%S 52488,15336,-20088,219375,-293625,-474552,1367631,-297,100872,

%T -105624,6021000,-6615000,40608000,-45360000,-423360000,69641775,

%U -72560097,110160000,-114912000,-1216512,1418946687,-1507379625,1450230912,-1533752064,2143550952,4566375

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

%C For successive x coordinates see A201047.

%C For successive y coordinates see A201269.

%C One elliptic curve with particular d can contain a finite number of extremal points.

%C Theorem (*Artur Jasinski*):

%C One elliptic curve cannot contain more than 1 extremal primary point with quadratic extension over rationals.

%C Consequence of this theorem is that any number in this sequence can't appear more than 1 time.

%C Conjecture (*Artur Jasinski*):

%C One elliptic curve cannot contain more than 1 point with quadratic extension over rationals.

%C 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 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 Conjecture (*Artur Jasinski*): Extremal points are k-th successive points with maximal coordinates x.

%F a(n) = (A201047(n))^3-(A201269(n))^2.

%Y Cf. A200218, A201047, A201269.

%K sign

%O 1,1

%A _Artur Jasinski_, Nov 29 2011